In this paper, by using IMPLICIT fourth order central DIFFERENCE METHOD and TLNS equations, the numerical solution of the steady axisymmetric viscous supersonic flow is implemented over blunt cone with shock-fitting METHOD. Because of using high order terms of Taylor series in discretization of derivatives, this METHOD has high accuracy and low numerical error (dispersion error) compared with low order METHOD. The boundary-closure scheme has an important role in stability of this METHOD. By using a coarse grid in this METHOD, the results of numerical solution are found to be very close to those obtained with a fine grid employing the second order (Beam-Warming) METHOD. Higher accuracy of this METHOD is identified relative to the second order METHOD when the grid is being refined. The convergence of this METHOD can be adjusted to accommodate the computational hardware capabilities.

In this study, one explicit and one IMPLICIT FINITE DIFFERENCE scheme is introduced for the numerical solution of time-fractional Riesz space diffusion equation. The time derivative is approximated by the standard Gr¨, unwald Letnikov formula of order one, while the Riesz space derivative is discretized by Fourier transform-based algorithm of order four. The stability and convergence of the proposed METHODs are studied. It is proved that the IMPLICIT scheme is unconditionally stable, while the explicit scheme is stable conditionally. Some examples are solved to illustrate the efficiency and accuracy of the proposed METHODs. Numerical results confirm that the accuracy of present schemes is of order one.

Fractional order partial differential equations are generalizations of classical partial differential equations. Increasingly, these models are used in applications such as fluid flow, finance and others. In this paper we examine some practical numerical METHODs to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a FINITE domain. Stability, consistency, and (therefore) convergence of the METHOD are examined. It is shown that the fractional METHOD based on the shifted Grunwald formula is unconditionally stable. This study concerns both theoretical and numerical aspects, where we deal with the construction and convergence analysis of the discretization schemes. A numerical example is presented and compared with exact solution for its order of convergence.

Introduction: In this paper, we have introduced a new METHOD for solving a class of the partial integro-differential equation with the singular kernel by using the FINITE DIFFERENCE METHOD. One of the best subjects in the numerical analysis is a FINITE DIFFERENCE METHOD (FDM). We used (FDM) to solve problems in mathematical physics, integral equations, and engineering, such as electromagnetic potential, fluid flow, radiation heats transfer, laminar boundary-layer theory and mass transport, Abel integral equations, and problem of mechanics or physics. Also in some physical problems such as fluid flow and heat transfer problems, the Laplace equations and the Poisson equations are describe by (FDM). In real life most phenomena are modelled by partial differential equations. Material and METHODs: First, we employing an algorithm for solving the problem based on the Crank-Nicholson scheme with given conditions. Furthermore, we discrete the singular integral for solving of the problem. Also, the numerical results obtained here can be compared with the cubic B-spline METHOD. Results and discussion: In addition, solving some examples demonstrates the validity and applicability of the approached METHOD, so that the results are reported in the tables and their figures are shown. The high speed of the calculations, and the assurance of having an approximate solution are obtain by proving the stability of the METHOD. Conclusion: The following conclusions were drawn from this research. Coefficients of the approximate function via Crank-Nicholson scheme are found very easily and therefore many calculations are reduced. The numerical results obtained here can be compared with the cubic B-spline METHOD The assurance of having an approximate solution are obtain by proving the stability of the METHOD.