Special Issue No. – 11, May 2024

Abstract:

By combining the ZZ transform with Adomian polynomials, the semi-analytical solutions to nonlinear Caputo partial fractional differential equations have been derived in this work. The Caputo sense has been applied to the fractional derivative. Using the proposed method, several fractional partial differential equations have been resolved. When compared to other similar procedures, it has been shown that applying the ZZ transform and breaking down the nonlinear components using Adomian polynomials is quite convenient.

Keywords:

ZZ Transform,Sumudu Transform,Adomian Polynomials,Caputo's system of Fractional Partial Differential Equations (FPDE),

Refference:

I. Adomian G., : ‘A new approach to nonlinear partial differential equations’. J. Math. Anal. Appl. Vol. 102, pp. 420–434, (1984). 10.1016/0022-247X(84)90182-3
II. Arshad S., A. Sohail, and K. Maqbool. : ‘Nonlinear shallow water waves: a fractional order approach’. Alexandria Engineering Journal. vol. 55(1), pp. 525–532, (2016). 10.1016/j.aej.2015.10.014
III. Arshad S., A. M. Siddiqui, A. Sohail, K. Maqbool, and Z. Li. : ‘Comparison of optimal homotopy analysis method and fractional homotopy analysis transform method for the dynamical analysis of fractional order optical solitons’. Advances in Mechanical Engineering. vol. 9(3), (2017) 10.1177/1687814017692
IV. Bhalekar S., and V. Daftardar-Gejji. : ‘Solving evolution equations using a new iterative method’. Numerical Methods for Partial Differential Equations: An International Journal, vol. 26(4), pp. 906–916, (2010). 10.1002/num.20463
V. Fethi et al., : ‘SUMUDU TRANSFORM FUNDAMENTAL PROPERTIES INVESTIGATIONS AND APPLICATIONS’. Journal of Applied Mathematics and Stochastic Analysis. Volume 2006, Article ID 91083, Pages 1–23. 10.1155/JAMSA/2006/91083
VI. Hilfer R., : ‘Applications of Fractional Calculus in Physics’. World Scientific, Singapore, Singapore. 2000. 10.1142/3779
VII. Jassim H. K., and H. A. Kadhim. : ‘Fractional Sumudu decomposition method for solving PDEs of fractional order’. J. Appl. Comput. Mech. Vol. 7, pp. 302–311. (2021). 10.22055/JACM.2020.31776.1920
VIII. Katatbeh Q. D., and F. B. M. Belgacem. : ‘Applications of the Sumudu transform to fractional differential equations’. Non-Linear Studies. vol. 18(1), pp. 99–112, (2011).
IX. Lakhdar R., and Mountassir H. Cherif. : ‘A Precise Analytical Method to Solve the Nonlinear System of Partial Differential Equations With the Caputo Fractional Operator’. Cankaya University Journal of Science and Engineering. vol. 19(1), pp. 29-39, (2022). https://dergipark.org.tr/tr/download/article-file/1953334
X. Lakhdar R., et al., : ‘An efficient approach to solving the Fractional SIR Epidemic Model with the Atangana-Baleanu-Caputo Fractional Operator’. Fractal and Fractional. Vol. 7, pp. 618, (2023). 10.3390/fractalfract7080618
XI. Liu Y., : ‘Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method’. Hindawi Publishing Corporation Abstract and Applied Analysis. Volume 2012, Article ID 752869, 14 pages. 10.1155/2012/752869
XII. Metzler R., and J. Klafter. : ‘The random walk’s guide to anomalous diffusion: a fractional dynamics approach’. Physics Reports, vol. 339(1), pp. 1-77, (2000). 10.1016/S0370-1573(00)00070-3
XIII. Moazzam A., and M. Iqbal. : ‘A NEW INTEGRAL TRANSFORMATION “ALI AND ZAFAR” TRANSFORMATION AND ITS APPLICATIONS IN NUCLEAR PHYSICS’. June 2022.
XIV. Podlubny I. : ‘Fractional Differential Equations’. Academic Press, New York, NY, USA. 1999.
XV. Riabi L. A., et al., : ‘HOMOTOPY PERTURBATION METHOD COMBINED WITH ZZ TRANSFORM TO SOLVE SOME NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS’. International Journal of Analysis and Applications. Volume 17(3), pp. 406-419. (2019). https://etamaths.com/index.php/ijaa/article/view/1875
XVI. Shah R., H. Khan M. Arif and P. Kumam. : ‘Application of Laplace-Adomian Decomposition Method for the Analytical Solution of Third-Order Dispersive Fractional Partial Differential Equations’. Entropy (Basel). Vol. 21(4), pp. 335, (2019). 10.3390/e21040335
XVII. Singh A., and S. Pippal. : ‘Solution of nonlinear fractional partial differential equations by Shehu transform and Adomian decomposition method (STADM)’. International Journal of Mathematics for Industry. 2350011 (18 pages). 10.1142/S2661335223500119
XVIII. Sontakke R., and R. Pandit. : ‘Convergence analysis and approximate solution of fractional differential equations’. : ‘Malaya Journal of Matematik’. Vol. 7(2), pp. 338-344, (2019). 10.26637/MJM0702/0029
XIX. Wang K., and S. Liu. : ‘A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation’. Springer Plus Vol. 5, 865 (2016). https://springerplus.springeropen.com/articles/10.1186/s40064-016-2426-8
XX. Wu G. C., and D. Baleanu. : ‘Variational iteration method for fractional calculus – a universal approach by Laplace transform’. Adv Differ Equ 2013, 18 (2013). 10.1186/1687-1847-2013-18
XXI. X. J. Yang, H. M. Srivastava, and C. Cattani. : ‘Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics’. Romanian Reports in Physics, vol. 67(3), pp. 752–761, 2015. https://rrp.nipne.ro/2015_67_3/A2.pdf
XXII. Zafar Z. U. A., : ‘Application of ZZ transform method on some fractional differential equations’. Int J Adv Eng Global Technol. Vol. 4(13), pp. 55–63, (2016).

View | Download

Abstract:

This study investigates the solution of complex mathematical problems of two-dimensional and three-dimensional telegraph equations. To solve these equations, we use a comprehensive approach that combines the Elzaki transform and the homotopy perturbation method (HPM) and provides a systematic and efficient means of obtaining exact solutions to these problems. Our methodology is rigorously tested in both 2 and 3 dimensions, demonstrating its effectiveness.

Keywords:

Telegraph equation,Homotopy Perturbation method,Elzaki transform,numerical problems,

Refference:

I. D. Konane, W. Y. S. B. Ouedraogo, T. T. Guingane, A. Zongo, Z. Koalaga, F. Zougmore : ‘An exact solution of telegraph equations for voltage monitoring of electrical transmission line’. Energy and Power Engineering. Vol. 14 (11), pp. 669-679, (2022). 10.4236/epe.2022.1411036.
II. F. Urena, L. Gavete J. J. Benito, Garcia, : ‘Solving the telegraph equation in 2-D and 3-D using generalized finite difference method (GFDM).’ Engineering Analysis with Boundary Elements. Vol. 112, pp. 13-24, (2019). 10.1016/j.enganabound.2019.11.010
III. J. H. He. : ‘Homotopy Perturbation Technique.’ Computer Methods in Applied Mechanics and Engineering. Vol. 178(3-4), pp. 257-262, (1999). 10.1016/S0045-7825(99)00018-3
IV. J. H. He. : ‘A Coupling method of Homotopy Technique and Perturbation technique for non-linear problems’. Int. J. of Non-Linear Mechanics. Vol. 35 (1), pp. 37-43, (2000). 10.1016/S0020-7462(98)00085-7
V. J. H. He. : ‘ New Interpretation of Homotopy Perturbation Method’. Int. J. of Modern Physics Letter B. Vol. 20 (18), pp. 2561-2568, (2006). 10.1142/S0217979206034819
VI. M. Dehghan A. Ghesmati. : ‘Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equations’. Eng. Anal. Boundary Elements. Vol. 34(4), pp. 324–336, (2010). 10.1016/j.enganabound.2009.10.010
VII. M. Dehghan, A. Shokri. : ‘Numerical method for solving the hyperbolic telegraph equation’. Numerical Methods for Partial Differential Equations. Vol. 24 (4), pp. 51–59, (2008). 10.1002/num.20306
VIII. M. H. Eljaily, T. M. Elzaki. : ‘Solution of linear and non linear Schrodinger equations by combine Elzaki transform and homotopy perturbation method’. American J. of Theoretical and Applied Statistics. Vol. 4(6), pp. 534-538, (2015). 10.11648/j.ajtas.20150406.24
IX. T. M. Elzaki. : ‘The new integral transforms Elzaki Transform’. Global J. of Pure and Applied Mathematics. Vol. 7(1), pp. 57-64, (2011). http://www.ripublication.com/gjpam.htm
X. T. M. Elzaki, and S. M. Elzaki. : ‘Application of new transform Elzaki Transform to partial differential equations’. Global J. of Pure and Applied Mathematics. Vol. 7(1), pp. 65-70, (2011). http://www.ripublication.com/gjpam.htm
XI. T. M. Elzaki, S. M. Elzaki. : ‘On the Connections between Laplace and Elzaki transforms’. Advances in Theoretical and Applied Mathematics. Vol. 6, pp. 1-10, (2011). http://www.ripublication.com/atam.htm
XII. P. Singh, D. Sharma. : ‘Convergence and error analysis of series solution of nonlinear partial differential equation’. Nonlinear Eng., Vol. 7, pp. 303-308, (2018). 10.1515/nleng-2017-0113
XIII. R. Jiwari, S. Pandit, R.C. Mittal. : ‘A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equations with Dirichlet and Neumann boundary conditions’. Applied Mathematics and Computation. Vol. 218 (13), pp. 7279-7294, (2012). 10.1016/j.amc.2012.01.006
XIV. R. K. Mohanty. : ‘New unconditionally stable difference schemes for the solution of multidimensional telegraphic equations’. Int. J. of Computer Mathematics. Vol. 86 (12), pp. 2061–2071 (2008). 10.1080/00207160801965271

