Abstract
In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHughβNagumo equation. Most of the developed methods in the literature for non-linear partial differential equations have not focused on optimizing the time step-size and a very small value must be considered to get accurate approximations. The motivation behind the development of this work is to overcome this trade-off up to much extent using a larger time step-size without compromising accuracy. The optimized hybrid block method considered is proved to be A-stable and convergent. Furthermore, the obtained numerical approximations have been compared with exact and numerical solutions available in the literature and found to be adequate. In particular, without using quasilinearization or filtering techniques, the results for small viscosity coefficient for Burgers equation are found to be accurate. We have found that the combination of the two considered methods is computationally efficient for solving non-linear PDEs.
Highlights
The non-linear PDE model equations arise very frequently in various areas of physical, chemical and biological sciences
A numerical solution is the rescuer in the situation, where it is impossible to find an analytical solution of the problem
Three PDEs containing non-linear terms are considered as they are famous for their numerous practical applications: (i) Burgers equation, (ii) Buckmaster equation, (iii) FitzHughβNagumo equation
Summary
The non-linear PDE model equations arise very frequently in various areas of physical, chemical and biological sciences. The FHN equation is used in the fields of neurophysiology, flame propagation, logistic population growth, nuclear reactor theory, branching brownian motion processes and catalytic chemical reactions (Bhrawy 2013; Wazwaz and Gorguis 2004) This equation has been solved using the Haar wavelet method (Hariharan and Kannan 2010), the homotopy analysis method (Abbasbandy 2008), a polynomial quadrature method (Jiwari et al 2014), the local meshless method (Ahmad et al 2019) and the collocation method using cubic B-splines with SSP-RK54 (Mittal and Tripathi 2015). We derive a one-step hybrid block method to solve first order initial value problems and is further applied to a system of first order differential equations obtained after applying a modified cubic B-splines collocation method (Mittal and Jain 2012) to PDE for space derivatives. The method proposed in this paper has the advantages that provides accurate results by taking few time steps as compared to numerical methods present in the literature, and produces efficient results even for small values of the kinematic coefficient in the case of the Burgers equation
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
