Uniqueness and Convergence of Solution of Multi-Term Fractional Order Fredholm Integro-Differential Equation

Authors

Keywords:

Fractional order , Uniqueness of solution , Convergence , Fredholm integro- differential equation

Abstract

This paper focuses on multi-term fractional order Fredholm integro-differential equation which was transformed to integral equation by using Riemann-Liouville fractional integral. The uniqueness of solution of the multi-term fractional order Fredholm integro-differential equation was proved using Banach contraction principle alongside the convergence of solution of the multi-term fractional order Fredholm integro-differential equation, where Cauchy convergence creteria was used. Examples were given to prove the solvability of the multi-term fractional order Fredholm integro-differential equation.

Dimensions

Agarwal, R. P. and Ahmad, B. (2011). Existence theory for anti-periodic boundary value problems of fractional

differential equations and inclusions. Computers & Mathematics with Applications, 62(3), 1200-1214.

Agarwal, R. P., Benchohra, M. and Hamani, S. (2010). A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions. Acta Applicandae Mathematicae, 109,

-1033

Aghajani, A., Banaś, J. and Jalilian, Y. (2011). Existence of solutions for a class of nonlinear Volterra singular integral equations. Computers & Mathematics with Applications, 62(3), 1215-1227.

Aghajani, A., Jalilian, Y., and Trujillo, J. J. (2012). On the existence of solutions of fractional integro-differential

equations. Fractional Calculus and Applied Analysis,

, 44-69

Almeida, R., Tavares, D. and Torres, D. F. (2019). The variable-order fractional calculus of variations. Cham, Switzerland: Springer International Publishing

Anguraj, A., Karthikeyan, P. and Trujillo, J. J. (2011). Existence of solutions to fractional mixed integro-differential equations with nonlocal initial condition. Advances in Difference Equations, 2011, 1-12.

Balachandran, K., and Kiruthika, S. (2011). Existence results for fractional integrodifferential equations with nonlocal

condition via resolvent operators. Computers & Mathematics with Applications, 62(3), 1350-1358.

Baleanu, D., Jajarmi, A., and Hajipour, M. (2017). A new formulation of the fractional optimal control problems

involving Mittag—Leffler nonsingular kernel. Journal of Optimization Theory and Applications, 175(3), 718-737.

Baleanu, D. and Lopes, A. M. (2019). Handbook of Fractional Calculus with Applications. Applications in Engineering, Life and Social Sciences, Part A, Southampton: Comput. Mech.Publicat., 7.

Baleanu, D. and Mustafa, O. G. (2010). On the global existence of solutions to a class of fractional differential equations. Computers & Mathematics with Applications, 59(5), 1835-1841.

Bulut, H., Sulaiman, T. A., Baskonus, H. M., Rezazadeh, H., Eslami, M. and Mirzazadeh, M. (2018). Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation. Optik, 172, 20-27.

Caballero, J., Harjani, J. and Sadarangani, K. (2011). On existence and uniqueness of positive solutions to a class of fractional boundary value problems. Boundary Value Problems, 1, 1-9.

Diethelm, K. and Ford, N. J. (2002). Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 265(2), 229-248.

Ertürk, V. S., Odibat, Z. M. and Momani, S. (2011). An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells. Computers & Mathematics with Applications, 62(3), 996-1002

Fallahgoul, H., Focardi, S. and Fabozzi, F. (2016). Fractional calculus and fractional processes with applications to financial economics: Theory and application. Academic Press

Ghomanjani, F. (2020). A new approach for solving linear fractional integro-differential equations and multi variable order fractional differential equations. Proyecciones (Antofagasta), 39(1), 199-218. Hajipour, M., Jajarmi, A. and Baleanu, D. (2018). An efficient nonstandard finite difference scheme for a class of fractional chaotic systems. Journal of Computational and Nonlinear Dynamics, 13(2)

Hamoud, A., Ghadle, K., andAtshan, S. (2019). The approximate solutions of fractional integro-differential equations by using modified Adomian decomposition method. Khayyam J. Mathematics, 5(1), 21-39

Hilfer, R. (Ed.). (2000). Applications of fractional calculus in physics. World scientific

Hu, Z. and Liu, W. (2011). Solvability for fractional order boundary value problems at resonance. Boundary Value Problems, 2011(1), 1-10.

Huang, L., and Bae, Y. (2018). Chaotic dynamics of the fractional-love model with an external environment. Entropy, 20(1), 53.

