Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition
Issue:
Volume 7, Issue 3, June 2021
Pages:
40-52
Received:
28 March 2021
Accepted:
24 May 2021
Published:
31 May 2021
DOI:
10.11648/j.ijtam.20210703.11
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Views:
Abstract: In this paper, the Lax-Wend off difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion, and the Lax-Wend off difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L2 and L∞ norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
Abstract: In this paper, the Lax-Wend off difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by ...
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Sensitivity Analysis of Linear Programming in Decision Making Model
Shek AhmedDepartment of Mathematics,
University of Barishal,
Barishal,
Jakia Sultana,
Tanzila Yeasmin Nilu,
Shamima Islam
Issue:
Volume 7, Issue 3, June 2021
Pages:
53-56
Received:
23 April 2021
Accepted:
11 May 2021
Published:
31 May 2021
DOI:
10.11648/j.ijtam.20210703.12
Downloads:
Views:
Abstract: The term Sensitivity Analysis (SA), sometimes called the post optimality analysis, refers to an analysis of the effect on the optimal solution of changes in the parameters of problem on the current optimal solution. Simplex method is an iterative procedure which gives the optimal solution to a Linear Programming Problem (LPP) in a finite number of steps or gives an indication that there is an unbounded solution whereas SA serves as an integral part of solving LPP and is normally carried out after getting optimal solution. In this research work, Sensitivity Analysis is used to understand the effect of a set of independent variables on some dependent variable under certain specific conditions. In order to determine the possible effect of independent parameters, we considered the changes in the input data of the optimal solution. This notion is actually based on the idea of Sensitivity Analysis. And it is found that all the possible alternative decision making converges in the neighborhood of the optimal solution. To avoid numerical complexity, we use LINDO software to show the changes in the input data and optimal solution.
Abstract: The term Sensitivity Analysis (SA), sometimes called the post optimality analysis, refers to an analysis of the effect on the optimal solution of changes in the parameters of problem on the current optimal solution. Simplex method is an iterative procedure which gives the optimal solution to a Linear Programming Problem (LPP) in a finite number of ...
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