View | Download

Abstract:

This article explores the impact of synchronization, both identical and non-identical supporting systems with six different Co-efficient Matrices, on the Rucklidge chaotic system. Two paired chaotic systems are proposed to synchronize using the Active Control Algorithm (ACA). Six sets of different control functions originating from identical and non-identical Master/Drive systems. All synchronizing design demonstrates that six sets of different control functions are always perfectly applied and chaotic systems are significantly synchronized with six different co-efficient Matrices. Parameters are similar across identical pairings of chaotic systems however must be different for non-identical pairs.  The feasibility and efficacy of synchronizing the state variables are derived from the error dynamics coefficient matrix. We analyze the effectiveness of synchronized identical and non-identical approaches to explore which control functions would provide better results. The non-identical pair is formed utilizing the Harb-Zohdy chaotic system with a unique initial value.  In addition, numerical simulations are offered to validate and expand upon the theoretical findings.

Keywords:

Rucklidge system,Synchronization,Identical pair,Non-identical pair,Co-efficient Matrices,Active Control Algorithm,

Refference:

I. A. Njah, U. Vincent. : ‘Chaos synchronization between single and double wells duffing–van der pol oscillators using active control’. Chaos, Solitons & Fractals. Vol.37(5),pp. 1356–136 (2008). 10.1016/j.chaos.2006.10.038
II. A. Tarammim, M. T. Akter. : ‘A comparative study of synchronization methods of rucklidge chaotic systems with design of active control and backstepping methods’. International Journal of Modern Nonlinear Theory and Application. Vol. 11(2), pp. 31–51, (2022). 10.4236/ijmnta.2022.112003
III. A. Tarammim, M. T. Akter. : ‘Shimizu–Morioka’s chaos synchronization: An efficacy analysis of active control and backstepping methods’. Frontiers in Applied Mathematics and Statistics. 9, 1100147, (2023). 10.3389/fams.2023.1100147
IV. B. T. Polyak, P. S. Shcherbakov. : ‘Hard problems in linear control theory: Possible approaches to solution’. Automation and Remote Control. Vol. 66 (5), pp. 681–718, (2005). 10.1007/s10513-005-0115-0
V. C. Xiu, R. Zhou, Y. Liu. : ‘New chaotic memristive cellular neural network and its application in secure communication system’. Chaos, Solitons & Fractals. 141, 110316 (2020). 10.1016/j.chaos.2020.110316
VI. C. Nishad, R. Prasad, P. Kumar, et al., : ‘Synchronization analysis chaos of fractional derivatives chaotic satellite systems via feedback active control methods’.AUTHOREA (2022). 10.22541/au.165527106.69915094/v1
VII. D. S. Scott. : ‘On the accuracy of the gerschgorin circle theorem for bounding the spread of a real symmetric matrix’. Linear algebra and its applications. Vol. 65 pp. 147–155, (1985). 10.1016/0024-3795(85)90093-X
VIII. Idowu Babatunde A., Olasunkanmi Isaac Olusola, O. Sunday Onma, Sundarapandian Vaidyanathan, Cornelius Olakunle Ogabi, and Olujimi A. Adejo. : ‘Chaotic financial system with uncertain parameters-its control and synchronisation’. International Journal of Nonlinear Dynamics and Control. Vol. 1 (3), pp. 271-286, (2019). 10.1504/IJNDC.2019.098682
IX. I. Pehlivan, Y. Uyaroglu, M. Yogun. : ‘Chaotic oscillator design and realizations of the rucklidge attractor and its synchronization and masking simulations’. Scientific Research and Essays. Vol. 5 (16), pp. 2210–2219, (2010).
X. J. Tang. : ‘Synchronization of different fractional order time-delay chaotic systems using active control’. Mathematical problems in Engineering. 2014 (2014). Article ID 262151. 10.1155/2014/262151
XI. L. M. Pecora, T. L. Carroll. : ‘Synchronization in chaotic systems’. Physical review letters. Vol. 64 (8), pp. 821, (1990). 10.1103/PhysRevLett.64.821
XII. M. T. Akter, A. Tarammim, S. Hussen. : ‘Chaos control and synchronization of modified lorenz system using active control and backstepping scheme’. Waves in Random and Complex Media. pp. 1–20, (2023). 10.1080/17455030.2023.2205529
XIII. M. Liu, Z. Wu, X. Fu. : ‘Dynamical analysis of a one-and two-scroll chaotic system’. Mathematics. Vol. 10 (24), 4682, (2022). doi.org/10.3390/math10244682
XIV. M. Marwan, S. Ahmad, M. Aqeel, M. Sabir. : ‘Control analysis of rucklidge chaotic system’. Journal of Dynamic Systems, Measurement, and Control. Vol. 141 (4), 041010 (2019). 10.1115/1.4042030
XV. R. Karthikeyan, V. Sundarapandian. : ‘Hybrid chaos synchronization of four scroll systems via active control’. Journal of Electrical Engineering. Vol. 65(2), pp. 97-103, (2014). 10.2478/jee-2014-0014
XVI. S. Vaidyanathan. : ‘A novel chemical chaotic reactor system and its output regulation via integral sliding mode control’. International Journal of ChemTech Research. Vol. 8(7), pp.146-158, https://sphinxsai.com/2015/ch_vol8_no11/3/(669-683)V8N11CT.pdf
XVII. S. Vaidyanathan. : ‘Lotka-volterra population biology models with negative feedback and their ecological monitoring’. International Journal Pharm Tech Res. Vol. 8 (5), pp. 974–981, (2015).
XVIII. S. Vaidyanathan, C. K. Volos, K. Rajagopal, I. Kyprianidis, I. Stouboulos. : ‘Adaptive backstepping controller design for the anti-synchronization of identical windmi chaotic systems with unknown parameters and its spice implementation’. Journal of Engineering Science & Technology Review. Vo. 8 (2), pp. 74-82, (2015).
XIX. S. Mobayen, S. Vaidyanathan, A. Sambas, S. Kacar, ̈U. C ̧ avu ̧so ̆glu. : ‘A novel chaotic system with boomerang-shaped equilibrium, its circuit implementation and application to sound encryption, Iranian Journal of Science and Technology’. Transactions of Electrical Engineering. Vol. 43, pp. 1–12. (2019). 10.1007/s40998-018-0094-0
XX. S. Vaidyanathan, A. Sambas, S. Kacar, U. Cavusoglu. : ‘A new finance chaotic system, its electronic circuit realization, passivity based synchronization and an application to voice encryption’. Nonlinear Engineering. Vol. 8 (1), pp. 193–205, (2019). 10.1515/nleng-2018-0012
XXI. S. Wang, S. Bekiros, A. Yousefpour, S. He, O. Castillo, H. Jahanshahi, : ‘Synchronization of fractional time-delayed financial system using a novel type-2 fuzzy active control method’. Chaos, Solitons & Fractals. 136, 109768 (2020). 10.1016/j.chaos.2020.109768
XXII. S. Vaidyanathan. : ‘Global chaos synchronization of rucklidge chaotic systems for double convection via sliding mode control’. International Journal of ChemTech Research. Vol. 8 (8), pp. 61–72 (2015).
XXIII. U. E. Kocamaz, Y. Uyaro ̆glu. : ‘Controlling rucklidge chaotic system with a single controller using linear feedback and passive control methods’. Nonlinear Dynamics. 75, pp. 63–72 (2014). 10.1007/s11071-013-1049-7
XXIV. U. Vincent. : ‘Chaos synchronization using active control and backstepping control: a comparative analysis’. Nonlinear Analysis: Modelling and Control. Vol. 13 (2), pp. 253–261, (2008).
XXV. Yao, Q., : ‘Synchronization of second-order chaotic systems with uncertainties and disturbances using fixed-time adaptive sliding mode control’. Chaos, Solitons & Fractals. 142, 110372 (2021). 10.1016/j.chaos.2020.110372
XXVI. Y. Lei, W. Xu, W. Xie. : ‘Synchronization of two chaotic four-dimensional systems using active control’. Chaos, Solitons & Fractals. Vol. 32 (5), pp. 1823–1829 (2007). 10.1016/j.chaos.2005.12.014
XXVII. Yang Xinsong, Yang Liu, Jinde Cao, and Leszek Rutkowski. : ‘Synchronization of coupled time-delay neural networks with mode-dependent average dwell time switching’. IEEE transactions on neural networks and learning systems. Vol. 31(12), pp. 5483-5496 (2020). 10.1109/TNNLS.2020.2968342
XXVIII. Zouari Farouk, and Amina Boubellouta. ‘Adaptive neural control for unknown nonlinear time-delay fractional-order systems with input saturation’. In Advanced Synchronization Control and Bifurcation of Chaotic Fractional-Order Systems, IGI Global. 54-98. (2018). 10.4018/978-1-5225-5418-9.ch003

