- Open Access
Numerical analysis for the Klein-Gordon equation with mass parameter
© The Author(s) 2017
Received: 21 June 2017
Accepted: 7 September 2017
Published: 16 September 2017
A numerical analysis of the well-known linear partial differential equation describing the relativistic wave is presented in this work. Three different operators of fractional differentiation with power law, exponential decay law and Mittag-Leffler law are employed to extend the Klein-Gordon equation with mass parameter to the concept of fractional differentiation. The three models are solved numerically. The stability and the convergence of the numerical schemes are investigated in detail.
The concept of differentiation as a convolution of some natural laws, like power law, exponential decay law and Mittag-Leffler law, is in fashion nowadays due to its ability as a mathematical tool to replicate the observed facts. These three major definitions are constructed as the convolution of derivative of a given differentiable function and power law, exponential decay law or Mittag-Leffler law [1–5]. This version is recognized as Caputo type but it is sometime criticized because it does not have an associated anti-derivative. The original version is the derivative of a convolution of a non-differentiable continuous function and power law, exponential decay law or Mittag-Leffler law. This last version is known as Riemann-Liouville approach that is obtained via the Abel integral and is considered as a real derivative with fractional order [3–5]. Nowadays research using many concepts of differentiations has been carried out and some good predictions have been obtained [6–11]. This derivative has been used in many research papers for theoretical purposes, and sometimes it is used to model some physical problems, but its numerical approximation is not popular in the literature. Recently, Atangana and Gomez did a work devoted to the derivation of the numerical approximation of the three Riemann-Liouville types of fractional derivatives . A great advantage of the fractional differentiation with non-singular and non-local kernel suggested by Atangana and Baleanu is that, when using the Laplace transform, we obtain the usual initial condition unlike Riemann-Liouville. In addition to this, the kernel is able to portray a full memory as there is no singularity associated to it. The kernel is more natural and is a combination of power law and exponential decay law which give this kernel the ability to describe phenomena with non-local fading memory [13–17]. In their work, they suggested the numerical approximation of these derivatives for first and second order. In this work, we analyze numerically the well-known Klein-Gordon equation with mass parameter, where the second time derivative will be replaced by Riemann-Liouville, Caputo-Fabrizio in Riemann-Liouville sense and Atangana-Baleanu in Riemann-Liouville sense fractional derivatives. We shall recall that the Klein-Gordon equation with mass parameter is a linear partial differential equation used to describe a relativistic wave, closely related to the Schrödinger equation [18–20].
2 Fractional order derivatives in Riemann-Liouville sense
In this section, we present the fractional order definitions in Riemann-Liouville sense.
3 Numerical analysis for the Klein-Gordon equation with mass parameter
3.1 Second approximation of Riemann Liouville approach and a stability analysis of the numerical scheme
3.1.1 Stability analysis of the numerical scheme
The Crank-Nicholson scheme for solving the Klein-Gordon equation with second order Riemann-Liouville is stable if inequality (22) is satisfied.
3.2 Second approximation of the Caputo-Fabrizio derivative in Riemann-Liouville sense and a stability analysis of the numerical scheme
3.2.1 Stability analysis of the numerical scheme for the Caputo-Fabrizio derivative in Riemann-Liouville sense
The Crank-Nicholson scheme for solving the Klein-Gordon equation with the second order Caputo-Fabrizio derivative in Riemann-Liouville sense is stable if inequality (38) is satisfied.
3.3 Second approximation of the Atangana-Baleanu derivative in Riemann-Liouville sense and a stability analysis of the numerical scheme
3.3.1 Stability analysis of the numerical scheme for the Atangana-Baleanu derivative in Riemann-Liouville sense
The Crank-Nicholson scheme for solving the Klein-Gordon equation with the second order Atangana-Baleanu derivative in Riemann-Liouville sense is stable if inequality (50) is satisfied.
In this paper the Klein-Gordon equation with mass parameter was considered. The time second derivative was replaced by three different fractional derivatives, namely, Riemann-Liouville power law fractional derivative, Riemann-Liouville exponential law fractional derivative and finally Riemann-Liouville Mittag-Leffler law fractional derivative. A second approximation of each derivative was presented and used to solve the corresponding model. In detail, the stability and convergence analysis of each numerical scheme were investigated.
The authors extend their sincere appreciations to the Deanship of Science Research at King Saud University for funding this prolific research group PRG-1437-35.
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