A Mixed Element Algorithm Based on the Modified L1 Crank–Nicolson Scheme for a Nonlinear Fourth-Order Fractional Diffusion-Wave Model

In this article, a new mixed finite element (MFE) algorithm is presented and developed to find the numerical solution of a two-dimensional nonlinear fourth-order Riemann–Liouville fractional diffusion-wave equation. By introducing two auxiliary variables and using a particular technique, a new coupled system with three equations is constructed. Compared to the previous space–time high-order model, the derived system is a lower coupled equation with lower time derivatives and second-order space derivatives, which can be approximated by using many time discrete schemes. Here, the second-order Crank–Nicolson scheme with the modified L1-formula is used to approximate the time direction, while the space direction is approximated by the new MFE method. Analyses of the stability and optimal L2 error estimates are performed and the feasibility is validated by the calculated data.

On a Multigrid Method for Tempered Fractional Diffusion Equations

In this paper, we develop a suitable multigrid iterative solution method for the numerical solution of second- and third-order discrete schemes for the tempered fractional diffusion equation. Our discretizations will be based on tempered weighted and shifted Grünwald difference (tempered-WSGD) operators in space and the Crank–Nicolson scheme in time. We will prove, and show numerically, that a classical multigrid method, based on direct coarse grid discretization and weighted Jacobi relaxation, performs highly satisfactory for this type of equation. We also employ the multigrid method to solve the second- and third-order discrete schemes for the tempered fractional Black–Scholes equation. Some numerical experiments are carried out to confirm accuracy and effectiveness of the proposed method.

Crank-Nicolson Scheme for Solving the Modified Nonlinear Schrodinger Equation

Purpose The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme. Design/methodology/approach The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities. Findings The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients. Originality/value The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.

Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation

This paper presents a numerical solution to the inverse problem consisting of recovering time-dependent thermal conductivity and heat source coefficients in the one-dimensional parabolic heat equation. This mathematical formulation ensures that the inverse problem has a unique solution. However, the problem is still ill-posed since small errors in the input data lead to a drastic amount of errors in the output coefficients. The finite difference method with the Crank-Nicolson scheme is adopted as a direct solver of the problem in a fixed domain. The inverse problem is solved subjected to both exact and noisy measurements by using the MATLAB optimization toolbox routine lsqnonlin , which is also applied to minimize the nonlinear Tikhonov regularization functional. The thermal conductivity and heat source coefficients are reconstructed using heat flux measurements. The root mean squares error is used to assess the accuracy of the approximate solutions of the problem. A couple of numerical examples are presented to verify the accuracy and stability of the solutions.