| Spectral and pseudo-spectral Chebyshev Polynomials’ First Derivative for solving several deferential problems: Real-life applications. |
| Paper ID : 1038-ISCHU |
| Authors |
|
Toqa Alaa-Eldeen *, Mamdouh Metwaly Elkady, Mohamed Ahmed Abdelhakem Mathematics Department, Faculty of Science, Helwan University, Cairo, Egypt |
| Abstract |
| This paper has presented methodologies that propose accurate and efficient numerical formulas to solve linear and non-linear ordinary differential equations (ODEs), several types of boundary value problems (BVPs), and integrato-differential equations (IDEs). The spectral Tau method (STM) functional design was established upon the first derivatives of Chebyshev polynomials (FDCHPs). A new linearization relation has been developed. Hence, this relation and another investigated one for integration have been used via the Tau method to solve non-linear ODES efficiently. Moreover, explicit algebraic systems for solving the Lane-Emden and Riccati equations are introduced. The leading coefficients, Gauss-Lobatto quadrature points, and Gauss-Lobatto quadrature weights for the FDCHPs are generated. The coefficients relation of the approximation solutions as a function of FDCHPs have been presented and proven. Unique alternative expressions of pseudo-spectral differentiation matrices and integration matrices called the first derivative Chebyshev polynomials differentiation matrices (FDCHPs-DM) and the first derivative Chebyshev polynomials integration matrices (FDCHPs-BM) are obtained using an explicit formula for the Chebyshev polynomials derivatives. The convergence and error analysis are explored in depth. Some enlightening BVPs, particularly the Lane-Emden and Riccati equations, and integro-differentil equations, are provided to guarantee that these approaches are authentic, reliable, and appropriate. |
| Keywords |
| First derivative of Chebyshev polynomials; Tau method; Gauss Lobatto quadrature; the quadrature weights; Pseudo-spectral method; differentiation matrix; integration matrices; error analysis; BVPs; Lane-Emden equation; Riccati equation; singularly perturbed equation; integro-differential equations. |
| Status: Abstract Accepted (Oral Presentation) |