Tridiagonal matrix algorithm

Abstract: In this paper, we have presented a special finite difference method for solving a singular perturbation problem with layer behaviour at one end. In this method, we have used a second order finite difference approximation for the second derivative, a modified second order upwind finite difference approximation for the first derivative and a second order average difference approximation for y to reduce the global error and retaining tridiagonal system. Then the discrete invariant imbedding algorithm is used to solve the tridiagonal system. This method controls the rapid changes that occur in the boundary layer region and it gives good results in both cases i.e., h ≤ e and e << h. The existence and uniqueness of the discrete problem along with stability estimates are discussed. Also we have discussed the convergence of the method. We have presented maximum absolute errors for the standard examples chosen from the literature.

Abstract: Traditionally, solving the parameter identification inverse problems of partial differential equations encountered many difficulties and insufficiency. In this paper, we propose an improved GEP (Gene Expression Programming) to identify the parameters in the reverse problems of partial differential equations based on the self-adaption, self-organization and self-learning characters of GEP. This algorithm simulates a parametric function itself of a partial differential equation directly through the observed values by fully taking into account inverse results caused by noises of a measured value. Modeling is unnecessary to add regularization in the modeling process aiming at special problems again. The experiment results show that the algorithm has good noise-immunity. In case there is no noise or noise is very low, the identified parametric function is almost the same as the original accurate value; when noise is very high, good results can still be obtained, which successfully realizes automation of the parameter modeling process for partial differential equations.

Abstract: In this paper, the cubic B-spline collocation scheme is implemented to find numerical solution of diffusion convection problem of chemical engineering. The scheme is based on the Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The numerical results are found to be in good agreement with the exact solutions. Results are also shown graphically and are compared with results given in the literature. Keywords: Diffusion; Cubic B-spline; Collocation; Tridiagonal system; Thomas algorithm.

Abstract: In this paper, a semi-implicit mask optimization method is investigated by quantifying the nonlinear diffusion quality of the mask pattern into the inverse optimization framework. The mask synthesis problem is addressed as a constrained time-dependent partial differential equation (PDE) which is further solved by the semi-implicit additive operator splitting (AOS) scheme. Thus the updating of the mask patterns is reduced to consecutive one-dimensional updates with respect to coordinate axes, where unconditional stable implicit difference schemes can be applied with larger time steps by solving a tridiagonal linear equation efficiently using the Thomas algorithm. Experimental results merit the superiority of the proposed semi-implicit AOS method with improved convergence performance.