A numerical approach for solving nonlinear fractional Volterra-Fredholm integro-differential equations with mixed boundary conditions (Q2819175)

This paper deals with the following fractional Volterra-Fredholm integro-differential equation NEWLINE\[NEWLINE (D^{\alpha}y)(x)=g(x)+\int\limits_{0}^{x}k_1(x,t)F_1[t,y(t)]dt+\int\limits_{0}^{1}k_2(x,t)F_2[t,y(t)]dt NEWLINE\]NEWLINE with mixed boundary conditions NEWLINE\[NEWLINE \sum\limits_{j=1}^{d}[a_{ij}y^{(j-1)}(0)+b_{ij}y^{(j-1)}(1)]=r_i, \quad i=1,2,\dots,d, NEWLINE\]NEWLINE where \(D^{\alpha}\) is Caputo fractional derivative, \(y:[0,1]\to \mathbb{R}\) is a continuous unknown function, \(g:[0,1]\to \mathbb{R}\) and \(k_i:[0,1]\times [0,1]\to \mathbb{R}\), \(i=1,2\), are continuous functions. \(F_i:[0,1]\times \mathbb{R} \to \mathbb{R}\), \(i=1,2\), are Lipschitz continuous functions.NEWLINENEWLINEUsing Legendre wavelets this problem is transformed into a system of algebraic equations which is solved by Newton's method. Convergence of the proposed numerical method is also discussed. Three illustrative numerical examples are adduced in conclusion.