An innovative computational approach for fuzzy space-time fractional Telegraph equation via the new iterative transform method

Document Type : Research Paper

Authors

1 Department of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.

2 Department of Mathematics, Loknete Vyankatrao Hiray Arts, Science and Commerce College, Nashik, Maharashtra, India.

Abstract

In this paper, the Fuzzy Sumudu Transform Iterative method (FSTIM) was applied to find the exact fuzzy solution of the fuzzy space-time fractional telegraph equations using the Fuzzy Caputo Fractional Derivative operator. The Telegraph partial differential equation is a hyperbolic equation representing the reaction-diffusion process in various fields. It has applications in engineering, biology, and physics. The FSTIM provides a reliable and efficient approach for obtaining approximate solutions to these complex equations improving accuracy and allowing for fine-tuning and optimization for better approximation results. The work introduces a fuzzy logic-based approach to Sumudu transform iterative methods, offering flexibility and adaptability in handling complex equations. This innovative methodology considers uncertainty and imprecision, providing comprehensive and accurate solutions, and advancing numerical methods. Solving the fuzzy space-time fractional telegraph equation used a fusion of the Fuzzy Sumudu transform and iterative approach. Solution of fuzzy fractional telegraph equation finding analytically and interpreting its results graphically. Throughout the article, whenever we draw graphs, we use Mathematica Software. We successfully employed FSTIM, which is elegant and fast to convergence.

