Numerical Solution of Linear Volterra-Fredholm Integro-Differential Equations using Chebyshev Polynomials
DOI:
https://doi.org/10.62054/ijdm/0204.07Keywords:
Integro-differential equation; Volterra-Fredholm integro-differential equation; Chebyshev polynomials; Numerical solution; Exact solutionAbstract
This paper presents a numerical technique for solving linear Volterra-Fredholm integro-differntial equations using Chebyshev polynomials as basis function. The method involves approximating the unknown function as a finite sum of Chebyshev polynomials. The integral terms, both Volterra-type and Fredholm-type are evaluated using Gauss-Chebyshev quadrature rules, which are particularly effective due to their compatibility with the orthogonality of Chebyshev polynomials. This approach transforms the original Volterra-Fredholm integro-differential equation into a system of algebraic equations. The resultant algebraic system of equations is then solved using symbolic computation capabilities in Maple 18. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed method, showing good agreement with known or reference solutions. Consistency, stability and convergence of the method developed were also analyzed. The technique is shown to be a reliable and practical tool for approximating solutions of linear Volterr-Fredholm integro-differential equations encountered in various scientific and engineering applications.
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Data were generated using Maple 18 software and the code developed can be obtained through the corresponding author.
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