Fractional Klein Gordon Equation Research Articles

Fractional Klein–Gordon equation on AdS2+1

We propose a covariant definition of the fractional Klein–Gordon equation (KGE) with long-range interactions independent of the metric of the underlying manifold. As an example we consider the fractional KGE on AdS2+1, computing the explicit kernel representation of the fractional Laplace–Beltrami operator as well as the two-point propagator of the fractional KGE. Our results suggest that the propagator only exists if the mass is small compared to the inverse AdS radius, presumably because the AdS space expands faster with distance as a flat space of the same dimension. Our results are expected to be useful in particular for new applications of the AdS/CFT correspondence within statistical mechanics and quantum information.

"General analytical solution of fractional Klein–Gordon equation in a spherical domain"

"Time-fractional Klein–Gordon equation in a sphere is considered for the case of central sym- metry under the time-variable Dirichlet condition. The time-fractional derivative with the power-law kernel is used. The Laplace transform and convenient transformations of the independent variable and unknown func- tion are used to determine the general analytical solution of the problem in the Laplace domain. In order to obtain the solution in the real domain, the inverse Laplace transforms of two functions of exponential type whose expressions are new in the literature have been determined. The similar solution for ordinary Klein– Gordon equation is a limiting case of general solution but a simpler form for this solution is provided. The convergence of general solution to the ordinary solution and the effects of fractional parameter on this solution are graphically underlined."

Novel Analysis of Fuzzy Fractional Klein-Gordon Model via Semianalytical Method

The current article discusses the new fuzzy iterative transform method, a hybrid methodology based on fuzzy logic and an iterative transformation technique. We demonstrate the consistency of our technique by employing the Caputo derivative under generalized Hukuhara differentiability to construct fractional fuzzy Klein-Gordon equations with the initial fuzzy condition. The series produced result was calculated and compared to the exact result’s recommended equations. Two problems were used to verify our method, with the results approximated in fuzzy form. The upper and lower half of the fuzzy results were approximated in each of the two examples using two distinct fractional orders between zero and one. Because it globalizes the dynamical behavior of the specified equation, it produces all forms of fuzzy results at any fractional order between 0 and 1. Since fuzzy numbers offer their results in a fuzzy form with lower and upper branches, the unknown amount also adds fuzziness. It is crucial to emphasize that the suggested fuzziness method is intended to demonstrate the efficiency and superiority of numerical solutions to nonlinear fractional fuzzy partial differential equations found in complex and physical structures.

Approximate Solution of Nonlinear Time-Fractional Klein-Gordon Equations Using Yang Transform

The algebras of the symmetry operators for the Klein–Gordon equation are important for a charged test particle, moving in an external electromagnetic field in a space time manifold on the isotropic hydrosulphate. In this paper, we develop an analytical and numerical approach for providing the solution to a class of linear and nonlinear fractional Klein–Gordon equations arising in classical relativistic and quantum mechanics. We study the Yang homotopy perturbation transform method (YHPTM), which is associated with the Yang transform (YT) and the homotopy perturbation method (HPM), where the fractional derivative is taken in a Caputo–Fabrizio (CF) sense. This technique provides the solution very accurately and efficiently in the form of a series with easily computable coefficients. The behavior of the approximate series solution for different fractional-order ℘ values has been shown graphically. Our numerical investigations indicate that YHPTM is a simple and powerful mathematical tool to deal with the complexity of such problems.

The Hamiltonian structure and fast energy-preserving algorithms for the fractional Klein-Gordon equation

In this paper, an energy-preserving difference scheme is proposed for solving the space fractional Klein-Gordon equation based on the Hamiltonian form of the equation. First, we study some properties of the fractional Laplacian operator and reformulate the equation as an infinite-dimension canonical Hamiltonian system. Then, we use the fractional centered difference formula to discrete the equation and derive a semi-discrete Hamiltonian system that can conserve the semi-discrete energy. Subsequently, a fully-discrete scheme is obtained by applying the averaged vector field method to the semi-discrete system. The resulting scheme's unconditional point-wise error estimate is discussed by using the “cut-off” technique. Furthermore, a fast solver is presented to reduce the computational complexity of the scheme based on the fast Fourier transformation technique in practical computation. Finally, some numerical experiments are displayed to demonstrate the efficiency and conservation of the constructed scheme in long time simulation.

Two-dimensional Gegenbauer wavelets for the numerical solution of tempered fractional model of the nonlinear Klein-Gordon equation

In the present article, we study a new version of the physical model, namely the Klein-Gordon equation involved with tempered fractional derivative. This model is numerically simulated for the tempered and the non-tempered fractional derivatives. Two-dimensional Gegenbauer Wavelets are implemented for the solution of the tempered fractional Klein-Gordon equation. The proposed method is a combination of different operational matrices of the two-dimensional Gegenbauer wavelets and the collocation nodes. The applicability and the accuracy of the suggested method has been examined by four illustrative examples and the obtained simulation results are in good agreement with the exact solution. The maximum absolute error, L2 norm error, and the root mean square error have been evaluated to analyze the precision of the proposed method. This method gives better and comparable results to the existing methods given in the literature.

