| Citation: |
Mohammed Abdel-Aty, Mohammed Abdou. ANALYTICAL AND NUMERICAL DISCUSSION FOR THE PHASE-LAG VOLTERRA-FREDHOLM INTEGRAL EQUATION WITH SINGULAR KERNEL[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3203-3220. doi: 10.11948/20220547
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ANALYTICAL AND NUMERICAL DISCUSSION FOR THE PHASE-LAG VOLTERRA-FREDHOLM INTEGRAL EQUATION WITH SINGULAR KERNEL
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Abstract
In this paper, we studied the existence and unique solution of the Volterra-Fredholm integral equation of the second kind (V-FIESK). The general singular kernel is considered to be in position with the Fredholm integral term. Singular kernel will tend to a logarithmic function under exceptional conditions and new discussions. The Volterra-Fredholm integral equation with the logarithmic form will be solved using Legendre polynomials, where the kernel of Volterra integral term is a positive continuous function in time. A system of infinite linear algebraic equations is obtained by solving the problem in series, where the convergence of this system is discussed. Finally, The error is calculated using Maple software after the numerical results have been acquired.
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References
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Mohammed Abdel-Aty, Mohammed Abdou. ANALYTICAL AND NUMERICAL DISCUSSION FOR THE PHASE-LAG VOLTERRA-FREDHOLM INTEGRAL EQUATION WITH SINGULAR KERNEL[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3203-3220. doi: 10.11948/20220547
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Figure 1.
Exact and approximate solution of Legendre polynomials for
$ t_{0}=0. $ -
Figure 2.
Exact and approximate solution of Legendre polynomials for
$ t_{1}=0.2. $ -
Figure 3.
Exact and approximate solution of Legendre polynomials for
$ t_{1}=0.4. $ -
Figure 4.
Exact and approximate solution of Legendre polynomials for
$ t_{1}=0.6. $ -
Figure 5.
Exact and approximate solution of Legendre polynomials for
$ t_{0}=0. $ -
Figure 6.
Exact and approximate solution of Legendre polynomials for
$ t_{1}=0.2. $ -
Figure 7.
Exact and approximate solution of Legendre polynomials for
$ t_{1}=0.4. $ -
Figure 8.
Exact and approximate solution of Legendre polynomials for
$ t_{1}=0.6. $