View | Download

Abstract:

Over the past few years, there has been increased awareness about the importance of protecting the environment particularly after global warming came up. The approach proposed here in this paper for reducing fuel consumption is the combination of clustering algorithms’ ideas with natural optimization techniques, aimed at efficient route optimization of vehicles. It uses clustering to group customer locations together that in turn allows the development of more efficient routes. The goal of this study is to reduce fuel consumption while optimizing travel plans. This study proposed a nature-inspired algorithm-based model for minimizing fuel consumption in the vehicle routing problem. K-means clustering and the genetic algorithm have been used in this study to find the optimized route with the minimum fuel consumption. It has been observed in this study that routing plans found by the proposed approach consume fewer units of fuel than those generated using optimization techniques which optimize distance covered. This indicates that such an approach could serve as a tool for minimizing fuel consumption in different enterprises.

Keywords:

Vehicle routing Problem,Fuel consumption,Genetic Algorithm,K-Means clustering,

Refference:

I. A. Garcia Najera. : ‘Multi-Objective evolutionary algorithms for vehicle routing problems’. University of Birmingham. pp 1-225, (2010).
II. D. J. Chang and E. K. Morlok. : ‘Vehicle speed profiles to minimize work and fuel consumption’. J. Transp. Eng.,Vol.-13, No.-13, pp 173–182, (2005).
https://doi.org/10.1061/(ASCE)0733-947X(2005)131:3(173)
III. D.K Shetty, R.K Raju, N.Ayedee, B.Singla, N.Naik and S. Pavithra. : ‘Assessment of e-marketplace in increasing the cost efficiency of road transport industry’. PalArch’s J. Archaeol. Egypt. Vol.-17, No.-9, pp 3799–3840, (2020).
https://archives.palarch.nl/index.php/jae/article/view/4508
IV. F. Russo and A. Comi. : ‘City characteristics and urban goods movements: A way to environmental transportation system in a sustainable city’. Procedia Soc. Behav. Sci., Vol.-39, pp 61–73., (2013).
https://doi.org/10.1016/j.sbspro.2012.03.091
V. F.A. Zainuddin, M.F. Abd Samad and D. Tunggal. : ‘A review of crossover methods and problem representation of genetic algorithm in recent engineering applications’. IJAST Vol.-29, No.10, pp 759–769. (2020).
VI. G. B Dantzig and J. H. Ramser’. ‘The truck dispatching problem’. sssManag. Sci.,Vol.- 6, No.-1, pp 80–91, (1959). 10.1287/mnsc.6.1.80
VII. Holland. : ‘Genetic algorithms’. Scientific American. vol.-267(1), pp 66-73, (1992). https://www.jstor.org/stable/24939139
VIII. H. Wu and S. C. Dunn. : ‘Environmentally responsible logistics systems’. Int. J. Phys. Distrib. Logist. Manag. Vol.- 25, No.-2, pp 20–38, (1995).
IX. I. Kara, B.Y. Kara and M. K. Yetis. : ‘Cumulative vehicle routing problems’. Vehicle routing problem, In-Teh, pp 85–98, (2008).
X. L. Schipper. : ‘Automobile use, fuel economy and CO2 emissions in industrialized countries: Encouraging trends through 2008?’. Transp. Policy,. Vol.-18, No.-2, pp 358–372, (2011). https://doi.org/10.1016/j.tranpol.2010.10.011
XI. M. F. Ibrahim, F.R. Nurhakiki, D.M Utama and A.A Rizaki. : ‘Optimised genetic algorithm crossover and mutation stage for vehicle routing problem pick-up and delivery with time windows’. Mater. sci. eng.,Vol.-1071, No.- 012025, pp 1-8, (2021). 10.1088/1757-899X/1071/1/012025
XII. M. K Kakkar, J. Singla, N. Garg, G. Gupta, P.Srivastava, A. Kuma. : ‘Class Schedule Generation using Evolutionary Algorithms’. J. Phys., Vol.-1950, No. 012067, pp 1-9, (2021). 10.1088/1742-6596/1950/1/012067
XIII. M.M. Solomon, Algorithms for the vehicle routing and scheduling problems with time window constraints’, Oper. Res.,Vol.- 35(2), (1987), pp 254–265. https://doi.org/10.1287/opre.35.2.254
XIV. P. Chandra and N. Jain. : ‘The logistics sector in India: Overview and challenges’. Indian Institute of Management Ahmedabad, India,(2007), pp. 105-269.
XV. P. Toth and D. Vigo. : ‘The vehicle routing problem’. SIAM, (2002).
XVI. R. Mittal, V. Malik, V. Singh, J. Singh and A. Kaur. : ‘Integrating genetic algorithm with random forest for improving the classification performance of web log data’. PDGC, pp. 77–181, (2020). 10.1109/PDGC50313.2020.9315807
XVII. Y. B. Kalpana and S. M. Nandhagopal. : ‘lulc image classifications using k-means clustering and knn algorithm’. Adv. Dyn. Syst. Appl., Vol.-30(10), pp 1640–1652, (2021) 10.46719/dsa202130.10.07
XVIII. Y. Kuo. : ‘Using simulated annealing to minimize fuel consumption for the time-dependent vehicle routing problem’. Comput. Ind. Eng, Vol.-59(1), pp 157–165, (2010). https://doi.org/10.1016/j.cie.2010.03.012
XIX. Y. Kuo and C. C Wang. : ‘Optimizing the VRP by minimizing fuel consumption, Manag’. Environ. Qual., Vol.-22(4), pp 440-450, (2011). https://doi.org/10.1108/14777831111136054
XX. Y. Suzuki. : ‘A new truck-routing approach for reducing fuel consumption and pollutants emission’. Transp. Res. D: Transp. Environ., Vol.-16(1), pp. 73–77, (2011). 10.1016/j.trd.2010.08.003

View | Download

Abstract:

This study utilizes an innovative approach by combining the Laplace transform with the variational iteration method to address one-dimensional heat equations encountered in diffusion phenomena. Initially, the heat equation is transformed into a modified form using the Laplace transformation. Subsequently, the variational iteration method is employed to obtain both numerical and approximate analytical solutions. In addition to graphical representations of the outcomes obtained using the suggested, the study includes practical instances to demonstrate the efficacy of the suggested approach.

