A New Solution for MKDV Equation Using Galerkin Finite Element Method

2017 ◽  
Vol 14 (1) ◽  
pp. 800-806 ◽  
Author(s):  
K. R Raslan ◽  
Z. F Abu Shaeer
Keyword(s):  
Finite Element ◽  
Mkdv Equation ◽  
Galerkin Finite Element Method ◽  
Shape Functions ◽  
Unconditionally Stable ◽  
Galerkin Finite Element ◽  
Numerical Tests ◽  
Set Up ◽  
Element Shape ◽  
Invariants Of Motion

A finite element solution of the modified Korteweg-de Vries (MKdV) equation, based on Galerkin’s method using cubic splines as element shape functions, is set up. A linear stability analysis shows the scheme is unconditionally stable. Numerical tests for single, two and three solitons are used to assess the performance of the proposed scheme. The four invariants of motion are evaluated to determine the conservation properties of the algorithm.

scholarly journals Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 469 ◽  
Author(s):  
Azhar Iqbal ◽  
Nur Nadiah Abd Hamid ◽  
Ahmad Izani Md. Ismail
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Spline Method ◽  
Galerkin Finite Element Method ◽  
Galerkin Finite Element ◽  
B Spline

This paper is concerned with the numerical solution of the nonlinear Schrödinger (NLS) equation with Neumann boundary conditions by quintic B-spline Galerkin finite element method as the shape and weight functions over the finite domain. The Galerkin B-spline method is more efficient and simpler than the general Galerkin finite element method. For the Galerkin B-spline method, the Crank Nicolson and finite difference schemes are applied for nodal parameters and for time integration. Two numerical problems are discussed to demonstrate the accuracy and feasibility of the proposed method. The error norms L 2 , L ∞ and conservation laws I 1 ,   I 2 are calculated to check the accuracy and feasibility of the method. The results of the scheme are compared with previously obtained approximate solutions and are found to be in good agreement.

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