| New Spectral Solutions for Some Differential and Integral Equations via Legendre’s Derivatives |
| Paper ID : 1034-ISCHU |
| Authors |
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Youssri Hassan Youssri1, Ahmed Mohamed Abbas *2, Mamdouh Metwaly Elkady3, Mohamed Ahmed Abdelhakem3 1Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt 2Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt. 3Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt |
| Abstract |
| This research presents a method to solve integral equations numerically. We expanded the exact unknown dependent variable - which is supposed to be a function of the independent variable - and approximated it as a linear combination of the basis functions (Legendre's derivative polynomials). Then, via the collocation method, we developed an integration matrix to expand the integrals as a linear combination of the basis functions, and we obtained a system of algebraic equations in the expansion coefficients. Finally, we solved the algebraic system using the Gauss elimination method if the system is linear or Newton's iteration method with zero initial guess if the system is non-linear to obtain the desired solution. In some problems, we may need a linearization formula to deal with non-linear integral equations. Our method can also solve ordinary differential equations since every initial value problem is equivalent to a corresponding Volterra integral equation, and every boundary value problem is equivalent to a corresponding Fredholm integral equation. So, we transform the ordinary differential equation into an integral equation with the aid of the conditions. The convergence and error analyses have been studied in detail. Some numerical test problems are performed to verify the efficiency and accuracy of the presented method. |
| Keywords |
| Legendre polynomials, spectral methods, collocation method, Lane-Emden equation, Bratu's equation, singularly perturbed equation. |
| Status: Abstract Accepted (Poster Presentation) |