Keywords:

Laplace Transform,heat equations,Variational iteration method,Partial differential equations,

Refference:

I. C. P. Murphy, D. J. Evans. : ‘Chebyshev series solution of two dimensional heat equation’. Mathematics and Computers in Simulation. Vol. 23(2), pp. 157-162 (1981). 10.1016/0378-4754(81)90053-7
II. E. F. Cara, A. Munch. : ‘Numerical exact controllability of the 1D heat equation: duality and Carleman weights’. Journal of Optimization Theory and Applications. Vol. 163(1), pp. 253-285, (2014). 10.1007/s10957-013-0517-z
III. E. Hesameddini and H. Latifizadeh. : ‘Reconstruction of variational iteration algorithms using the Laplace transform’. International Journal of Nonlinear Sciences and Numerical Simulation. Vol. 10(11-12), pp. 1377-1382, (2009). 10.1515/IJNSNS.2009.10.11-12.1377
IV. G. Singh, I. Singh. : ‘New Laplace variational iterative method for solving 3D Scrodinger equations’. Journal of Mathematical and computational science,Vol. 10(5), pp. 2015-2024 (2020). 10.28919/jmcs/4792
V. J. Douglas, D. W. Peaceman. : ‘Numerical solution of two dimensional heat flow problems’. Alche Journal. Vol. 1(4), pp. 505-512 (1955). 10.1002/aic.690010421
VI. J. H. He. : ‘Variational iteration method – a kind of non-linear analytical technique: some examples’. International Journal of Non-Linear Mechanics. Vol. 34(4), pp. 699-708 (1999). 10.1016/S0020-7462(98)00048-1
VII. J. Kouatchou. : ‘Finite Difference and Collocation Methods for solution of Two dimensional Heat Equation’. Numerical Methods for Partial Differential Equations. Vol. 17(1), pp. 54-63 (2001). 10.1002/1098-2426(200101)17:13.0.CO;2-A
VIII. K. A. Koroche. : ‘Numerical solution for one dimensional linear types of parabolic partial differential equation and application to heat equation’. Mathematics and Computer Science. Vol. 5(4), pp. 76-85 (2020). 10.11648/j.mcs.20200504.12
IX. K. D. Sharma, R. Kumar, M. K. Kakkar, S. Ghangas. : ‘Three dimensional waves propagation in thermo-viscoelastic medium with two temperature and void’. In IOP Conference Series: Materials Science and Engineering. 1033(1), 012059-73 (2021). 10.1088/1757-899X/1033/1/012059
X. M. Aneja, M. Gaur, T. Bose, P. K. Gantayat, R. Bala. : ‘Computer-based numerical analysis of bioconvective heat and mass transfer across a nonlinear stretching sheet with hybrid nanofluids’. In International Conference on Frontiers of Intelligent Computing: Theory and Applications, pp. 677-686 (2023). 10.1007/978-981-99-6702-5_55
XI. M. Dehghan. : ‘The one-dimensional heat equation subject to a boundary integral specification’. Chaos, Solitons & Fractals. Vol. 32(2), pp. 661-675, (2007). 10.1016/j.chaos.2005.11.010
XII. S. A. Khuri and A. Sayfy. : ‘A Laplace variational iteration strategy for the solution of differential equations’. Applied Mathematics Letters. Vol. 25 (12), pp. 2298–2305 (2012). 10.1016/j.aml.2012.06.020
XIII. T. Luga, T. Aboiyar, S. O. Adee. : ‘Radial basis function methods for approximating the two dimensional heat equations’. International Journal of Engineering Applied Sciences and Technology. Vol. 4(2), pp. 7-15 (2019). 10.33564/IJEAST.2019.v04i02.002
XIV. Z. Hammouch and T. Mekkaoui. : ‘A Laplace-variational iteration method for solving the homogeneous Smoluchowski coagulation equation’. Applied Mathematical Sciences. Vol. 6(18), pp. 879-886 (2012).

View | Download

Abstract:

Transportation problems are one of the most important classes of linear programming problems. This manages a product's transportation from its point of origin to its final destination. The primary objective is to meet destination requirements while minimizing shipping expenses. This work presents a two-stage fuzzy transportation cost-related problem and uses a parametric approach to derive a fuzzy solution. A novel method is suggested to address a two-phase fuzzy transportation issue where the transport cost is expressed in terms of fuzzy trapezoidal figures. This approach is particularly effective because it is easy to comprehend. By supporting decision-makers during the process and offering a simple and cost-effective solution, the suggested strategy assists decision-makers with logistics-related problems

Keywords:

trapezoidal uncertain number,two-stage uncertain transportation problem,optimal transportation cost solution,

Refference:

I. A. A. Noora, and P. Karami. : ‘Ranking functions and its applications to fuzzy DEA’. Int. Math. Forum. Vol. 3(30), pp. 1469-1480, (2008). https://www.m-hikari.com/imf-password2008/29-32-2008/nooraIMF29-32-2008.pdf.
II. A. N. Gani, and K. A. Razak. : ‘Two Stage Fuzzy Transportation Problem’. J. Phys. Sci., Vol. 10, pp. 63–69, (2016). http://inet.vidyasagar.ac.in:8080/jspui/handle/123456789/720.
III. B. Choudhary. : ‘Optimal solution of transportation problem based on revised distribution method’. IJIRSET. Vol. 5(8), pp. 254-257, (2016). https://www.ijirset.com/upload/2016/august/109_Optimal.pdf.
IV. E. Hosseini. : ‘Three new methods to find initial basic feasible solution of transportation problems’. Appl. Math. Sci., Vol. 11(37), pp. 1803-1814, (2017). 10.12988/ams.2017.75178.
V. F. L. Hitchcock. : ‘The distribution of a product from several sources to numerous localities’. J Math Phys., Vol. 20(1-4), pp. 224–230, (1941). 10.1002/sapm1941201224.
VI. G. Gupta, S. Singh and D. Rani. : ‘A note on zero suffix method for the optimal solution of the transportation problems’. Natl Acad Sci Lett., Vpl. 41, pp. 293-294, (2018). 10.1007/s40009-018-0653-y.
VII. G. Singh and A. Singh. : ‘Soft Computing Approach for Three-Level Time Minimization Transportation Problem’. ICTACS (IEEE). pp. 236-241, (2022), 10.1109/ICTACS56270.2022.9988352.
VIII. H. A. Hussein and M. A. K. Shiker. : ‘A modification to Vogel’s approximation method to solve transportation problems’. J Phys Conf Ser., Vol. 1591(1), 012029, (2020). 10.1088/1742-6596/1591/1/012029.
IX. H. I. Calvete, C. Gale, J. A. Iranzo, and P. Toth. : ‘A metaheuristic for the two-stage fixed-charge transportation problem’. Comput Oper Res. Vol. 95, pp. 113-122, (2018). 10.1016/j.cor.2018.03.007.
X. J. Singla, G. Gupta, M. K. Kakkar and N. Garg. : ‘A novel approach to find initial basic feasible solution of transportation problems under uncertain environment. Proceedings of AIP., 2357, 110007, (2022). doi:10.1063/5.0080755.
XI. J. Singla, G. Gupta, M. K. Kakkar and N. Garg. : ‘Revised algorithm of Vogel’s approximation method (RA-VAM): an approach to find basic initial feasible solution of transportation problem’. ECST., Vol. 107(1), pp. 8757, (2022). 10.1149/10701.8757ecst.
XII. K. Karagul and Y. Sahin. : ‘A novel approximation method to obtain initial basic feasible solution of transportation problem’. J. King Saud Univ. Eng. Sci., Vol. 32(3), pp. 211-218, (2020). 10.1016/j.jksues.2019.03.003.
XIII. L. A. Zadeh. : ‘Fuzzy sets’. Inf. Control. Vo.8, pp. 338–353, (1965), 10.1016/S0019-9958(65)90241-X.
XIV. M. M. Hossain, M. M. Ahmed, M. A. Islam and S. I. Ukil. : ‘An effective approach to determine an initial basic feasible solution: A TOCM-MEDM Approach’. OJOp., Vol. 9(2), pp. 27-37, (2020). 10.4236/ojop.2020.92003.
XV. M. Malireddy. : ‘A New Algorithm for initial basic feasible solution of Transportation Problem’. Int. j. eng. sci. invention res. Dev., Vol. 7(8), pp. 41‒43, (2018).
https://www.ijesi.org/papers/Vol(7)i8/Version-4/E0708044143.pdf.
XVI. M. Mondal, and D. Srivastava.: ‘A genetic algorithm-based approach to solve a new time-limited travelling salesman problem’. IJDST. Vol. 14(2), pp. 1-14, (2023), 10.4018/IJDST.317377.
XVII. M. Mondal, and D. Srivastava. : ‘Solving a Multi-Conveyance Travelling Salesman Problem using an Ant Colony Optimization Method’. Indian J Sci Technol., Vol. 15(45), pp. 2468-2475, (2022). 10.17485/IJST/v15i45.1506.
XVIII. M. Sathyavathy and M. Shalini. : ‘Solving transportation problem with four different proposed mean method and comparison with existing methods for optimum solution’. J. Phys. Conf. Ser., Vol. 1362(1), 012088, (2019). doi:10.1088/1742-6596/1362/1/012088.
XIX. N. Hashmi, S.A. Jalil and S. Javaid. : ‘A model for two-stage fixed charge transportation problem with multiple objectives and fuzzy linguistic preferences’. Soft Comput. Vol. 23, pp. 12401-12415, 2019.
10.1007/s00500-019-03782-1.
XX. O. Mutlu, K. Karagül, and Y. Şahin. : ‘Avoid maximum cost method for determining the initial basic feasible solution of the transportation problem’. PAJES. Vol. 28(4), pp. 569-576, (2022). 10.5505/pajes.2022.61426.
XXI. P. Pandian and G. Natarajan. : ‘A new method for finding an optimal solution for transportation problems’. IJMSEA. Vol. 4(2), pp. 59–65, (2010). https://www.m-hikari.com/ams/ams-2010/ams-37-40-2010/pandianAMS37-40-2010.pdf.
XXII. R. Kumar, R. Gupta and O. Karthiyayini. : ‘A new approach to find the initial basic feasible solution of a transportation problem’. Int. j. res. Granthaalayah. Vol. 6(5), pp. 321-325, (2018). 10.5281/zenodo.1283413.
XXIII. S. Jamali, A. S. Soomro and M. M Shaikh. : ‘The minimum demand method–a new and efficient initial basic feasible solution method for transportation problems’. J. mech. continua math. Sci., Vol. 15(19), pp. 94-109, (2020). 10.26782/jmcms.2020.10.00007.
XXIV. S. Singh, G. Gupta and D. Rani. : ‘An alternate for the initial basic feasible solution of category 1 uncertain transportation problems’. Proc. Natl. Acad. Sci. India – Phys. Sci., Vol.90, pp. 157-167, (2020). 10.1007/s40010-018-0557-8.
XXV. Sonia and R. Malhotra. : ‘A polynomial algorithm for a two – stage time minimizing transportation problem’. OPSEARCH. Vol. 39(5-6), pp. 251 – 266, (2002). 10.1007/BF03399188.
XXVI. T. Can and H. Koçak. : ‘Tuncay can’s approximation method to obtain initial basic feasible solution to transport problem’. Appl. Comput. Math., Vol. 5(2), pp. 78-82, (2016), 10.11648/j.acm.20160502.17.
XXVII. T.C. Koopmans. : ‘Optimum utilization of the transportation system’. Econometrica. Vol. 17, pp. 136–146, (1949). 10.2307/1907301.
XXVIII. V. J. Sudhakar and V. N. Kumar. : ‘A different approach for solving two stage fuzzy transportation problems’. International Journal of Contemporary Mathematical Sciences. Vol. 6(11), pp. 517–526, (2011). https://www.m-hikari.com/ijcms-2011/9-12-2011/sudhakarIJCMS9-12-2011.pdf.
XXIX. V. Jaiswal, A. K. Agrawal and A. Pandey. : ‘Solving fuzzy transportation problem by various methods and their comparison in fuzzy environment’. Eur. Chem. Bull., Vol. 12, pp. 2079-2084, (2023). 10.48047/ecb/2023.12.si5a.087
XXX. W. Ritha and J.M. Vinotha. : ‘Multiobjective two-stage fuzzy transportation problem’. J. Phys. Sci., Vol. 13, pp. 107–120, (2009). http://inet.vidyasagar.ac.in:8080/jspui/handle/123456789/793.

