Abstract
Numerical calculation of the fractional integrals and derivatives is the code tosearch fractional calculus and solve fractional differential equations. The exactsolutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhanced numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give two examples to demonstrate how efficiently our theory can be implemented in practice.
Highlights
Fractional differential equations have been studied by many investigators in recent years
Coimbra [1] applied a consistent approximation with first-order accurate for the solution of variable order differential equations
No one had tried to find the numerical solutions of the variable order fractional differential equations by the reproducing kernel method (RKM)
Summary
Fractional differential equations have been studied by many investigators in recent years. Different authors have presented different definitions of variable order differential operators. To get the numerical solutions of variable order fractional differential equations is quite compelling. There are few studies of variable order fractional differential equations. Coimbra [1] applied a consistent approximation with first-order accurate for the solution of variable order differential equations. Lin et al [2] worked the stability and the convergence of an explicit finite-difference approximation for the variable-order fractional diffusion equation with a nonlinear source term. No one had tried to find the numerical solutions of the variable order fractional differential equations by the reproducing kernel method (RKM). The aim of our work is to investigate the efficiency of RKM to solve variable-order fractional differential equations.
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