scholarly journals One-Dimensional Heat Equation Subject to Both Neumann and Dirichlet Initial Boundary Conditions and Used a Spline Collocation Method

10.29007/w6lj ◽  
2018 ◽  
Author(s):  
Nilesh Patel ◽  
Jigisha Pandya
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Collocation Method ◽  
Initial Boundary ◽  
Mathematical Tool ◽  
Spline Collocation ◽  
Discrete Approximations ◽  
Spline Collocation Method ◽  
One Dimensional ◽  
Good Agreement

In this paper, one-dimensional heat equation subject to both Neumann and Dirichlet initial boundary conditions is presented and a Spline Collocation Method is utilized for solving the problem. Also, Spline provides continuous solution in contrast to finite difference method, which only provides discrete approximations. It is found that this method is a powerful mathematical tool and can be applied to large class of linear and nonlinear problem in different fields of science and technology. Numerical results obtained by the present method are in a good agreement with the analytical solutions available in the literature.

The Crank-Nicolson Hermite Cubic Orthogonal Spline Collocation Method for the Heat Equation with Nonlocal Boundary Conditions

2013 ◽  
Vol 5 (04) ◽  
pp. 442-460 ◽  
Author(s):  
B. Bialecki ◽  
G. Fairweather ◽  
J.C. López-Marcos
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Collocation Method ◽  
Error Estimates ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Maximum Norm ◽  
Spline Collocation ◽  
Spline Collocation Method ◽  
Orthogonal Spline Collocation

AbstractWe formulate and analyze the Crank-Nicolson Hermite cubic orthogonal spline collocation method for the solution of the heat equation in one space variable with nonlocal boundary conditions involving integrals of the unknown solution over the spatial interval. Using an extension of the analysis of Douglas and Dupont [23] for Dirichlet boundary conditions, we derive optimal order error estimates in the discrete maximum norm in time and the continuous maximum norm in space. We discuss the solution of the linear system arising at each time level via the capacitance matrix technique and the package COLROW for solving almost block diagonal linear systems. We present numerical examples that confirm the theoretical global error estimates and exhibit superconvergence phenomena.

scholarly journals The exponential cubic B-spline collocation method for the Kuramoto-Sivashinsky equation

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 853-861 ◽  
Author(s):  
Ozlem Ersoy ◽  
Idiris Dag
Keyword(s):  
Collocation Method ◽  
Numerical Solutions ◽  
Spline Approximation ◽  
Algebraic Equations ◽  
Spline Collocation ◽  
Fully Integrated ◽  
B Spline ◽  
Spline Collocation Method ◽  
Sivashinsky Equation ◽  
Good Agreement

In this study the Kuramoto-Sivashinsky (KS) equation has been solved using the collocation method, based on the exponential cubic B-spline approximation together with the Crank Nicolson. KS equation is fully integrated into a linearized algebraic equations. The results of the proposed method are compared with both numerical and analytical results by studying two text problems. It is found that the simulating results are in good agreement with both exact and existing numerical solutions.

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