View | Download

Abstract:

The present paper is a comparative analysis of a two-unit autoclave system in a manufacturing plant. Most of the studies have been done by considering standby units to remain as good as new ones in this mode, but practically they may be corrupted by any environmental issues. This fact makes us concerned about the standby unit. Two stochastic models were developed based on such concern.  Model 1 is constructed based on basically two possibilities; firstly, the standby unit is inspected after a fixed amount of time to check its feasibility. Secondly, either it will be repaired or replaced. Replacement is instant. Model 2 is constructed based on the same assumptions but replacement is not instant, it takes some random amount of time to be replaced. Stochastic analysis uses Markov processes to investigate how these dynamic factors interact and impact system profitability, availability, and dependability. By studying various scenarios about repair prices, replacement costs, inspection frequency, and fluctuating demand patterns, this research provides vital insights into the most effective approaches for handling redundant units and preserving system functionality. The results guide managing complex decision-making processes for safeguarding and maximizing system functionality, which has practical ramifications for sectors and systems that depend on redundancy to guarantee continuity and reliability.

Keywords:

Stochastic Model,Reliability,semi-Markov Process,Regenerative Point Technique,Varied Production,comparative analysis,Innovation,

Refference:

I. Gao S., & Wang, J., : ‘Reliability and availability analysis of a retrial system with mixed standbys and an unreliable repair facility’. Reliability Engineering & System Safety. 205:107240, (2021). 10.1016/j.ress.2020.107240
II. Juybari M. N., Hamadani, A. Z., & Ardakan, M. A., : ‘Availability analysis and cost optimization of a repairable system with a mix of active and warm-standby components in a shock environment’. Reliability Engineering & System Safety. Vol. 237:109375, (2023). 10.1016/j.ress.2023.109375
III. Kakkar MK., Bhatti, J., Gupta, G., : ‘Reliability Optimization of an Industrial Model Using the Chaotic Grey Wolf Optimization Algorithm’. In Manufacturing Technologies and Production Systems CRC Press. pp. 317-324, (2023).
IV. Kakkar M. K., Bhatti, J., Gupta, G., & Sharma K. D., : ‘Reliability analysis of a three unit redundant system under the inspection of a unit with correlated failure and repair times’. AIP Conf. Proc., 2357, 100025. (2022). 10.1063/5.0080964
V. Kumar A., Garg, R., & Barak, M. S., : ‘Reliability measures of a cold standby system subject to refreshment’. International Journal of System Assurance Engineering and Management. Vol. 14(1), pp. 147-55, (2023). 10.1007/s13198-021-01317-2
VI. Levitin G., Xing, L., & Dai, Y., : ‘Cold standby systems with imperfect backup’. IEEE Transactions on Reliability. Vol. 65(4), pp. 1798-809. (2015), 10.1109/TR.2015.2491599
VII. Levitin G., Xing, L., & Xiang, Y., : ‘Optimizing preventive replacement schedule in standby systems with time consuming task transfers’. Reliability Engineering & System Safety. 205:107227, (2021). 10.1016/j.ress.2020.107227
VIII. Malhotra R., & Taneja, G., : ‘Stochastic analysis of a two-unit cold standby system wherein both units may become operative depending upon the demand’. Journal of Quality and Reliability Engineering. Article ID 896379, (2014). 10.1155/2014/896379
IX. Malhotra R., & Taneja, G., : ‘Comparative study between a single unit system and a two-unit cold standby system with varying demand’. Springer plus, Vol. 4(1), pp. 1-7, (2015). 10.1186/s40064-015-1484-7
X. Malhotra R., : ‘Reliability and availability analysis of a standby system with activation time and varying demand’. In Engineering Reliability and Risk Assessment. Elsevier. pp. 35-51, (2023). 10.1016/B978-0-323-91943-2.00004-6
XI. Malhotra R., Alamri, F. S., & Khalifa, H. A., : ‘Novel Analysis between Two-Unit Hot and Cold Standby Redundant Systems with Varied Demand’. Symmetry. Vol. 15(6), 1220, (2023). 10.3390/sym15061220
XII. Malhotra R., & Kaur, H., : ‘Reliability of a Manufacturing Plant with Scheduled Maintenance, Inspection, and Varied Production’. In Manufacturing Engineering and Materials Science. CRC Press. pp. 254-264, (2023).
XIII. Ram M., Singh, S. B., & Singh V. V., : ‘Stochastic analysis of a standby system with waiting repair strategy’. IEEE Transactions on Systems, man, and cybernetics: Systems. Vol. 43(3), pp. 698-707. (2013). 10.1109/TSMCA.2012.2217320
XIV. Shekhar C., Devanda M., & Kaswan S., : ‘Reliability analysis of standby provision multi‐unit machining systems with varied failures, degradations, imperfections, and delays’. Quality and Reliability Engineering International. Vol. 39(7), pp. 3119-39, (2023). 10.1002/qre.3421
XV. Taneja G., Goyal A., & Singh D. V., : ‘Reliability and cost-benefit analysis of a system comprising one big unit and two small identical units with priority for operation/repair to big unit’. Vol. 5(3), pp. 235-248, (2011). https://www.sid.ir/paper/322548/en
XVI. Wang J., Xie N., & Yang N., : ‘Reliability analysis of a two-dissimilar-unit warm standby repairable system with priority in use’. Communications in Statistics-Theory and Methods. Vol. 50(4), pp. 792-814. (2021). 10.1080/03610926.2019.1642488
XVII. Wang W., Wu Z., Xiong J., & Xu Y., : ‘Redundancy optimization of cold-standby systems under periodic inspection and maintenance’. Reliability engineering & system safety. Vol. 180, pp. 394-402. (2018). 10.1016/j.ress.2018.08.004
XVIII. Yang D. Y., & Tsao C. L., : ‘Reliability and availability analysis of standby systems with working vacations and retrial of failed components’. Reliability Engineering & System Safety. Vol. 182, pp. 46-55, (2019). 10.1016/j.ress.2018.09.020

View | Download

Abstract:

This study adopts a novel approach that integrates the Laplace transform and the variational iteration method to tackle the (1+1)-Dimensional Telegraph equations, representing the current or voltage flow in electrical circuits. The methodology commences with transforming the telegraph equation into a modified format using Laplace transformation. Following this, the variational iteration method is utilized to derive both numerical and approximate analytical solutions. The paper incorporates practical examples to demonstrate the effectiveness of the proposed approach, supplemented with graphical illustrations depicting the outcomes achieved through the suggested techniques.

