Homotopy Analysis Method for Solving Fuzzy Fractional Volterra-Fredholmintegro-differential Equations

Qin Chen

doi:10.54097/sy0pgm77

Authors

Qin Chen

DOI:

https://doi.org/10.54097/sy0pgm77

Keywords:

Volterra-Fredholm integro-differential Equations; Fixed Point Theorem; Homotopy Analysis Method; Caputo Fractional Derivative.

Abstract

The fuzzy fractional Volterra-Fredholm integro-differential equation is introduced by using the fuzzy Caputo derivative under the generalized Hukuhara difference, and the existence and uniqueness of the solution of this equation are proved by using the fixed point theorem. The homotopy analysis method is used to study the numerical solutions of linear and nonlinear fuzzy fractional integro-differential equations. Several numerical examples are given to illustrate the effectiveness and applica-bility of the method.

Downloads

Download data is not yet available.

References

A. Carpinteri, F. Mainardi. Fractals and fractional calculus in continuum me- chanics, Springer, Vienna, 378(1997), 277-290.

D. Baleanu, K. Diethelm, E. Scalas, J. Trujillo. Fractional calculus: models and numerical methods, Series on Complexity, Nonlinearity and Chaos, 3(2012), 39- 133.

A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and applications of frac- tional differential equations. Elsevier, Amsterdam, 204(2006), 1-523.

N. H. Abel. Solution de quelques problemes a l´aide d´ıntegrales definites, Christiania Grondahl, Norway, 1(1881), 16-18.

A. A. Hamoud, I. Y. A. Alasadi. Existence and stability results for fractional Volterra-Fredholm integro-dierential equation with mixed conditions, Advances in Dynamical Systems and Applications, 16(2021), 217-236.

S. Andr´as. Weakly singular Volterra and Fredholm-Volterra integral equations, Studia Universitatis Babes-Bolyai. Mathematica, 48(2003), 147-155.

S. Abbas, M. Benchohra. Fractional order Riemann-Liouville integral equations with multiple time delays, Applied Mathematics E-Notes [electronic only], 12(2012), 79-87.

A. Atangana, N. Bildik. Existence and numerical solution of the Volterra fractional integral equations of the second kind, Mathematical Problems in Engineering, 2013(2013), 1-11.

V. Laksmikantam. Theory of fractional functional differential equations, Non- linear Analysis: Theory, Methods and Applications, 69(2008), 3337-3343.

S. S. L. Chang, L. A. Zadeh. On fuzzy mapping and control, IEEE Transactions on Systems, Man, and Cybernetics, 2(1972), 30-34.

L. A. Zadeh. The concept of linguistic variable and its application to approximate reasoning. Information Sciences, 8 (1975), 199-249.

[M. Mizumoto, K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems Comput Controls, 7(1976), 73-81.

D. Dubois, H. Prade. Towards fuzzy differential calculus, Part I: integration of fuzzy mappings, class of second-order. Fuzzy Sets and Systems, 8(1982), 1-17.

M. Puri, D. Ralescu. Differentials of fuzzy functions. Journal of Mathematical Analysis and Applications, 91 (1983), 552-558.

B. Bede, S. G. Gal. Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems, 151(2005), 581-599.

S. J. Liao. On the homotopy analysis method for nonlinear problem, Applied Mathematics and Computation, 147(2004), 499-513.

A. A. Hamoud, K. P. Ghadle. Homotopy analysis method for the first order fuzzy Volterra-Fredholm integro-differential equations. Indonesian Journal of Electrical Engineering and Computer Science, 11(2018). 857-867.

E. Hussain, A. Ali. Homotopy analysis method for solving fuzzy integro- differential equations, Modern Applied Science, 7 (2013), 15-25.

I. K. Hanan. The Homotopy analysis method for solving multi-fractional orderintegro-differential equations, Journal of Al-Nahrain University Science, 14(2011), 139-143.

A. A. Hamoud, K. P. Ghadle, S. M. Atshan. The approximate solutions offractional integro-differential equations by using modified Adomian decomposition method. Khayyam Journal of Mathematics, 5(2019), 21-39.

M. SH. Bani Issa, A. A. Hamoud, K. P. Ghadle. Numerical solutions of fuzzy integro-differential equations of the second kind. Journal of Mathematics and Computer Science, 23(2021), 67-74.

S. Salahshour, A. Ahmadian, C. S. Chan. A novel technique for solving fuzzy differential equations of fractional order using laplace and integral transforms, 2016 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), 2016, 1473-1477.

S. Salahshour, T. Allahviranloo, S. Abbasbandy. Solving fuzzy fractional differ- ential equations by fuzzy Laplace transforms, Commun Nonlinear Sci Numer Simulat, 17(2012), 1372-1381.

H. Zureigat, A. I. Ismail, S. Sathasivam. Numerical solutions of fuzzy fractional diffusion equations by an implicit finite difference scheme, Neural Computing and Applications, 31 (2018), 4085-4094.

A. A. Hamoud, A. D. Azeez, K. P. Ghadle. A study of some iterative methods for solving fuzzy Volterra-Fredholm integral equations. Indonesian Journal of Electrical Engineering and Computer Science, 11(2018), 1228-1235.

S. Arshad, V. Lupulescu. On the fractional differential equations with uncer- tainty, Nonlinear Analysis: Theory, Methods and Applications, 74(2011), 85-93.

S. Arshad, V. Luplescu. Fractional differential equation with fuzzy initial conditon, Electronic Journal of Differential Equations, 34(2011), 1-8.

N. Ahmad, A. Ullah, A. Ullah, S. Ahmad, K. Shah , I. Ahmad. On analysis of the fuzzy fractional order Volterra-Fredholm integro-differential equation. Alexandria Engineering Journal, 60(2020), 1827-1838.

T. Allahviranloo, A. Armand, Z. Gouyandeh, H. Ghadiri. Existence and unique- ness of solutions for fuzzy fractional Volterra-Fredholm integro-differential equa- tions. Journal of Fuzzy Set Valued Analysis, 2013(2013), 1-9.

A. Armand, Z. Gouyandeh. Fuzzy fractional integro-differential equations under generalized Capotu differentiability, Annals of Fuzzy Mathematics and Infor- matics, 10(2015), 789-798.

S. Salahshour, A. Ahmadian, N. Senu, D. Baleanu, P. Agarwal. On analytical solutions of the fractional differential equation with uncertinity: Application to the basset problem, Entropy, 17(2015), 855-902.

D. Dubois, H. Prade. Operations on fuzzy numbers. International Journal of Systems Science, 9(1978), 613-626.

S. Hajighasemi. Fuzzy Fredholm-Volterra integral equations and existance and uniqueness of solution of them, Australian Journal of Basice and Applied Sci- ence, 5(2011), 1-8.

B. Bede, L. Stefanini. Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, In press, 230(2013), 119-141.

T. Allahviranloo, S. Salahshour, S. Abbasbandy. Explicit solutions of fractional differential equations with uncertainty. Soft Comput, 16(2012), 297-302.

R. L. Pouso. Schauder’s fixed-point theorem: new applications and a new ver- sion for discontinuous operators. Boundary Value Problems. 2012(2012), 1-14.

M. R. M. Shabestari, R. Ezzati, T.Allahviranloo . Numerical solution of fuzzy fractionalintegro-differential equation via two-dimensional Legendre Wavelet method, Journal of Intelligent and Fuzzy Systems, 34(2018), 2453-2465.