Efficiency of the Spectral Wavelets Approach in Solving Systems of Differential Equations

Abstract

This thesis is a contribution to numerical studies on systems of ordinary differential equations, fractional differential equations, integrodifferential equations and partial differential equations using different types of Chebyshev wavelets jointly with the spectral tau and collocation methods. Numerical tests are provided and the obtained result was compared with the published data in the literature moreover convergence analysis, error estimate and stability were discussed in details whenever it is possible. A detailed introduction to spectral methods, wavelets, and fractional calculus are given. The system of integrodifferential equations was solved using second kind Chebyshev wavelets. Coupled system of fractional differential equations was solved using third kind Chebyshev wavelets. The SIRC and Prey-Predator models were solved using first kind Chebyshev wavelets. Finally, the telegraph equation, KdV and Burger time fractional differential equations were solved using four different kinds of Chebyshev wavelets. All computations in this thesis were performed using Mathematica 9. Keywords: Chebyshev Wavelets; Spectral Methods; Error Analysis; Fractional Differential Equations; SIRC Model; Telegraph Equation