Volume 9, Issue 4

In this paper, a nonlinear Volterra | Fredholm integro-differential equation is solved by using the modified Adomian decomposition method (MADM). The approximate solution of this equation is calculated in the form of a series in which its components are computed easily. The accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results. The existence, uniqueness and convergence and an error bound of the proposed method are proved. Some examples are presented to illustrate the efficiency and the performance of the modified decomposition method.

The iteration algorithm of finding bifurcation points of simple eigenvalue curves of the linear algebraic two-parameter eigenvalue problem is considered. The algorithm is based on the efficient numerical procedure of calculation of the derivative of matrix determinant. Numerical results are given.

Under some assumptions, we prove that the solution of a discrete form of the above problem quenches in a finite time and estimate its numerical quenching time. We also show that the numerical quenching time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.

A mixed hybrid finite element method has been applied to a variational inequality with a potential second-order quasi-linear differential operator. The Lagrange multiplier method for a dual problem has been used to construct this finite element scheme. The existence and uniqueness of a solution for the resulting finite- dimensional problem has been proved, the solution iterative methods are discussed. The non-overlapping domain decomposition method combined with the mixed hybrid finite element approximation is analyzed.

We consider residual-based stabilised finite element methods for the generalised Oseen problem. The unique solvability based on a modified stability condition and an error estimate are proved for inf-sup stable discretisations of velocity and pressure. The analysis highlights the role of an additional stabilisation of the incompressibility constraint. It turns out that the stabilisation terms of the streamline-diffusion (SUPG) type play a less important role. In particular, there exists a conditional stability problem of the SUPG stabilisation if two relevant problem parameters tend to zero. The analysis extends a recent result to general shape-regular meshes and to discontinuous pressure interpolation. Some numerical observations support the theoretical results.

An upwind and a Lax-Wendroff scheme are introduced for the solution of a one-dimensional non-local problem modelling ohmic heating of foods. The schemes are studied regarding their consistency, stability, and the rate of convergence for the cases that the problem attains a global solution in time. A high resolution scheme is also introduced and it is shown that it is total-variation-stable. Finally some numerical experiments are presented in support of the theoretical results.

The Fourier pseudo-spectral method has been studied for a one- dimensional coupled system of viscous Burgers equations. Two test problems with known exact solutions have been selected for this study. In this paper, the rate of con- vergence in time and error analysis of the solution of the first problem has been studied, while the numerical results of the second problem obtained by the present method are compared to those obtained by using the Chebyshev spectral collocation method. The numerical results show that the proposed method outperforms the conventional one in terms of accuracy and convergence rate.

In this paper, we consider a general form of two-dimensional linear Volterra integro-differential equations(TDLVIDE) of the second order with some sup- plementary conditions and develop the operational Tau method with standard base for obtaining a numerical solution.