Keywords

Main Subjects

References

  • [1] R. P. Agarwal, D. Baleanu, J. J. Nieto, D. F. Torres, and Y. Zhou, A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, Journal of Computational and Applied Mathematics, 339 (2018), 329.
  • [2] M. Z. Ahmad and N. A. bin Abdul Rahman, Explicit solution of fuzzy differential equations by mean of fuzzy sumudu transform, International Journal of Applied Physics and Mathematics, 5, 2 (2015), 86-93.
  • [3] E. K. Akgul, A. Akgul, and M. Yavuz, New illustrative applications of integral transforms to financial models with different fractional derivatives, Chaos, Solitons and Fractals, 146 (2021), 117.
  • [4] M. K. Alaoui, F. M. Alharbi, and S. Zaland, Novel analysis of fuzzy physical models by generalized fractional fuzzy operators, Journal of Function Spaces, 2022 (2022), Article ID 9824568.
  • [5] M. N. Alam, O. A. Ilhan, J. Manafian, M. I. Asjad, H. Rezazadeh, and H. M. Baskonus, New results of some of the conformable models arising in dynamical systems, Advances in Mathematical Physics, (2022), 113.
  • [6] M. N. Alam, O. A. Ilhan, M. S. Uddin, and M. A. Rahim, Regarding on the results for the fractional clannish random walkers parabolic equation and the nonlinear fractional cahn-Allen equation, Advances in Mathematical Physics, 2022 (2022), 112.
  • [7] M. N. Alam, S. Islam, O. A. Ilhan, and H. Bulut, Some new results of nonlinear model arising in incompressible visco-elastic kelvin-voigt fluid, Mathematical Methods in the Applied Sciences, 45 (2022), 10347-10362.
  • [8] M. N. Alam, Soliton solutions to the electric signals in telegraph lines on the basis of the tunnel diode, Partial Differential Equations in Applied Mathematics, 7 (2023), 100491.
  • [9] M. N. Alam, An analytical technique to obtain traveling wave solutions to nonlinear models of fractional order, Partial Differential Equations in Applied Mathematics, 8 (2023), 100533.
  • [10] M. N. Alam and S. M. R. Islam, The agreement between novel exact and numerical solutions of nonlinear models, Partial Differential Equations in Applied Mathematics, 8 (2023), 100584.
  • [11] M. N. Alam, H. S. Akash, U. Saha, M. S. Hasan, M. W. Parvin, and C. Tunc, Bifurcation analysis and solitary wave analysis of the nonlinear fractional soliton neuron model, Iran. J. Sci., 47 (2023), pages 17971808.
  • [12] M. N. Alam, Exact solutions to the foam drainage equation by using the new generalized (g/g)-expansion method, Results in Physics, 5 (2015), 168-177.
  • [13] F. A. Alawad, E. A. Yousif, and A. I. Arbab, A new technique of laplace variational iteration method for solving space-time fractional telegraph equations, International Journal of Differential Equations, 2013 (2013) Article ID 256593.
  •  [14] N. H. Aljahdaly, R. P. Agarwal, R. Shah, and T. Botmart, Analysis of the time fractional-order coupled burgers equations with non-singular kernel operators, Mathematics, 9(18) (2021), 2326.
  • [15] T. Allahviranloo and M. B. Ahmadi, Fuzzy Laplace transforms, Soft Computing, 14(3), 235243.
  • [16] S. A. Altaie, N. Anakira, A. Jameel, O. Ababneh, A. Qazza, and A. K. Alomari, Homotopy analysis method analytical scheme for developing a solution to partial differential equations in fuzzy environment, Fractal and Fractional, 6 (2022), 419.
  • [17] R. Alyusof, S. Alyusof, N. Iqbal, and S. K. Samura, Novel evaluation of fuzzy fractional biological population model, Journal of Function Spaces, 2022 (2022).
  • [18] M. Arfan, K.Shah, A. Ullah, and T. Abdeljawad, Study of fuzzy fractional order diffusion problem under the mittag-leffler kernel law, Phys. Scr. , 96 (2021), 074002.
  • [19] M. A. Asiru, Classroom note: Application of the sumudu transform to discrete dynamic systems, International Journal of Mathematical Education in Science and Technology, 34 (2010), 944949.
  • [20] Z. Ayati and J. Biazar, On the convergence of homotopy perturbation method, Journal of the Egyptian Mathematical Society, 23(2) (2015),424428.
  • [21] S. Bhalekar and V. Daftardar-Gejji, Solving evolution equations using a new iterative method, Numerical Methods for Partial Differential Equations, 26 (2010), 906916.
  • [22] S. Chakraverty, S. Tapaswini, and D. Behera, Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications, John Wiley & Sons, 2016.
  • [23] V. Daftardar-Gejji and S. Bhalekar, Solving fractional boundary value problems with dirichlet boundary conditions using a new iterative method, Computers & Mathematics with Applications, 59 (2010), 18011809.
  • [24] L. Debnath, Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, 2003 (2003), Article ID 753601, 30.
  • [25] E. ElJaoui, S. Melliani, and L. S. Chadli, Solving second-order fuzzy differential equations by the fuzzy laplace transform method, Adv. Differ. Equ., 2015 (2015), 114.
  • [26] A. Z. Fino and H. Ibrahim, Analytical solution for a generalized space-time fractional telegraph equation, Mathematical Methods in the Applied Sciences, 36 (2013), 18131824.
  • [27] M. Garg, P. Manohar, and S. L. Kalla, Generalized differential transform method to space-time fractional telegraph equation, International Journal of Differential Equations, 2011 (2011), Article ID 548982, 9.
  • [28] M. Garg, A. Sharma, and P. Manohar, Solution of generalized space-time fractional telegraph equation with composite and riesz-feller fractional derivatives, International Journal of Pure and Applied Mathematics, 83 (2013), 685691.
  • [29] Z. Hammouch, M. Yavuz, and N. Ozdemir, Numerical solutions and synchronization of a variable-order fractional chaotic system, Mathematical Modelling and Numerical Simulation with Applications, 1 (2021), 1123.
  • [30] A. Harir, S. Melliani, and L. S. Chadli, Fuzzy space-time fractional telegraph equations, International Journal of Mathematics Trends and Technology, 64 (2018), 101108.
  • [31] A. K. Haydar, Fuzzy sumudu transform for fuzzy nth-order derivative and solving fuzzy ordinary differential equations, Int. J. Sci. Res, 4 (2015), 13721378.
