Generalized Reproducing Kernel Methods for Solving Higher-Order Quasi-Linear Fractional Partial Differential Equations

Abstract

Higher-order quasi-linear fractional partial differential equations are very hard to solve analytically and computationally because they have iterations with weak regularity, boundary constraints, nonlinear solution-dependent operator terms, nonlocal memory effects, and higher-order differential structures. The performance of classical discretization and decomposition-based methods may get worse when fractional operators, high-order spatial derivatives, and quasi-linear interactions need to be treated at the same time. With this paper, we create an extended reproducing kernel method for fixing higher-order quasi-linear fractional partial differential equations that have Caputo-type time-fractional derivatives and boundary-constrained spatial operators. Using a problem-adapted reproducing kernel Hilbert space, the proposed framework goes beyond standard reproducing kernel methods. The starting and border conditions are built directly into the acceptable approximation space. This paper uses an operator equation to describe the fractional differential operator, the higher-order spatial operator, and the quasi-linear term. It also uses an iterative linearization approach to handle the nonlinear component while keeping the kernel-based estimate structure. There are finite-dimensional estimated solutions, the reproducing-kernel series converges under completeness and Lipschitz-type assumptions, the system is stable when the source and border data change, and residual-based error control is used. Numerical validation is structured through linear, quasi-linear, and variable-coefficient benchmark problems, using , , RMSE, residual norms, iteration counts, and CPU time as performance indicators. It was shown that the generalized framework is a good way to solve higher-order quasi-linear fractional PDEs that is also correct, doesn't change at the boundaries, and is fast on computers. It is also sensitive to kernel conditioning, node distribution, and nonlinear iteration stability.