Idczak, D. and Kamocki, R. (2011). On the existence and uniqueness and formula for the solution of RL fractional Cauchy problem in ℝn. Fractional Calculus and Applied Analysis, 14, 538-553.

Kilbas, A. A. and Marzan, S. A. (2005). Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differential Equations, 41, 84-89.

Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006).

Theory and applications of fractional differential equations (Vol. 204). Elsevier

Kostić, M. (2011). Abstract time-fractional equations: existence and growth of solutions. Fractional Calculus and Applied Analysis, 14(2), 301-316.

Magin, R. L. (2012). Fractional calculus in bioengineering: A tool to model complex dynamics. In Proceedings of the 13th International Carpathian Control Conference (ICCC) (pp. 464-469). IEEE.

Mahmudov, N. I. (2017). Finite-approximate controllability of evolution equations. Appl. Comput. Math, 16(2), 159-167.

Mainardi, F. (2022). Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models. World Scientific.

Ming, H., Wang, J. and Fečkan, M. (2019). The application of fractional calculus in Chinese economic growth models. Mathematics, 7(8), 665.

Nemati, S., Lima, P. M. and Torres, D. F. (2021). Numerical solution of variable-order fractionaldifferential equations using Bernoulli polynomials. Fractal and Fractional, 5(4), 219.

Oldham, K. B. (2010). Fractional differential equations in electrochemistry. Advances in engineering software, 41(1), 9-12.

Pilipović, S. and Stojanović, M. (2006). Fractional differential equations through Laguerre expansions in abstract spaces: Error estimates. Integral Transforms and Special Functions, 17(12), 877-887.

Rostamy, D., Alipour, M., Jafari, H. and Baleanu, D. (2013). Solving multi-term orders fractional differential equations by operational matrices of BPs with convergence analysis. Romanian Reports in Physics, 65(2), 334-349.

Rudin, W. (1953). Principle of Mathematical Analysis. McGraw Hill. New York, NY

Rui, W. (2011). Existence of solutions of nonlinear fractional differential equations at resonance. Electronic Journal of Qualitative Theory of Differential Equations, 66, 1-12.

Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional integrals and derivatives (Vol. 1). Yverdon-les-Bains, Switzerland: Gordon and breach science publishers, Yverdon.

Singh, J., Kumar, D. and Baleanu, D. (2017). On the analysis of chemical kinetics system pertaining to a

fractional derivative with Mittag-Leffler type kernel. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(10), 03113.

Stoenoiu, C. E., Bolboaca, S. D. and Jäntschi, L. (2008). Model formulation & interpretation-from experiment to theory. Int. J. Pure Appl. Math, 47, 9-1

Tarasov, V. E. (2020). Mathematical economics: application of fractional calculus. Math., 8(5), 660.

Tian, Y. and Bai, Z. (2010). Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Computers & Mathematics with Applications, 59(8), 2601-2609

Wei, Z., Li, Q. and Che, J. (2010). Initial value problems for

fractional differential equations involving

Riemann--Liouville sequential fractional derivative. J.

Math. Analysis and Applic., 367(1), 260-272.

Yang, F., and Zhu, K. Q. (2011). On the definition of fractional derivatives in rheology. Theoretical and Applied Mechanics Letters, 1(1), 012007.

Yuste, S. B. and Acedo, L. (2005). An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equation SIAM J. on Numerical Analysis, 42(5), 1862-1874.

Zabadal, J., Vilhena, M. and Livotto, P. (2001). Simulation of chemical reactions using fractional derivatives. Nuovo Cimento. B, 116(5), 529-545

Zeidler, E. (1986). Nonlinear Functional Analysis and its Applications, Part 1: Fixed-Point Theorems. Springer

Zheng, L. and Zhang, X. (2017). Modeling and analysis of modern fluid problems. Academic Press. London, UK.

Zhou, Y. (2014). Basic theory of fractional differential equations. World Scientific, Singapore.

Published

24-08-2024

How to Cite

Uniqueness and Convergence of Solution of Multi-Term Fractional Order Fredholm Integro-Differential Equation. (2024). FULafia Journal of Science and Technology , 8(2), 1-8. https://doi.org/10.62050/fjst2024.v8n2.318

Issue

Vol. 8 No. 2 (September, 2024): Fulafia Journal of Science and Technology (FJST)

Section

Physical Sciences
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How to Cite

Uniqueness and Convergence of Solution of Multi-Term Fractional Order Fredholm Integro-Differential Equation. (2024). FULafia Journal of Science and Technology , 8(2), 1-8. https://doi.org/10.62050/fjst2024.v8n2.318

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