Solution of Benjamin-Bona-Mahony-Burgers equation using collocation method with quintic Hermite splines

2020 ◽  
Vol 154 ◽  
pp. 1-16 ◽  
Author(s):  
Shelly Arora ◽  
Rajiv Jain ◽  
V.K. Kukreja
Keyword(s):  
Collocation Method ◽  
Burgers Equation ◽  
Hermite Splines

scholarly journals The Application of the Collocation Method Using Hermite Cubic Splines to Nonlinear Transient One-Dimensional Heat Conduction Problems

1975 ◽  
Vol 97 (4) ◽  
pp. 562-569 ◽  
Author(s):  
T. C. Chawla ◽  
G. Leaf ◽  
W. L. Chen ◽  
M. A. Grolmes
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Finite Difference ◽  
Differential Equations ◽  
Collocation Method ◽  
Gaussian Quadrature ◽  
Time Step ◽  
One Dimensional ◽  
Hermite Splines ◽  
Nonlinear Transient

A collocation method for the solution of one-dimensional parabolic partial differential equations using Hermite splines as approximating functions and Gaussian quadrature points as collocation points is described. The method consists of expanding dependent variables in terms of piece-wise cubic Hermite splines in the space variable at each time step. The unknown coefficients in the expansion are obtained at every time step by requiring that the resultant differential equation be satisfied at a number of points (in particular at the Gaussian quadrature points) in the field equal to the number of unknown coefficients. This collocation procedure reduces the partial differential equation to a system of ordinary differential equations which is solved as an initial value problem using the steady-state solution as the initial condition. The method thus developed is applied to a two-region nonlinear transient heat conduction problem and compared with a finite-difference method. It is demonstrated that because of high-order accuracy only a small number of equations are needed to produce desirable accuracy. The method has the desirable characteristic of an analytical method in that it produces point values as against nodal values in the finite-difference scheme.

scholarly journals Wavelet-Based Homotopy Analysis Method for Nonlinear Matrix System and Its Application in Burgers Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xing Ruyi
Keyword(s):  
Collocation Method ◽  
Homotopy Analysis Method ◽  
Burgers Equation ◽  
Nonlinear Problems ◽  
Nonlinear Pdes ◽  
Analysis Method ◽  
Precise Integration ◽  
Wavelet Collocation Method ◽  
Homotopy Analysis ◽  
Shannon Wavelet

To generalize the homotopy analysis method (HAM) to multidegree-of-freedom nonlinear system, the adaptive precise integration method (APIM) is introduced into the HAM, with which the almost exact value of the exponential matrix can be obtained. Combining the interval interpolation wavelet collocation method, HAM-based APIM can be employed to solve the nonlinear PDEs. As an example, Burgers equation is spatially discretized by the interval quasi-Shannon wavelet collocation method and solved by the proposed method to illustrate the effectiveness and great potential of the homotopy analysis method in nonlinear problems.

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