Spectral Chebyshev wavelet method for state-dependent fractional integro-differential equations in adaptive tumor growth modeling

Abstract

This paper develops a Chebyshev Wavelet Operational Matrix Method (CWOMM) for solving a novel class of nonlinear mixed Volterra–Fredholm fractional integro-differential equations with state-dependent kernels and time-varying coefficients. Unlike classical formulations, in which the Volterra and Fredholm kernels depend only on the independent variables, the kernels considered here depend on the unknown solution itself; this state dependence enables the modeling of adaptive biological processes such as acquired drug resistance and is the principal novelty of the present formulation. Within the Caputo fractional derivative framework, we establish existence and uniqueness results using fixed-point theory in weighted Banach spaces under explicit contraction and compactness hypotheses, valid for both \(0<\alpha <1\) and \(1<\alpha <2\) . The method transforms the integro-differential equation into a matrix algebraic system via operational matrices for fractional integration and state-dependent kernel operators, solved by a frozen-kernel Newton iteration with diagonal or row-equilibration preconditioning. Under a fully reproducible benchmark protocol, CWOMM outperforms L1 finite-difference and spectral collocation methods in both accuracy and computational efficiency. The framework is applied to a novel fractional tumor growth model with adaptive treatment resistance, using parameters representative of experimentally calibrated NSCLC xenograft dynamics and supported by a fully verified physical-to-scaled time transformation. Results support CWOMM as an accurate and efficient tool for biological systems governed by memory, hereditary effects, and state-dependent feedback mechanisms.