The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations

A numerical method for fractional pantograph differential equations based on Taylor wavelets

“…Let k be a positive integer. For each n = 1 , … , 2 k − 1 and non-negative integer m , the Taylor wavelets ψ n , m ( t ) , is defined on the interval [ 0 , 1 ) by (Keshavarz et al, 2018)…”

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

“…TWs n,m (x) = (k,n, m, x) involve four arguments:n = n − 1, n = 1, 2, … , 2 k−1 , k is assumed any positive integer, m is the degree of the Taylor polynomials, and x is the normalized time. Keshavarz et al have defined them on the interval [0, 1) as follows 32 :…”

“…Wavelet basis can be used to reduce the under study problem to a system of algebraic equations by estimating the integrals using operational matrices. Taylor wavelets (TWs) have been constructed by Keshavarz and et al 32 The TW method exhibits several advantageous features:…”

“…The simplicity and small computation costs compared with Chebyshev, Legendre, and Bernoulli wavelets. 32 In this paper, we construct TWs operational matrix of fractional integration, for the first time, to solve the fractional integro-differential equations with weakly singular kernels. Our method consists of reducing the given weakly singular fractional integro-differential equation to a set of algebraic equations by expanding the equation as TWs with unknown coefficients and employing fractional operational matrix.…”

Matrix-valued Bratu equation and the exact solution of its initial value problem

“…16 Because of its simplicity, this equation is widely used to test numerical nonlinear eigenvalue solvers. 1,14 The Bratu equation for a real function uðsÞðs 2 RÞ is written as u 00 ðsÞ þ e uðsÞ ¼ 0;…”