Novel algorithms to approximate the solution of nonlinear integro-differential equations of Volterra-Fredholm integro type

Article Mathematics, Applied

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

Hawsar Ali Hama Rashid et al.

Summary: This paper investigates the solvability of boundary value problems for a nonlinear integro-differential equation. A suitable transformation is used to convert the problem into an equivalent nonlinear Volterra-Fredholm integral equation (NVFIE). The existence and uniqueness of continuous solutions for the NVFIE are studied under certain given conditions using the Krasnoselskii fixed point theorem and Banach contraction principle. Finally, the NVFIE is numerically solved and the rate of convergence is studied using modified Adomian decomposition method and Liao's homotopy analysis method. Some examples are provided to support the findings.

AIMS MATHEMATICS (2023)

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Article Mathematics, Applied

Existence and uniqueness of solutions for nonlinear Volterra-Fredholm integro-differential equation of fractional order with boundary conditions

Zaid Laadjal et al.

Summary: In this paper, the existence and uniqueness of a nonlinear Volterra-Fredholm integro-differential equation of Caputo fractional order subject to the boundary value conditions are established using standard fixed point theorems. An example is presented to illustrate the main result.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES (2021)

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Article Multidisciplinary Sciences

Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations

Hari Mohan Srivastava et al.

Summary: This article introduces two classes of lacunary fractional spline functions and a new higher-order lacunary fractional spline method using the Liouville-Caputo fractional Taylor expansion. The method's existence and uniqueness are derived, as well as error bounds for approximating the unique positive solution. Applications of the method are illustrated through Liouville-Caputo fractional differential equations (FDEs), showing the practicality and effectiveness of the proposed method. Recent developments in the theory and applications of FDEs in real-life situations are also highlighted.

SYMMETRY-BASEL (2021)

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Article Mathematics

Adomian Decomposition and Fractional Power Series Solution of a Class of Nonlinear Fractional Differential Equations

Pshtiwan Othman Mohammed et al.

Summary: This paper investigates nonlinear fractional differential equations with unknown analytical solutions, using Adomian decomposition and fractional power series methods to approximate the solutions. The two approaches are illustrated and compared through four numerical examples.

MATHEMATICS (2021)

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Article Mathematics