Keywords:

Telegraph Equations,Laplace Transform,Variational iteration method,

Refference:

I. A. S. Arife and A. Yildirim. : ‘New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step’. World Applied Sciences Journal. Vol. 13(10), pp. 2186–2190 (2011).
II. A. Saadatmandi, M. Dehghan. : ‘Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method’. Numerical Methods Partial Differ. Equation. Vol. 26(1), pp. 239-252 (2010). 10.1002/num.20442
III. B. Blbl, M. Sezer. : ‘A Taylor matrix method for the solution of a two-dimensional linear hyperbolic equation’. Applied Mathematics Letter. Vol. 24(10), pp. 1716-1720 (2011). 10.1016/j.aml.2011.04.026
IV. E. Hesameddini and H. Latifizadeh. : ‘Reconstruction of variational iteration algorithms using the Laplace transform’. International Journal of Nonlinear Sciences and Numerical Simulation. Vol. 10(11-12), pp. 1377-1382 (2009). 10.1515/IJNSNS.2009.10.11-12.1377
V. F. Gao, C. Chi. : ‘Unconditionally stable difference schemes for a one space-dimensional linear hyperbolic equation’. Appl. Math. Comput., Vol. 187(2), pp. 1272-1276 (2007). 10.1016/j.amc.2006.09.057
VI. G. C. Wu. : ‘Laplace transform overcoming principle drawbacks in application of the variational iteration method to fractional heat equations’. Thermal Science. Vol. 16(4), pp. 1257-1261, (2012). 10.2298/TSCI1204257W
VII. H. Y. Martinez, J. F. Gomez-Aguilar. : ‘Laplace variational iteration method for modified fractional derivatives with non-singular kernel’. Journal of Applied and Computational Mechanics. Vol. 6(3), pp. 684-698 (2020). 10.22055/JACM.2019.31099.1827
VIII. I. Singh, S. Kumar. : ‘Wavelet methods for solving three- dimensional partial differential equations’. Mathematical Sciences. Vol. 11, pp. 145-154 (2017). 10.1007/s40096-017-0220-6
IX. I. Singh, S. Kumar. : ‘Numerical solution of two- dimensional telegraph equations using Haar wavelets’. Journal of Mathematical Extension. Vol. 11(4), pp. 1-26 (2017). 10.1007/s40096-017-0220-6
X. J. H. He. : ‘Variational iteration method-a kind of non-linear analytical technique: some examples’. International Journal of Non-Linear Mechanics. Vol. 34(4), pp. 699-708 (1999). https://tarjomefa.com/wp-content/uploads/2018/06/9165-English-TarjomeFa.pdf
XI. K. D. Sharma, R. Kumar, M. K. Kakkar, S. Ghangas. : ‘Three dimensional waves propagation in thermo-viscoelastic medium with two temperature and void’. In IOP Conference Series: Materials Science and Engineering Vol. 1033(1), pp. 012059-73 (2021). 10.1088/1757-899X/1033/1/012059
XII. M. Aneja, M. Gaur, T. Bose, P. K. Gantayat, R. Bala. : ‘Computer-based numerical analysis of bioconvective heat and mass transfer across a nonlinear stretching sheet with hybrid nanofluids’. In International Conference on Frontiers of Intelligent Computing: Theory and Applications. pp. 677-686 (2023). 10.1007/978-981-99-6702-5_55
XIII. M. Dehghan, A. Ghesmati. : ‘Solution of the second-order one dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method’. Eng. Anal. Bound. Elem., Vol. 34(1), pp. 51-59 (2010). 10.1016/j.enganabound.2009.07.002
XIV. M. Dehghan, and M. Lakestani. : ‘The use of chebyshev cardinal functions for solution of the second‐order one‐dimensional telegraph equation’. Numerical Methods for Partial Differential Equations. Vol. 25(4), pp. 931-938 (2009). 10.1002/num.20382
XV. M. Javidi, N. Nyamoradi. : ‘Numerical solution of telegraph equation by using LT inversion technique’. International Journal of Advanced Mathematical Sciences. Vol. 1(2), pp. 64-77 (2013). 10.14419/ijams.v1i2.780
XVI. M. Lakestani, B. N. Saray. : ‘Numerical solution of telegraph equation using interpolating scaling functions’. Comput. Math. Appl., Vol. 60(7), pp. 1964-1972 (2010). 10.1016/j.camwa.2010.07.030
XVII. R. C. Mittal, R. Bhatia. : ‘Numerical solution of second order one dimensional hyperbolic telegraph equation by cubic B-spline collocation method’. Appl. Math. Comput., Vol. 220, pp. 496-506 (2013). 10.1016/j.amc.2013.05.081
XVIII. R. Jiwari R, S. Pandit S, R.C. Mittal. : ‘A differential quadrature algorithm for the numerical solution of the second-order one dimensional hyperbolic telegraph equation’. International Journal of Nonlinear Science. Vol. 13(3), pp. 259-266 (2012).
XIX. S. Yüzbaşı, M. Karaçayır. : ‘A Galerkin-type method to solve one-dimensional telegraph equation using collocation points in initial and boundary conditions’. International Journal of Computational Methods. Vol. 25(15), 1850031 (2018). 10.1142/S0219876218500317
XX. T. M. Elzaki, E. M. A. Hilal. : ‘Analytical Solution for Telegraph Equation by Modified of Sumudu Transform “Elzaki Transform”. Mathematical Theory and Modeling. Vol. 2(4), pp. 104-111 (2012). https://core.ac.uk/download/pdf/234678972.pdf
XXI. T. M. Elzaki. : ‘Solution of nonlinear partial differential equations by new Laplace variational iteration method’. Differential Equations: Theory and Current Research. (2018).
XXII. T. S. Jang. : ‘A new solution procedure for the nonlinear telegraph equation’. Communications in Nonlinear Science and Numerical Simulation. Vol. 29(1-3), pp. 307-326 (2013). 10.1016/j.cnsns.2015.05.004
XXIII. Z. Hammouch and T. Mekkaoui. : ‘A Laplace-variational iteration method for solving the homogeneous Smoluchowski coagulation equation’. Applied Mathematical Sciences. Vol. 6(18), pp. 879-886 (2012). https://www.m-hikari.com/ams/ams-2012/ams-17-20-2012/hammouchAMS17-20-2012.pdf

View | Download

Abstract:

This research paper focuses on the comparison study of wavelet solutions for solving logistic differential equations. For this purpose, we are utilizing Chebyshev wavelets of the second kind and Haar wavelets. Various numerical tests have been conducted to demonstrate the ease of use, precision, and effectiveness of the solutions provided by various wavelet techniques. The implications of these results are discussed within the broader context of mathematical and scientific research.

Keywords:

Wavelets,Chebyshev wavelets of the second kind,Haar wavelets,Operational metrics of integrations,Logistic differential equations,Numerical examples.,

Refference:

