The numerical solution of the nonlinear Klein–Gordon and Sine–Gordon equations using the Chebyshev tau meshless method

A Space-Time Fully Decoupled Wavelet Integral Collocation Method with High-Order Accuracy for a Class of Nonlinear Wave Equations

“…In this study, we consider a numerical solution of the general nonlinear wave problem αu tt + βu t − γ∆u t − λ∆u + ρ∇u = g(u, ∇u, ∆u) + f , x ∈ Ω, t ≥ 0 u(x, 0) = g 1 (x), u t (x, 0) = g 2 (x), u(x, t) = h(x, t) for x ∈ Γ (1) in which α, β, γ, and ρ are constants; Γ is the boundary of the problem domain Ω = [0, T] 2 ; u is the unknown function of time and space; g can be an arbitrary continuous function; f is the source term; g 1 (x), g 2 (x), and h(x, t) describe the initial and boundary conditions. The general form (1) covers several types of well-known wave problems arising in many scientific and engineering fields, such as nonlinear optics [1][2][3], solid state physics [4][5][6][7][8], and quantum field theory [9][10][11]. For instance, when α = 1, γ = 0, β= 0, λ= 1, ρ = 0, and g(u) = au + bu c with constants a, b, and c, Equation (1) degrades to the so-called nonlinear Klein-Gordon equation:…”

Legendre spectral element method for solving sine-Gordon equation

“…Other numerical methods have also been used to solve the sine-Gordon equation such as Chebyshev tau meshless method [25], meshless method of lines [26], high-accuracy mul-tiquadric quasi-interpolation [27], reduced differential transform method [28], pseudospectral method [29], modified cubic B-spline differential quadrature method [30], modified cubic B-spline collocation method [31], etc.…”

A numerical meshless method of soliton-like structures model via an optimal sampling density based kernel interpolation

“…In a strong form formulation, it is assumed that the approximate unknown function should have sufficient degree of consistency, so that it is differentiable up to the order of the PDEs, and a series of meshless strong form approaches were presented in Refs. [10,[21][22][23][24][25][26][27][28][29][30]. Unfortunately, a strong form of equation is difficult for practical engineering problems that are usually complex in nature.…”