  • [32] N. V. Hoa, H. Vu, and T. M. Duc, Fuzzy fractional differential equations under CaputoKatugampola fractional derivative approach, Fuzzy Sets and Systems, 375 (2019), 7099.
  • [33] R. W. Ibrahim, Complex transforms for systems of fractional differential equations, Adv. Differ. Equ., 2012 (2012), 15.
  • [34] H. Jafari, Iterative method for non-adapted fuzzy stochastic differential equations, Russ Math., 65 (2021),24-34.
  • [35] A. Khastan, F. Bahrami, and K. Ivaz, New results on multiple solutions for nth-order fuzzy differential equations under generalized differentiability, Boundary Value Problems, 2009 (2009), 113.
  • [36] A. Kilbas, H. Srivastava, and J. Trujillo, Theory and applications of fractional differential equations, NorthHolland Mathematics Studies, book Series, 2006.
  • [37] A. Kochubei and Y. Luchko, Fractional differential equations, Mathematics in Science and Engineering, New York, 2019.
  • [38] K. A. Kshirsagar, V. R. Nikam, S. B. Gaikwad, and S. A. Tarate, The double fuzzy elzaki transform for solving fuzzy partial differential equations, Journal of the Chungcheong Mathematical Society, 35 (2022), 177196.
  • [39] K. A. Kshirsagar, V. R. Nikam, S. B. Gaikwad, and S. A. Tarate, Solving Fuzzy Caputo-Fabrizio Fractional OneDimensional Heat Equations by the Fuzzy Laplace Transform Iterative Method, Tuijin Jishu/Journal of Propulsion Technology, 44(3) (2023), 362-374.
  • [40] K. A. Kshirsagar, V. R. Nikam, S.B. Gaikwad, and S. A. Tarate, Fuzzy Laplace-Adomian Decomposition Method for Approximating Solutions of Time Fractional Klein-Gordan Equations in a Fuzzy Environment, European Chemical Bulletin, 12(8) (2023), 5926-5943.
  • [41] Y. Liu, Approximate solutions of fractional nonlinear equations using homotopy perturbation transformation method, Abstract and Applied Analysis, 2012 (2012), Article ID 752869.
  • [42] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, john-wily and sons. Inc. New York, 1993.
  • [43] S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Applied Mathematics and Computation, 170 (2005), 11261134.
  • [44] Z. Odibat, S. Momani, and V. S. Erturk, Generalized differential transform method: Application to differential equations of fractional order, Applied Mathematics and Computation, 197 (2008), 467477.
  • [45] M. Osman, Y. Xia, M. Marwan, and O. A. Omer, Novel approaches for solving fuzzy fractional partial differential equations, Fractal and Fractional, 6 (2022).
  • [46] P. Ravi, V. L. Agarwal, and J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis Theory, Methods, and Applications, 72 (2010), 28592862.
  • [47] N. A. A. Rahman and M. Z. Ahmad, Solving fuzzy fractional differential equations using fuzzy sumudu transform, J. Nonlinear Sci. Appl, 10 (2017), 26202632.
  • [48] N. A. A. Rahman, Fuzzy sumudu decomposition method for solving differential equations with uncertainty, AIP Conference Proceedings, 2184 (2019), 060042.
  • [49] N. A. Rahman, Fuzzy sumudu decomposition method for fuzzy delay differential equations with strongly generalized differentiability, Mathematics and Statistics, 8 (2020), 570576.
  • [50] S. Rashid, R. Ashraf, and F. S. Bayones, A novel treatment of fuzzy fractional swiftHohenberg equation for a hybrid transform within the fractional derivative operator, Fractal and Fractional, 5 (2021), 209.
  • [51] Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Applied Mechanics Reviews, 50 (1997),1567.
  • [52] A. Sevimlican, An approximation to the solution of space and time fractional telegraph equations by hes variational iteration method, Mathematical Problems in Engineering, (2010), 2010.
  • [53] K. Shah, A. R. Seadawy, and M. Arfan, Evaluation of one dimensional fuzzy fractional partial differential equations, Alexandria Engineering Journal, 59 (2020), 33473353.
  • [54] R. Shah, A. S. Alshehry, and W. Weera, A semi-analytical method to investigate fractional-order gas dynamics equations by shehu transform, Symmetry, 14 (2022), 1458.
  • [55] I. Talib, M. N. Alam, D. Baleanu, D. Zaidi, and A. Marriyam, A new integral operational matrix with applications to multi-order fractional differential equations, AIMS Mathematics, 6 (2021), 87428771.
  • [56] S. Tapaswini and D. Behera, Analysis of imprecisely defined fuzzy space-fractional telegraph equations, Pramana - J. Phys., 94 (2020), 110.
  • [57] S. A. Tarate, A. P. Bhadane, S. B. Gaikwad, and K. A. Kshirsagar, Sumudu-iteration transform method for fractional telegraph equations, J. Math. Comput. Sci., 12 (2022), Article ID 127.
  • [58] S. Tarate, A. Bhadane, S. Gaikwad, and K. Kshirsagar, Solution of time-fractional equations via sumudu-adomian decomposition method, Computational Methods for Differential Equations, 11(2) (2023), 345356.
  • [59] S. Tarate, A. Bhadane, S. Gaikwad, and K. Kshirsagar, A Semi-Analytic Solution For Time-Fractional Heat Like And Wave Like Equations Via Novel Iterative Method, European Chemical Bulletin 12 (Special issue 8) (2023), 6164-6187.
  • [60] S. A. Tarate, A. P. Bhadane, S. B. Gaikwad, and K. A. Kshirsagar, Duality Relations of Fractional order Transforms, Tuijin Jishu/Journal of Propulsion Technology, 44(3) (2023), 375-384.
  • [61] L. Verma and R. Meher, Effect of heat transfer on JefferyHamel cu/agwater nanofluid flow with uncertain volume fraction using the double parametric fuzzy homotopy analysis method, Eur. Phys. J. Plus, 137 (2022), 372.
  • [62] A. Yildirim, The homotopy perturbation method for solving the modified Korteweg-de Vries equation, Z. Naturforsch, 63 (2008), 621626.
  • [63] Z. Zhao and C. Li, Fractional difference/finite element approximations for the timespace fractional telegraph equation, Applied Mathematics and Computation, 219 (2012), 29752988.
Computational Methods for Differential Equations
Volume 12, Issue 4
October 2024
Pages 763-779

Files

History

  • Receive Date: 24 June 2022
  • Revise Date: 20 January 2024
  • Accept Date: 09 February 2024

Share

How to cite

Statistics