I. Alsaedi, D. Baleanu, S. Etemad, & S. Rezapour. : ‘On coupled systems of time-fractional differential problems by using a new fractional derivative’. Journal of Function Spaces, 2016, (2016). 10.1155/2016/4626940
II. A. Turan Dincel, & S. N. Tural Polat. : ‘Fourth kind Chebyshev Wavelet Method for the solution of multi-term variable order fractional differential equations’. Engineering Computations. Vol. 39(4), pp. 1274-1287, (2022). https://www.emerald.com/insight/content/doi/10.1108/EC-04-2021-0211/full/html
III. B. Sripathy, P. Vijayaraju, & G. Hariharan. : ‘Chebyshev wavelet-based approximation method to some non-linear differential equations arising in engineering’. Int. J. Math. Anal., Vol. 9(20), pp. 993-1010, (2015) 10.12988/ijma.2015.5393
IV. C. H. Hsiao, & S. P. Wu. : ‘Numerical solution of time-varying functional differential equations via Haar wavelets’. Applied Mathematics and Computation. Vol. 188(1), pp. 1049-1058, (2007) https://doi.org/10.1016/j.amc.2006.10.070
V. Debnath Lokenath. : ‘A brief historical introduction to fractional calculus’. International Journal of Mathematical Education in Science and Technology. Vol. 35, pp. 487-501, (2004) 10.1080/00207390410001686571
VI. E. N. Petropoulou. : ‘A discrete equivalent of the logistic equation’. Advances in Difference Equations, 2010, pp. 1-15, (2010) doi:10.1155/2010/457073
VII. F. A. Shah, R. Abass, & L. Debnath. : ‘Numerical solution of fractional differential equations using Haar wavelet operational matrix method’. International Journal of Applied and Computational Mathematics. Vol. 3, pp. 2423-2445, (2017) https://link.springer.com/article/10.1007/s40819-016-0246-8
VIII. G. Hariharan, & K. Kannan. : ‘Review of wavelet methods for the solution of reaction–diffusion problems in science and engineering’. Applied Mathematical Modelling. Vol. 38(3), pp. 799-813, (2014) https://doi.org/10.1016/j.apm.2013.08.003
IX. H. Jafari, S. A. Yousefi, M. A. Firoozjaee, S. Momani, & C. M Khalique. : ‘Application of Legendre wavelets for solving fractional differential equations’. Computers & Mathematics with Applications’. Vol. 62(3), pp. 1038-1045, (2011). https://doi.org/10.1016/j.camwa.2011.04.024
X. I. Aziz, & R. Amin. : ‘Numerical solution of a class of delay differential and delay partial differential equations via Haar wavelet’. Applied Mathematical Modelling. Vol. 40(23-24), pp. 10286-10299, (2016) 10.1016/j.apm.2016.07.018
XI. K. Srinivasa, & R. A. Mundewadi. : ‘Wavelets approach for the solution of nonlinear variable delay differential equations’. International Journal of Mathematics and Computer in Engineering. Vol. 1(2), pp. 139-148, (2023). 10.2478/ijmce-2023-0011
XII. M. E. Benattia, & K. Belghaba. : ‘Numerical solution for solving fractional differential equations using shifted Chebyshev wavelet’. Gen. Lett. Math., Vol. 3(2), pp. 101-110, (2017). https://www.refaad.com/Files/GLM/GLM-3-2-3.pdf
XIII. M. Faheem, A. Raza, & A. Khan. : ‘Wavelet collocation methods for solving neutral delay differential equations’. International Journal of Nonlinear Sciences and Numerical Simulation. Vol. 23(7-8), pp. 1129-1156, 10.1515/ijnsns-2020-0103
XIV. M. Ghergu, and V. Radulescu. : ‘Existence and nonexistence of entire solutions to the logistic differential equation’. In Abstract and Applied Analysis. pp. 995-1003, (2003). 10.1155/S1085337503305020
XV. M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, & F. Mohammadi. : ‘Wavelet collocation method for solving multiorder fractional differential equations’. Journal of Applied mathematics. 2012. 10.1155/2012/542401
XVI. M. H. Heydari, M. R. Mahmoudi, A. Shakiba, & Z. Avazzadeh. : ‘Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion’. Communications in Nonlinear Science and Numerical Simulation. Vol. 64, pp. 98-121, (2018). https://doi.org/10.1016/j.cnsns.2018.04.018
XVII. M. M. Khader and M. M. Babatin. : ‘Numerical study of fractional Logistic differential equation using implementation of Legendre wavelet approximation’. Journal of Computational and Theoretical Nanoscience. Vol. 13(1), pp. 1022-1026, (2016). 10.1166/jctn.2016.4510
XVIII. M. S. Hafshejani, S. K. Vanani, & J. S. Hafshejani. : ‘Numerical solution of delay differential equations using Legendre wavelet method’. World Applied Sciences Journal. Vol. 13, pp. 27-33, (2011). https://www.researchgate.net/profile/Mahmoud-Sadeghi-6/publication/351267182_Numerical_Solution_of_Delay_Differential_Equations_Using_Legendre_Wavelet_Method/links/608e63c1458515d315ee5639/Numerical-Solution-of-Delay-Differential-Equations-Using-Legendre-Wavelet-Method.pdf
XIX. N. A. Kudryashov. : ‘Logistic function as solution of many nonlinear differential equations’. Applied Mathematical Modelling. Vol. 39(18), pp. 5733-5742, (2015) 10.1016/j.apm.2015.01.048
XX. N. I. Mahmudov, M. Awadalla, and K. Abuassba. : ‘Nonlinear sequential fractional differential equations with nonlocal boundary conditions’. Advances in Difference Equations, 2017, pp. 1-15, (2017) https://link.springer.com/article/10.1186/s13662-017-1371-3
XXI. P. Manchanda, & M. Rani. : ‘Second kind Chebyshev wavelet method for solving system of linear differential equations’. Int. J. Pure Appl. Math., Vol. 114, pp. 91-104, (2017) 10.12732/ijpam.v114i1.8
XXII. S. C. Shiralashetti, B. S. Hoogar, & S. Kumbinarasaiah. : ‘Hermite wavelet-based method for the numerical solution of linear and nonlinear delay differential equations’. Int. J. Eng. Sci. Math., Vol. 6(8), pp. 71-79, (2017) https://www.researchgate.net/profile/Kumbinarasaiah-S/publication/322438497_Hermite_Wavelet_Based_Method_for_the_Numerical_Solution_of_Linear_and_Nonlinear_Delay_Differential_Equations/links/5cb854f292851c8d22f3511d/Hermite-Wavelet-Based-Method-for-the-Numerical-Solution-of-Linear-and-Nonlinear-Delay-Differential-Equations.pdf
XXIII. S. G. Hosseini, & F. Mohammadi. : ‘A new operational matrix of derivative for Chebyshev wavelets and its applications in solving ordinary differential equations with non-analytic solution’. Applied mathematical sciences. Vol. 5(51), pp. 2537-2548, (2011). https://m-hikari.com/ams/ams-2011/ams-49-52-2011/mohammadiAMS49-52-2011.pdf
XXIV. S. Kumbinarasaiah, & K. R. Raghunatha. : ‘The applications of Hermite wavelet method to nonlinear differential equations arising in heat transfer’. International Journal of Thermofluids. Vol. 9, 100066, (2021). 10.1016/j.ijft.2021.100066
XXV. S. S. Zeid. : ‘Approximation methods for solving fractional equations, Chaos’. Solitons & Fractals. Vol. 125, pp. 171-193, (2019) 10.1016/j.chaos.2019.05.008
XXVI. T. Abdeljawad, R. Amin, K. Shah, Q. Al-Mdallal, & F. Jarad. : ‘Efficient sustainable algorithm for numerical solutions of systems of fractional order differential equations by Haar wavelet collocation method’. Alexandria Engineering Journal. Vol. 59(4), pp. 2391-2400, (2020). 10.1016/j.aej.2020.02.035
XXVII. V. Bevia, J. Calatayud, J. C. Cortés, & M. Jornet. : ‘On the generalized logistic random differential equation: Theoretical analysis and numerical simulations with real-world data’. Communications in Nonlinear Science and Numerical Simulation. Vol. 116, 106832, (2023). 10.1016/j.cnsns.2022.106832
XXVIII. X. J. Yang, D. Baleanu, & H. M. Srivastava. : ‘Local fractional integral transforms and their applications’. Academic Press. (2015).
XXIX. X. Xu, & F. Zhou. : ‘Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations’. International Journal of Nonlinear Sciences and Numerical Simulation. (0), (2022). 10.1515/ijnsns-2021-0413.
XXX. Y. Y. Yameni Noupoue, Y. Tandoğdu, & M. Awadalla. : ‘On numerical techniques for solving the fractional logistic differential equation’. Advances in Difference Equations. 2019, pp. 1-13, (2019). https://link.springer.com/article/10.1186/s13662-019-2055-y

View | Download

Abstract:

Parkinson's disease (PD) is chronic, permanent, and life-threatening. Neurologically protective treatments for PD rely on early detection. Recent studies have demonstrated that clinical data, cerebrospinal Fluid (CSF) based proteomes, and gene mutations are important biomarkers for accurate and early detection of PD. This study aims to investigate the heterogeneous data comprised of CSF-based clinical data, CSF-based proteomic analysis data as well as the mutation information of the genes, Glucose Beta Acid (GBA), leucine-rich kinase (LRRK2) to classify controls into PD-affected and Healthy Control (HC). The dataset contains 1103 controls (569 PD affected and 534 HC). Automated Machine Learning (AutoML) framework using PyCaret is utilized. The study has proposed an Extra Tree Classifier (ETC) as a feature selection mechanism to select features that significantly affect the PD classification. Selected features are further used to train Random Forest (RF), Logistic Regression (LR), and Decision Tree (DT) classifiers. Accuracy, sensitivity, specificity, area under the receiver operating characteristic curve (AUC-ROC), and the confusion matrix are used to evaluate the performance of classifiers. RF has depicted the best performance in terms of accuracy value of 96.12%, sensitivity of 95.59%, and specificity of 95.34% while LR has shown the highest AUC value of 98.33. RF has made the highest number of correct predictions 316 out of 331.

