Numerical computation of fractional partial differential equations with variable coefficients utilizing the modified fractional Legendre wavelets and error analysis

This paper presents a numerical scheme to address a type of fractional spatial–temporal telegraph equations with variable coefficients by applying the modified fractional Legendre wavelets. Compared with other common Legendre wavelets method, a new fractional Legendre wavelet is constructed. The modified fractional Legendre wavelet has two parameters of t and α, and each variable is expressed as tα. The differential operational matrices of modified fractional Legendre wavelets are established by building the relations between the modified Legendre wavelets and modified polynomials functions. The original differential system is transferred into a linear algebraic system of equations by introducing the modified Legendre wavelet vectors and their fractional differential operational matrices. The obtained system can be easily solved by any mathematical technique and tool. The error analysis theorem of the system is detailed investigated by introducing the variables x and t and the parameters α and β. Several numerical experiments, combined with their error analysis, are provided to authenticate that the proposed method is practicable and efficient.