Stability Analysis and Approximation of Time Fractional Hyperbolic Integro Partial Differential Equations

This study investigates fractional order hyperbolic integro-differential equations involving Caputo fractional derivatives, emphasizing the need for accurate approximate solutions. To address this, first- and second-order finite-difference schemes are proposed for the associated initial boundary value problem. Detailed error analyses, presented through tables and figures, demonstrate the accuracy of the numerical results. Furthermore, Von Neumann stability analysis is carried out, highlighting its role in ensuring the reliability and consistency of the devel- oped schemes. Numerical experiments validate the effectiveness of the methods by confirming excellent agreement with exact solutions. Comparative evaluations across different fractional orders are performed using identical grid numbers, with illustrative plots showing behavior for large spatial and temporal vari- ables. The findings emphasize that finite-difference approaches, supported by stability analysis, offer a reliable and efficient framework for solving Caputo time- fractional hyperbolic integro-differential equations, underlining their applicability in broader fractional calculus and differential equation contexts.