combining the reproducing kernel method with taylor series expansion to solve systems of nonlinear fractional volterra integro-differential equations

In this article, we present a novel approach for solving systems of nonlinear fractional volterra integro-differential equations $(nfvi-des)$ by reproducing the hilbert kernel method. kernel methods are powerful tools for addressing both linear and nonlinear problems. the reproducing kernel method stands out for its wide-ranging applications in solving complex scientific challenges. our method combines the reproducing kernel method with a truncated taylor series expansion, resulting in a more precise solution. this transformation converts the original $nfvi-des$ into a system of nonlinear fractional differential equations. our numerical results showcase this approach’s effectiveness and align with theorems about error analysis.