Abstract
In this work, a new numerical method for the fractional diffusion-wave equation and nonlinear Fredholm and Volterra integro-differential equations is proposed. The method is based on Euler wavelet approximation and matrix inversion of an M×M collocation points. The proposed equations are presented based on Caputo fractional derivative where we reduce the resulting system to a system of algebraic equations by implementing the Gaussian quadrature discretization. The reduced system is generated via the truncated Euler wavelet expansion. Several examples with known exact solutions have been solved with zero absolute error. This method is also applied to the Fredholm and Volterra nonlinear integral equations and achieves the desired absolute error of 0×10−31 for all tested examples. The new numerical scheme is exceptional in terms of its novelty, efficiency and accuracy in the field of numerical approximation.
Highlights
Fractional calculus is very useful and widely used in many applications in science, numerical computations and engineering, where the mathematical modeling of several real world problems is presented in terms of fractional differential equations, see, e.g., [1,2,3,4,5,6,7,8].For example, the authors in [8] approximated the Caputo fractional derivative by quadratic segmentary interpolation
That raised a new approach of approximating fractional derivatives and provides some insights for a new applications where the numerical resolution of ordinary fractional differential equations is achieved
We propose a new numerical method based on Euler wavelets with different sets of collocation points
Summary
Citation: Mohammad, M.; Trounev, A.; Alshbool, M. A Novel Numerical and Mohammed Alshbool 1,3, * Department of Mathematics & Statistics, Zayed University, Abu Dhabi 144543, United Arab Emirates Department of Applied Mathematics, Abu Dhabi University, Abu Dhabi P.O. Box 59911, United Arab Emirates
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