Abstract
In this paper, a Crank–Nicolson finite difference scheme based on cubic B-spline quasi-interpolation has been derived for the solution of the coupled Burgers equations with the Caputo–Fabrizio derivative. The first- and second-order spatial derivatives have been approximated by first and second derivatives of the cubic B-spline quasi-interpolation. The discrete scheme obtained in this way constitutes a system of algebraic equations associated with a bi-pentadiagonal matrix. We show that the proposed scheme is unconditionally stable. Numerical examples are provided to verify the efficiency of the method.
Highlights
IntroductionBurgers equations are coupled partial differential equations which are capable of describing realitic polydispersive supensions
Introduction e coupled Burgers equations are coupled partial differential equations which are capable of describing realitic polydispersive supensions. e coupled Burgers equation perdicts an interesting phenomenon, which is called phase shifts [1]. is equation is one of the fundamental models in fluid mechanics and arises in gas dynamics, chromatography, and flood waves in rivers [2]. e coupled viscous Burges equation is given by ut − uxx + ηuux + α1(uv)x 0, vt − vxx + ηvvx + α2(uv)x 0, x ∈ [a, b], t ∈ [0, T], x ∈ [a, b], t ∈ [0, T], (1)
Spline is a special function defined piecewise by polynomials. e spline approximation first appeared in a paper by Schoenberg [4]
Summary
Burgers equations are coupled partial differential equations which are capable of describing realitic polydispersive supensions. E coupled Burgers equation perdicts an interesting phenomenon, which is called phase shifts [1]. Spline is a special function defined piecewise by polynomials. Spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. Applications of spline function in fractional partial differential equations can be found in [5,6,7,8,9,10,11,12,13,14,15]. We consider the following coupled Burgers equation with time fractional derivative: zcu z2u zu z(uv) ztc − zx2 + ηu zx + α1 zx q1, (4)
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