Numerical solution of fractional differential equation by wavelets and hybrid functions

Abstract

In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By the use of these methods we translate an FDE to an algebraic linear equations which can be solved. Methods have been tested by some numerical examples.

Infinite Existence Solutions of Fractional Systems with Lipschitz Nonlinearity

“…In fact since (2) holds for every x 1 , x 2 ∈ R and h 1 ð0Þ = h 2 ð0Þ = 0, one has jh i ðxÞj ≤ L i jxj p−1 , i = 1, 2, for all x ∈ ℝ. It follows from 5 Journal of Function Spaces (14), (20), and Lemma 5 that…”

“…FDEs have attracted considerable interest due to their ability to model complex phenomena in several fields of science, engineering, physics, biology, and economics (see [1][2][3][4][5][6][7]). In summary, many improvements have been made in the theory of partial calculus and partial differential equations and partial and ordinary differential equations (see [8][9][10][11][12][13][14][15][16][17][18], [2,5]). Numerous studies have explored the existence and solutions of different nonlinear elementary and boundary value problems through the use of various nonlinear analysis tools and techniques (see, for example, [7,[19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]).…”

Multiplicity Solutions of Fractional Impulsive p -Laplacian Systems: New Result

“…In viscoelasticity, electrochemistry, power, porous media, and electromagnetism, for instance, see and the references therein. Many articles have recently investigated the existence of solutions to boundary value problems for FDEs, and we refer the reader to one of them [2,[18][19][20][34][35][36][37][38][39][40][41][42][43][44][45][46] and the references therein. For example, Kamache et al [40] investigated the existence of three solutions for a class of fractional p-Laplacian systems using a variational structure and critical point theory.…”