Orthogonal polynomials

Legendre polynomials

Physics

Legendre wavelet

Mathematical analysis

Legendre function

Fractional calculus

Artificial intelligence

Nonlinear system

Wavelet transform

Classical orthogonal polynomials

Algebraic equation

Applied mathematics

Mathematics

Associated Legendre polynomials

Computer science

Discrete wavelet transform

Wavelet

Quantum mechanics

Differential operator

Gegenbauer polynomials

Applied Mathematical Modelling

In this study, a new numerical method for the solution of the linear and nonlinear distributed fractional differential equations is introduced. The fractional derivative is described in the Caputo sense. The suggested framework is based upon Legendre wavelets approximations. For the first time, an exact formula for the Riemann–Liouville fractional integral operator for the Legendre wavelets is derived. We then use this formula and the properties of Legendre wavelets to reduce the given problem into a system of algebraic equations. Several illustrative examples are included to observe the validity, effectiveness and accuracy of the present numerical method.

Legendre wavelets approach for numerical solutions of distributed order fractional differential equations