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In this work, we combined the Haar wavelet collocation method with the backward Euler difference formula to determine the approximate solutions of the modified unstable nonlinear Schrödinger equation. The backward Euler difference formula estimates the time derivative term and the Haar wavelet collocation method estimate the space derivative terms of the modified unstable nonlinear Schrödinger equation. This approach reduces the modified unstable nonlinear Schrödinger equation into a finite system of linear equations. In addition, we substantiate the efficiency and accuracy of the method graphically and numerically with the help of four examples.
In this study, numerical solutions are obtained for the time-dependent two-dimensional nonlinear parabolic partial differential equations (PDEs) with initial and Dirichlet boundary conditions. In assessing spatial derivatives, we employ the modified Galerkin method with the aid of Green's theorem, which minimizes the derivatives' order and incorporates boundary conditions. In the trial function, we use bivariate Bernstein polynomial bases. All the initial and boundary conditions are handled carefully by suitable transformation. Further, we exploit an iterative α-family approximation, especially the Crank Nicolson scheme, to take care the time derivative. Applying the proposed technique to a variety of nonlinear 2D parabolic PDEs, such as the 2D Burger's equation and the 2D Convection-Diffusion Reaction equation, the numerical results are presented in the form of tables and figures. The numerical results provide conclusive evidence that the technique being proposed is accurate and effective.
This paper discusses the construction of polynomialand non-polynomial splines of the fourth order of approximation.The behavior of the Lebesgue constants for the left, the right, andthe middle continuous cubic polynomial splines are considered.The non-polynomial splines are used for the construction of thespecial central difference approximation. The approximation offunctions, and the solving of the boundary problem with thepolynomial and non-polynomial splines are discussed. Numericalexamples are done.
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