Numerical solutions of a generalized time fractional partial integro-differential equation of Volterra type via an extended cubic B-spline technique

In this work, we investigate numerical solutions of a generalized time-fractional partial integro-differential equation (TFPIDE) of Volterra type. The equation incorporates Caputo fractional derivatives and convolution-type integral terms, making it suitable for modeling physical processes with memory and non-local effects. To approximate the solution, we employ a temporal discretization based on the L1-2 scheme for the Caputo derivative and a composite trapezoidal rule for the integral term. Spatial discretization is carried out using the extended cubic B-spline (ECBS) collocation method, which generalizes classical cubic B-splines and provides enhanced flexibility and accuracy. Convergence analysis is established, proving that the proposed scheme achieves second-order accuracy in space and optimal error bounds in time. Several numerical examples are presented to demonstrate the efficiency, stability, and accuracy of the method.