A new approach of solving fractional fredholm-volterra integrodifferential equations using the Differential Transform Method

Abstract

This study presents two simple and effective approaches for solving linear and nonlinear fractional Fredholm–Volterraintegro-differential equations (FFVIDEs) with the Caputo fractional derivative.The first approach ,called the Adaptive Residual- Corrected Differential Transform Method(ARCDTM),extends the standard Differential Transform Method by using the residual of the original equation to improve the truncated series solution. The correction is applied adaptively until a prescribed accuracy is reached ,which helps control the truncation error with out rquiring linearization of non linear terms.The second approach, referred to as the Fractional Multi-Stage Differential Transform Method (FMS-DTM), divides the computational interval into smaller sub intervals and constructs a local fractional series at each stage. This provides better control of the approximation when the solution is considered over a relatively large interval. Several linear and non linear FFVIDEs are considered, including problems with polynomial and non-polynomial non linear it is and examples with known and unavailable exact solutions. The numerical results are examined in terms of accuracy, residual error, convergence, and computational effort. The results indicate that the proposed approaches provide accurate and flexible approximations while retaining the straightforward recurrence structure of the differential transform method.