Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation

A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Gr¨unwald-Letnikov definition is used for the time-fractional derivative. The stability and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented to test that the numerical scheme is accurate and feasible

The Fourier Spectral Method For The Sivashinsky Equation.

In this paper, a Fourier spectral method for solving the Sivashinsky equation with periodic boundary conditions is developed. We establish semi-discrete and fully discrete schemes of the Fourier spectral method. A fully discrete scheme is constructed in such a way that the linear part is treated implicitly and the nonlinear part explicitly. We use an energy estimation method to obtain error estimates for the approximate solutions. We also perform some numerical experiments

Numerical Solution Of A Linear Goursat Problem Stability,Consistency And Convergence .

The Goursat problem, associated with hyperbolic partial differential equations, arises in several areas of applications. These include mathematical modeling of reacting gas flows and supersonic flow

Solving Polynomial Equations using Modified Super Ostrowski Homotopy Continuation Method

Homotopy continuation methods (HCMs) are now widely used to find the roots of polynomial equations as well as transcendental equations.  HCM can be used to solve the divergence problem as well as starting value problem. Obviously, the divergence problem of traditional methods occurs when a method cannot be operated at the beginning of iteration for some points, known as bad initial guesses. Meanwhile, the starting value problem occurs when the initial guess is far away from the exact solutions.   The starting value problem has been solved using Super Ostrowski homotopy continuation method for the initial guesses between . Nevertheless, Super Ostrowski homotopy continuation method was only used to find out real roots of nonlinear equations.  In this paper, we employ the Modified Super Ostrowski-HCM to solve several real life applications which involves polynomial equations by expanding the range of starting values. The results indicate that the Modified Super Ostrowski-HCM performs better than the standard Super Ostrowski-HCM. In other words, the complex roots of polynomial equations can be found even the starting value is real with this proposed scheme

Heat transfer analysis for falkner-skan boundary layer flow past a stationary wedge with slips boundary conditions considering temperature-dependent thermal conductivity

We studied the problem of heat transfer for Falkner-Skan boundary layer flow past a stationary wedge with momentum and thermal slip boundary conditions and the temperature dependent thermal conductivity. The governing partial differential equations for the physical situation are converted into a set of ordinary differential equations using scaling group of transformations. These are then numerically solved using the Runge-Kutta-Fehlberg fourth-fifth order numerical method. The momentum slip parameter δ leads to increase in the dimensionless velocity and the rate of heat transfer whilst it decreases the dimensionless temperature and the friction factor. The thermal slip parameter leads to the decrease rate of heat transfer as well as the dimensionless temperature. The dimensionless velocity, rate of heat transfer and the friction factor increase with the Falkner-Skan power law parameter m but the dimensionless fluid temperature decreases with m. The dimensionless fluid temperature and the heat transfer rate decrease as the thermal conductivity parameter A increases. Good agreements are found between the numerical results of the present paper with published results

Cubic Trigonometric B-Spline Approach to Numerical Solution of Wave Equation

The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature

Cubic Trigonometric B-Spline Approach to Numerical Solution of Wave Equation

The generalized wave equation models various problems in sciences and engineering. In this paper, a new three-time level implicit approach based on cubic trigonometric B-spline for the approximate solution of wave equation is developed. The usual finite difference approach is used to discretize the time derivative while cubic trigonometric B-spline is applied as an interpolating function in the space dimension. Von Neumann stability analysis is used to analyze the proposed method. Two problems are discussed to exhibit the feasibility and capability of the method. The absolute errors and maximum error are computed to assess the performance of the proposed method. The results were found to be in good agreement with known solutions and with existing schemes in literature

Numerical Solution Of The Goursat Problem.

The Goursat problem, associated with hyperbolic partial differential equations, arises in several areas of applications. Several finite difference schemes have been proposed to solve the Goursat problem