EXISTENCE, UNIQUENESS, AND STABILITY OF SOLUTIONS TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN FUNCTIONAL SPACES

Abstract

Background: Nonlinear fractal fractional PDEs and problems involving these operators lead to difficult mathematical problems, even for more classical functional spaces, such as Gelfand or Sobolev spaces, let alone in advanced functional frameworks as the fractional Sobolev spaces. These models describe physical processes which are characterized by memory and spatial nonlocality, and thus require a rigorous treatment of the solution-properties concerning existence, uniqueness and stability. Aim: The study seeks to study the existence and uniqueness as well as the stability of solutions of nonlinear PDEs in fractional Sobolev spaces via variational methods, fixed-point theory and stability notions including Ulam–Hyers. It also investigates the effect of boundary conditions and of numerical implementation on the theoretical accuracy. Method: A combined analytical and numerical technique was used. The theoretical part based on fractional Sobolev embeddings, eigenvalue decomposition and variational methods in the presence of Neumann and Robin boundary condition. Numerics support this claim, a theoretical study using reproducing kernel Hilbert space methods was performed and the stability was investigated based on Hardy–Sobolev inequalities. The results were tabulated in six charts to account for differences among function types and boundary regimes. Results: The results verify that on some function spaces, the nonlinear PDEs in fractional Sobolev spaces possess the unique and stable solution. The stability of approximate Ulam-type solutions was guaranteed against perturbations in the sense of Hyers. The effect of the Neumann and the Robin boundary conditions was considerable on the regularity of the solution, and numerical simulations confirmed theoretical results with very small errors. Conclusion: The work justifies the theoretical background of fractional PDEs in functional spaces and their utilization in the modeling of practical systems with memory and nonlocal effects. These results open the door for further studies on coupled systems and non-regular domains.