Document Details
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Document Type |
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Thesis |
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Document Title |
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Numerical Treatment for Solving Singular Mixed Integral Equations and Integro-Differential Equations المعالجة العددية لحل معادلات تكاملية مختلطة شاذة ومعادلات تكاملية - تفاضلية |
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Subject |
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Faculty of Sciences |
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Document Language |
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Arabic |
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Abstract |
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Our goal in this thesis is to obtain an approximate solution of the mixed two dimensional Volterra-Fredholm integral equations and integro-differential equations by using methods recently developed. The contents of the thesis are divided into two parts. Part I contains four chapters that handle the linear and nonlinear integral equations by using the modern mathematical methods. Part II contains two chapters that handle the linear and nonlinear integro -differential equations and also use the modern mathematical methods, that are
1) Adomian decomposition method (ADM).
2) Variational iteration method (VIM).
3) Homotopy perturbation method (HPM).
So we presented a detailed study of the recently developed methods, namely the ADM, VIM, and HPM and that used to handle the mixed Volterra-Fredholm integral equation (V-FIE) and Volterra-Fredholm integro-differential equation (V-FIDE). We proved the existence and uniqueness of solution of mixed V-FIE and V-FIDE, convergence of the ADM, VIM and HPM are proved.
Also we presented a detailed study of the coupling of VIM with ADM for solving nonlinear mixed V-FIDE. In this approach, a new formula is called variational Adomian decomposition method (VADM) and variational accelerated Adomian decomposition method (VAADM), based on form of Adomian polynomials and on form of accelerated Adomian polynomials. Also we presented a detailed study of the coupling of VIM with HPM for solving nonlinear mixed V-FIDE. In this approach, a new formula is called variational homotopy perturbation method (VHPM) and variational accelerated homotopy perturbation method (VAHPM), this approach based on form He's polynomials and on form of accelerated He's polynomials. The existence solution and convergence of iterative methods are proved. We considered and discussed many numerical examples to illustrate the validity and efficiency of these methods.
The outcome of this thesis is written in seven published papers their details are as follows:
(1) Hendi, F. A. and Al-Qarni, M. M. (2016), Comparison Between Adomian’s Decomposition Method and Toeplitz Matrix Method for Solving Linear Mixed Integral Equation with Hilbert Kernel, American Journal of Computational Mathematics, 6, 177-183.
(2) Hendi, F. A. and Al-Qarni, M. M. (2016), Numerical Treatment of Nonlinear Volterra-Fredholm Integral Equation with A Generalized Singular Kernel, American Journal of Computational Mathematics, 6, 245-250.
(3) Hendi, F. A. and Al-Qarni, M. M. (2016), Numerical Solution of Nonlinear Mixed Integral Equations with Singular Volterra Kernel , Int. J. Adv. Appl. Math. and Mech. 3(4), 41-48.
(4) Hendi, F. A. and Al-Qarni, M. M. (2017), Numerical Solution of Nonlinear Mixed Integral Equation with A Generalized Cauchy Kernel , Applied Mathematics, 8, 209-214 .
(5) Hendi, F. A. and Al-Qarni, M. M. (2017), The Homotopy Perturbation Method for Solving Nonlinear Volterra-Fredholm Integral Equation with Singular Volterra Kernel, Far East Journal of Applied Mathematics, Accepted.
(6) Hendi, F. A. and Al-Qarni, M. M. (2017), The Variational Adomian Decomposition Method for Solving Nonlinear Two-Dimensional Volterra -Fredholm Integro-Differential Equations, Submitted for publication .
(7) Hendi, F. A. and Al-Qarni, M. M. (2017), An Accelerated Homotopy Perturbation Method for Solving Nonlinear Two- Dimensional Volterra-Fredholm Integro-Differential Equations, ADV MATH PHYS, Accepted. |
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Supervisor |
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Prof. Dr. FATHEAH AHMAD ALHENDI |
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Thesis Type |
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Doctorate Thesis |
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Publishing Year |
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1438 AH
2017 AD |
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Added Date |
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Thursday, June 1, 2017 |
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Researchers
| منال محمد القرني | AL-QARNI, MANAL MOHAMMED | Researcher | Doctorate | |
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