A unified Haar wavelet collocation framework for fractional volterra integro-differential equations with application to tumor-immune dynamics modeling.

This paper develops a novel Haar wavelet collocation technique (HWCT) for solving a general class of fractional Volterra integro-differential equations (FVIDEs). The method employs operational matrices of integration and fractional integration to systematically handle both the fractional derivative and Volterra integral terms within a unified framework. By approximating the highest-order derivative using a truncated Haar series and recovering the solution through successive integration, the complex FVIDE is transformed into an easily solvable linear algebraic system. A rigorous convergence analysis establishes that the method achieves second-order accuracy under appropriate regularity conditions. The efficacy of the proposed approach is demonstrated through several numerical examples, showing excellent agreement with exact solutions. Furthermore, we present a significant application to fractional-order tumor-immune dynamics, modeling the complex interactions between tumor growth, immune response, and immunotherapy. The model captures memory effects and distributed time delays inherent in biological systems, providing insights for optimizing cancer treatment protocols. Numerical simulations confirm the method’s computational efficiency and accuracy, making it a valuable tool for solving challenging problems in mathematical biology and engineering.