A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain

A Study for Coupled Systems of Nonlinear Boundary Value Problem

This chapter deals with the existence and uniqueness of solutions for a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions and for the system of two-point boundary value problem when we take the case of integer derivative. The existence results for the fist problem are obtained by using Leray-Shauder nonlinear alternative and Banach contraction principle and for the second problem, we derive explicit eigenvalue intervals of λ for the existence of at least one positive solution by using Krasnosel’skii fixed point theorem. An illustrative examples is presented at the end for each problem to illustrate the validity of our results

Non-negative solutions of systems of ODEs with coupled boundary conditions

We provide a new existence theory of multiple positive solutions valid for a wide class of systems of boundary value problems that possess a coupling in the boundary conditions. Our conditions are fairly general and cover a large number of situations. The theory is illustrated in details in an example. The approach relies on classical fixed point index