Numerical solution of distributed-order time fractional Klein–Gordon–Zakharov system

An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

“…Therefore, it is absolutely necessary to pay attention to the use of numerical methods. Many authors and researchers have been proposed different numerical methods for the numerical solution of partial differential equations with fractional distributed order operators, such as standard quadrature method [15], implicit finite difference method [16], compact difference method [17], implicit numerical method [18], weighted and shifted Grünwald difference method [19], finite element method [20], Chebyshev collocation method [21], Petrov-Galerkin and spectral collocation methods [22], mid-point quadrature method [23], Legendre wavelets method [24], improved meshless method [25], finite volume method [26] and combination of alternating direction implicit difference, Laplace transform and Hankel transform [27], matrix transfer technique [28], fourth-order compact difference scheme [29], Chebyshev cardinal polynomials [30], and extrapolation method [31]. Recently, the use of wavelets has led to better numerical methods in fractional calculus.…”

New Algorithms for Dealing with Fractional Initial Value Problems

“…In [7], the authors combined fixed-point theory with a set of falling fractional functions in a Banach space to prove the existence and uniqueness of solutions of a class of fractional difference equations. In [8], the authors introduced the distributed-order time fractional Klein-Gordon-Zakharov system by substituting the second-order temporal derivative with a distributed-order fractional derivative. In [9], the authors defined the distributed-order time fractional version of the Schrödinger problem by replacing the first-order derivative in the classical problem with the fractional derivative.…”