Abstract
This work concerns for solving of coupled Burgers’ equations (CBEs) in 2D and 3D via DQM based on cubic trigonometric B-spline (CTB) shape functions. In the method, the shape functions are modified and used for the integration of space derivative. Consequently, the CBEs are transformed into the integral equations. These integral equations are solved by an “optimal strong stability-preserving Runge-Kutta method (SSP-RK54)”. Three examples are taken for analysis. The assessment of the present results are done with a number of already presented results in the literature. We initiated that the present method generates more precise results. Straightforward algorithm, little amount of computational cost and less error norms are the major achievements of the method. Therefore, the present method possibly will be very valuable optional method for the computation of nonlinear PDEs. Moreover, the analysis of method’s stability is also done.
Highlights
The coupled Burgers’ equations (CBEs) are transformed into the form of integral equations
In this work, the CBEs in two and three dimensions are solved via DQM based on modified form of TCB shape functions
These integral equations are solved by SSP-RK54 method
Summary
There is a significant role of the nonlinear Burgers’ equations to model the fluid dynamics problems to study turbulence, shock wave structures, mass transport, etc. The Burgers’ equation may be treated as a straightforward mathematical model, which is generally used for a range of applications. Bahadir (2003) used a fully implicit FDM to solve 2D CBEs while Srivastava et al (2013b), Srivastava et al (2013c) and Shukla et al (2014) used implicit, implicit logarithm FD methods and modified cubic B-spline DQM respectively. Srivastava et al (2013a) generated an exact solution of 3D CBEs while Shukla et al (2016) employed MCB-DQM to solve it. Baishya (2019) reproduced the advantage of a new technique using Hermite orthogonal basis elements to solve DEs. Vaid and Arora (2019) presented a collocation method based on trigonometric cubic B-spline functions to approximate a singular perturbed delay DE while. The solutions of CBEs in 2D and 3D are obtained via DQM based on a modified form of CTB shape functions. Where c 1 i a 1 , a 1 ,..., a 1 T , and A is the coefficient matrix which is given by i1 i2
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