Existence Of Numerical Solution Research Articles

Spectral Jacobi approximations for Boussinesq systems

AbstractThis paper is concerned with the numerical approximation of initial‐boundary‐value problems of a three‐parameter family of Bona–Smith systems, derived as a model for the propagation of surface waves under a physical Boussinesq regime. The work proposed here is focused on the corresponding problem with Dirichlet boundary conditions and its approximation in space with spectral methods based on Jacobi polynomials, which are defined from the orthogonality with respect to some weighted inner product. Well‐posedness of the problem on the corresponding weighted Sobolev spaces is first analyzed and existence and uniqueness of solution, locally in time, are proved. Then, the spectral Galerkin semidiscrete scheme and some detailed comments on its implementation are introduced. The existence of numerical solution and error estimates on those weighted Sobolev spaces are established. Finally, the choice of the time integrator to complete the full discretization takes care of different stability issues that may be relevant when approximating the semidiscrete system. Some numerical experiments illustrate the results.

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

AbstractIn this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.

A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz sense. Initial and homogeneous Dirichlet boundary data are imposed on a multidimensional spatial domain. To approximate the solutions, we employ a finite difference methodology. We rigorously establish the existence of numerical solutions along with the main numerical properties. Concretely, we show that the scheme is consistent in both space and time as well as stable and convergent. Numerical simulations in the one-dimensional scenario are presented in order to show the performance of the scheme. For the sake of convenience, A MATLAB code of the numerical model is provided in the appendix at the end of this work.

Weak Galerkin finite element method for a class of time fractional generalized Burgers' equation

AbstractIn this article, we use the weak Galerkin (WG) finite element method to study a class of time fractional generalized Burgers' equation. The existence of numerical solutions and the stability of fully discrete scheme are proved. Meanwhile, by applying the energy method, an optimal order error estimate in discrete L2 norm is established. Numerical experiments are presented to validate the theoretical analysis.

A finite difference scheme for the nonlinear time‐fractional partial integro‐differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time‐fractional integro‐differential equation. This model involves two nonlocal terms in time, ie, a Caputo time‐fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray‐Schauder theorem. And we obtain the discrete L2 stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.

Numerical simulation and convergence analysis of a high-order conservative difference scheme for SRLW equation

Coupled with the Richardson extrapolation, a new conservative Crank–Nicolson finite difference scheme, which has the accuracy of O(τ2+h4) without refined mesh for the symmetric regularized long wave equation is proposed. The corresponding conservative quantities are discussed, and the existence of numerical solution is proved by the Browder fixed point theorem. The convergence, unconditional stability and uniqueness of the scheme are also derived using the energy method. Numerical results are given to verify the accuracy and the efficiency of proposed algorithm.

Meshfree method for the non-stationary incompressible Navier-Stokes equations

We consider the solvability and the error estimates of numerical solutions of the non-stationary incompressible Stokes and Navier-Stokes equations by the meshfree method. The moving least square reproducing kernel method or the MLSRK method is employed for the space approximations. The existence of numerical solutions and the $L^2$-type error estimates are obtained. As a numerical example, we compare the numerical solutions of the Stokes and the Navier-Stokes equations with the exact solutions. Also we solve the non-stationary Navier-Stokes driven cavity flow using the MLSRK method.

Petrov–Galerkin methods for nonlinear systems without monotonicity

Petrov–Galerkin method is investigated for solving nonlinear systems without monotonicity. A new concept of ordered pair of supersolution and subsolution is proposed. The existence of numerical solutions is studied. A new monotone iteration is provided for solving the resulting problem. Two conditions are given, ensuring the geometric convergence rate. The numerical results show the advantages of this method.

Petrov-Galerkin methods for systems of nonlinear reaction-diffusion equations

Petrov-Galerkin method is proposed for solving systems of nonlinear reaction-diffusion equations. A new concept of ordered pair of supersolution and subsolution is introduced, and then the existence of numerical solutions is studied. A new monotone iteration is provided for solving the resulting problem. The convergence of the Petrov-Galerkin scheme is discussed. The numerical results show the advantages of such method.

Existence of numerical solutions and the order of linear circuits with dependent sources

A necessary and sufficient condition for the existence of a solution to a set of linear circuit equations with dependent sources is shown to be that a submatrix derived from the coefficients of dissipative elements has rank equal to the number of dissipative elements. This result is particularly useful in computer-aided circuit analysis to determine a cause when a numerical solution does not converge. In the development a general systematic technique for reducing circuit equations to normal form is presented. It is also demonstrated that when dependent sources are present, the order of a circuit is not necessarily equal to the number of independent energy-storing elements. The techniques and conclusions are illustrated with some simple nontrivial examples.