Abstract
The main aim of this work is to consider a meshfree algorithm for solving Burgers’ equation with the quartic B-spline quasi-interpolation. Quasi-interpolation is very useful in the study of approximation theory and its applications, since it can yield solutions directly without the need to solve any linear system of equations and overcome the ill-conditioning problem resulting from using the B-spline as a global interpolant. The numerical scheme is presented, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the time derivative of the dependent variable. Compared to other numerical methods, the main advantages of our scheme are higher accuracy and lower computational complexity. Meanwhile, the algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.
Highlights
Burgers’ equation plays a significant role in various fields, such as turbulence problems, heat conduction, shock waves, continuous stochastic processes, number theory, gas dynamics, and propagation of elastic waves [1,2,3,4,5]
For an interval I = [a, b], we introduce a set of -spaced knots of partition Ω = {x0, x1, . . . , xn}
To show the efficiency of the present method for our problem in comparison with the exact solution, we use the following norms to assess the performance of our scheme: L∞
Summary
Burgers’ equation plays a significant role in various fields, such as turbulence problems, heat conduction, shock waves, continuous stochastic processes, number theory, gas dynamics, and propagation of elastic waves [1,2,3,4,5]. The onedimensional Burgers’ equation first suggested by Bateman [6] and later treated by Burgers [1] has the form. U (a, t) = β1, u (b, t) = β2, t ≥ 0, where β1, β2, and f(x) will be chosen in a later section. Some analytic solutions consist of infinite series, converging very slowly for small viscosity coefficient λ. Finite difference methods were presented to solve the numerical solution of Burgers’ equation in [8,9,10,11]. Finite element methods for the solution of Burgers’ equation were introduced in [12,13,14,15]. Various powerful mathematical methods such as Galerkin finite element method [16, 17], spectral collocation method [18, 19], sinc differential quadrature method [20], factorized diagonal padeapproximation [21], B-spline collocation method [22], and reproducing kernel function method [23] have been used in attempting to solve the equation
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