Keywords:

Parkinson's Disease,CSF,Feature Selection,Extra Tree Classifier,Machine Learning,Random Forest,Logistic Regression.,

Refference:

I. Abdulhay Enas, N. Arunkumar, Kumaravelu Narasimhan, Elamaran Vellaiappan, and V. Venkatraman. : ‘Gait and tremor investigation using machine learning techniques for the diagnosis of Parkinson disease’. Future Generation Computer Systems. Vol. 83, pp. 366-373, (2018). 10.1016/j.future.2018.02.009
II. Bloem, Bastiaan R., Michael S. Okun, and Christine Klein. : ‘Parkinson’s disease’. The Lancet. Vol. 397, no. 10291, pp. 2284-2303, (2021). 10.1016/S0140-6736(21)00218-X
III. Bonifati Vincenzo. : ‘Genetics of Parkinson’s disease–state of the art, 2013’. Parkinsonism & related disorders. Vol. 20, pp. S23-S28, (2014). 10.1016/S1353-8020(13)70009-9
IV. Caramia Carlotta, Diego Torricelli, Maurizio Schmid, Adriana Muñoz-Gonzalez, Jose Gonzalez-Vargas, Francisco Grandas, and Jose L. Pons. : ‘IMU-based classification of Parkinson’s disease from gait: A sensitivity analysis on sensor location and feature selection’. IEEE journal of biomedical and health informatics. Vol. 22(6), pp. 1765-1774, (2018). 10.1109/JBHI.2018.2865218
V. Chapuis S., Ouchchane L., Metz O., Gerbaud L., & Durif F., : ‘Impact of the motor complications of Parkinson’s disease on the quality of life’. Movement disorders: official journal of the Movement Disorder Society. Vol. 20(2), pp. 224-230, (2005). 10.1002/mds.20279
VI. Dhiman Poonam, Vinay Kukreja, Poongodi Manoharan, Amandeep Kaur, M. M. Kamruzzaman, Imed Ben Dhaou, and Celestine Iwendi. : ‘A novel deep learning model for detection of severity level of the disease in citrus fruits’. Electronics. Vol. 11(3), pp. 495, (2022), 10.3390/electronics11030495
VII. Kaiser Sergio, Luqing Zhang, Brit Mollenhauer, Jaison Jacob, Simonne Longerich, Jorge Del-Aguila, Jacob Marcus et al., : ‘A proteogenomic view of Parkinson’s disease causality and heterogeneity’. npj Parkinson’s Disease. Vol. 9(1), pp. 24, (2023).10.1038/s41531-023-00461-9
VIII. Kaushal Chetna, and Anshu Singla. : ‘Automated segmentation technique with self‐driven post‐processing for histopathological breast cancer images’. CAAI Transactions on Intelligence Technology. Vol. 5(4), pp. 294-300, (2020). 10.1049/trit.2019.0077
IX. Kurt Imran, Mevlut Ture, and A. Turhan Kurum. : ‘Comparing performances of logistic regression, classification and regression tree, and neural networks for predicting coronary artery disease’. Expert systems with applications. Vol. 34(1), pp. 366-374, (2008). 10.1016/j.eswa.2006.09.004
X. Markello Ross D., Golia Shafiei, Christina Tremblay, Ronald B. Postuma, Alain Dagher, and Bratislav Misic. : ‘Multimodal phenotypic axes of Parkinson’s disease’. npj Parkinson’s Disease. Vol. 7(1), pp. 6, (2021). 10.1038/s41531-020-00144-9
XI. Oh Shu Lih, Yuki Hagiwara, U. Raghavendra, Rajamanickam Yuvaraj, N. Arunkumar, M. Murugappan, and U. Rajendra Acharya. : ‘A deep learning approach for Parkinson’s disease diagnosis from EEG signals’. Neural Computing and Applications. Vol. 32, (2020). 10927-10933. 10.1007/s00521-018-3689-5
XII. Pantaleo Ester, Alfonso Monaco, Nicola Amoroso, Angela Lombardi, Loredana Bellantuono, Daniele Urso, Claudio Lo Giudice et al. : ‘A machine learning approach to Parkinson’s disease blood transcriptomics’. Genes. Vol. 13(5), pp. 727, (2022). 10.3390/genes13050727
XIII. Parmar Aakash, Rakesh Katariya, and Vatsal Patel. : ‘A review on random forest: An ensemble classifier’. International conference on intelligent data communication technologies and internet of things (ICICI). pp. 758-763, 2018. Springer International Publishing. 2019. 10.1007/978-3-030-03146-6_86
XIV. Patrician Patricia A., : ‘Multiple imputation for missing data’. Research in nursing & health. Vol. 25(1), pp. 76-84, (2002). 10.1002/nur.10015
XV. Pol Urmila R., and Tejshree U. Sawant. : ‘Automl: Building An Classfication Model With Pycaret’. Ymer. Vol. 20, pp. 547-552, (2021).
XVI. Priyanka and Dharmender Kumar. : ‘Decision tree classifier: a detailed survey’. International Journal of Information and Decision Sciences. Vol. 12(3), pp. 246-269, (2020). 10.1504/IJIDS.2020.108141
XVII. Rasheed Jawad, Alaa Ali Hameed, Naim Ajlouni, Akhtar Jamil, Adem Özyavaş, and Zeynep Orman. : ‘Application of adaptive back-propagation neural networks for Parkinson’s disease prediction’. 2020 International Conference on Data Analytics for Business and Industry: Way Towards a Sustainable Economy (ICDABI), IEEE, pp. 1-5. 2020. 10.1109/ICDABI51230.2020.9325709
XVIII. Sabherwal G., Kaur A., : ‘Machine learning and deep learning approach to Parkinson’s disease detection: present state-of-the-art and a bibliometric review’. Multimedia Tools and Applications. (2024). 10.1007/s11042-024-18398-3
XIX. Sachdeva Ravi Kumar, Tushar Garg, Gagandeep Singh Khaira, Dikshant Mitrav, and Rakesh Ahuja. : ‘A Systematic Method for Lung Cancer Classification’. 10th International Conference on Reliability, Infocom Technologies and Optimization (Trends and Future Directions)(ICRITO), 2022. IEEE, 2022. pp. 1-5. 10.1109/ICRITO56286.2022.9 964778
XX. Sakar C. Okan, and Olcay Kursun. ‘Telediagnosis of Parkinson’s disease using measurements of dysphonia’. Journal of medical systems. Vol. 34 pp. 591-599, (2010). 10.1007/s10916-009-9272-y
XXI. Schapira Anthony HV, K. Ray Chaudhuri, and Peter Jenner. : ‘Non-motor features of Parkinson disease’. Nature Reviews Neuroscience. Vol. 18(7), pp. 435-450, (2017). 10.1038/nrn.2017.62
XXII. Sharaff Aakanksha, and Harshil Gupta. : ‘Extra-tree classifier with metaheuristics approach for email classification’. Advances in Computer Communication and Computational Sciences: Proceedings of IC4S 2018. Springer Singapore, 2019. pp. 189-197. 10.1007/978-981-13-6861-5_17
XXIII. Shetty Sachin, and Y. S. Rao. : ‘SVM based machine learning approach to identify Parkinson’s disease using gait analysis’. 2016 International conference on inventive computation technologies (ICICT). IEEE, vol. 2, pp. 1-5. 2016. 10.1109/INVENTIVE.2016.7824836
XXIV. Sihombing Denny Jean Cross, Jawangi Unedo Dexius, Jonson Manurung, Mendarissan Aritonang, and Harni Seven Adinata. : ‘Design and Analysis of Automated Machine Learning (AutoML) in PowerBI Application Using PyCaret’. 2022 International Conference of Science and Information Technology in Smart Administration (ICSINTESA). IEEE. pp. 89-94. 2022. 10.1109/ICSINTESA56431.2022.10041543
XXV. Tsukita Kazuto, Haruhi Sakamaki-Tsukita, Sergio Kaiser, Luqing Zhang, Mirko Messa, Pablo Serrano-Fernandez, and Ryosuke Takahashi. : ‘High-throughput CSF proteomics and machine learning to identify proteomic signatures for Parkinson disease development and progression’. Neurology. Vol. 101(14), pp. e1434-e1447, (2023). 10.1212/WNL0000000000207725
XXVI. Yaman Orhan, Fatih Ertam, and Turker Tuncer. : ‘Automated Parkinson’s disease recognition based on statistical pooling method using acoustic features’. Medical Hypotheses. Vol. 135, 109483 (2020). 10.1016/j.mehy.2019. 109483
XXVII. Zhu Biqing, Dominic Yin, Hongyu Zhao, and Le Zhang. : ‘The immunology of Parkinson’s disease’. Seminars in Immunopathology, vol. 44(5), pp. 659-672. Berlin/Heidelberg: Springer Berlin Heidelberg, 2022. 10.1007/s00281-022-00947-3

View | Download