Approximate solutions for a class of nonlinear Volterra-Fredholm integro-differential equations under Dirichlet boundary conditions

Hawsar Ali Hama Rashid; Mudhafar Fattah Hama; Hawsar Ali Hama Rashid; Mudhafar Fattah Hama

doi:10.3934/math.2023022

AIMS Mathematics

2023, Volume 8, Issue 1: 463-483. doi: 10.3934/math.2023022

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Approximate solutions for a class of nonlinear Volterra-Fredholm integro-differential equations under Dirichlet boundary conditions

Hawsar Ali Hama Rashid ¹,
Mudhafar Fattah Hama ^{2
,
,}

1.
College of Education- Department of Mathematics, University of Sulaimani, 46001 Sulaimani, KRG- Iraq
2.
College of Science- Department of Mathematics, University of Sulaimani, 46001 Sulaimani, KRG- Iraq

Received: 11 August 2022 Revised: 22 September 2022 Accepted: 26 September 2022 Published: 08 October 2022
MSC : 45B05, 45D05, 45J05, 45L05

This paper studies the solvability of boundary value problems for a nonlinear integro-differential equation. Converting the problem to an equivalent nonlinear Volterra-Fredholm integral equation (NVFIE) is driven by using a suitable transformation. To investigate the existence and uniqueness of continuous solutions for the NVFIE under certain given conditions, the Krasnoselskii fixed point theorem and Banach contraction principle have been used. Finally, we numerically solve the NVFIE and study the rate of convergence using methods based on applying the modified Adomian decomposition method, and Liao's homotopy analysis method. As applications, some examples are provided to support our work.
- boundary value problem,
- integro-differential equations,
- existence,
- uniqueness,
- modified Adomian's decomposition method,
- homotopy analysis method
Citation: Hawsar Ali Hama Rashid, Mudhafar Fattah Hama. Approximate solutions for a class of nonlinear Volterra-Fredholm integro-differential equations under Dirichlet boundary conditions[J]. AIMS Mathematics, 2023, 8(1): 463-483. doi: 10.3934/math.2023022

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Abstract

This paper studies the solvability of boundary value problems for a nonlinear integro-differential equation. Converting the problem to an equivalent nonlinear Volterra-Fredholm integral equation (NVFIE) is driven by using a suitable transformation. To investigate the existence and uniqueness of continuous solutions for the NVFIE under certain given conditions, the Krasnoselskii fixed point theorem and Banach contraction principle have been used. Finally, we numerically solve the NVFIE and study the rate of convergence using methods based on applying the modified Adomian decomposition method, and Liao's homotopy analysis method. As applications, some examples are provided to support our work.

References

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