Numerical solutions of nonlinear time fractional Klein-Gordon equation via natural transform decomposition method and iterative Shehu transform method

The time-fractional Klein-Gordon equation was analyzed in this article using the natural transform decomposition method (NTDM) and the iterative Shehu transform method (ISTM) with a singular kernel derivative. We employed natural transform (NT) and Shehu transform (ST), followed by inverse natural transform and inverse Shehu transform respectively, to obtain the solution. The numerical simulations are provided to ensure that the methods under consideration are efficient. The current framework captures the behaviour of the obtained findings for various fractional orders. The findings of this study indicate that the proposed methods are effective and reliable in analyzing fractional differential equations.

Two Fibonacci operational matrix pseudo-spectral schemes for nonlinear fractional Klein–Gordon equation

This paper is devoted to developing spectral solutions for the nonlinear fractional Klein–Gordon equation. The typical collocation method and the tau method are employed for obtaining the desired numerical solutions. In order to do this, a new operational matrix of fractional derivatives of Fibonacci polynomials is established. The idea behind the derivation of this matrix is based on utilizing the connection formula between the Fibonacci and Chebyshev polynomials. The introduced operational matrix is used along with the weighted residual quadrature spectral method and the collocation method to convert the nonlinear fractional Klein–Gordon equation into a system of algebraic equations. By solving the resulting system, we obtain a semi-analytic solution. The convergence and error analysis of the method are discussed. Some numerical results and discussions are presented aiming to illustrate the wide applicability and accuracy of the proposed algorithms.

Numerical Solution of Space-Time Fractional Klein-Gordon Equation by Radial Basis Functions and Chebyshev Polynomials

Numerical Solution of Space-Time Fractional Klein-Gordon Equation by Radial Basis Functions and Chebyshev Polynomials

Fractional Klein-Gordon equation with singular mass. II: hypoelliptic case

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solution are also proved, under suitable considerations. This extends and improves the results obtained in the first part [Altybay et al. Fractional Klein-Gordon equation with singular mass. Chaos Solitons Fractals. 2021;143:Article ID 110579] which was devoted to the classical Euclidean Klein-Gordon equation.

Space–time fractional Klein-Gordon equation: Symmetry analysis, conservation laws and numerical approximations

In this paper the symmetry operators of the fractional Klein-Gordon (KG) equation are found. The generalized fractional Noether’s theorem is used in order to find the conservation laws of the considered equation. Finally, the Chebyshev spectral collocation method is extended to space–time-fractional case for giving some numerical results and conservation laws for the equation.

A Linearised Three-Point Combined Compact Difference Method with Weighted Approximation for Nonlinear Time Fractional Klein-Gordon Equations

A Linearised Three-Point Combined Compact Difference Method with Weighted Approximation for Nonlinear Time Fractional Klein-Gordon Equations

Fractional Klein-Gordon equation with singular mass

We consider a space-fractional wave equation with a singular mass term depending on the position and prove that it is very weak well-posed. The uniqueness is proved in some appropriate sense. Moreover, we prove the consistency of the very weak solution with classical solutions when they exist. In order to study the behaviour of the very weak solution near the singularities of the coefficient, some numerical experiments are conducted where the appearance of a wall effect for the singular masses of the strength of δ2 is observed.

An Extension of the Double G ′ / G , 1 / G -Expansion Method for Conformable Fractional Differential Equations

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic (CRWP) equation, nonlinear time fractional SharmaTassoOlver (STO) equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the G ′ / G , 1 / G -expansion method is found to be ensuing, effective, and capable to provide functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the G ′ / G , 1 / G -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs.

An Efficient Computational Method for the Time-Space Fractional Klein-Gordon Equation

In this paper, we present a computational method to solve fractional Klein-Gordon equation (FKGE). The proposed technique is the grouping of orthogonal polynomial matrices and collocation method. The benefit of the computational method is that it reduces the FKGE into a system of algebraic equations which makes the problem straightforward and easy to solve. The main reason for using this technique is high accuracy and low computational cost compared to some other methods. The main solution behaviours of these equations are due to fractional orders which are explained graphically.Numerical results obtained by proposed computational method are also compared with the exact solution. The results obtained by suggested technique reveals that the method is very useful for solving FKGE.

DEVELOPMENT AND ANALYSIS OF NEW APPROXIMATION OF EXTENDED CUBIC B-SPLINE TO THE NONLINEAR TIME FRACTIONAL KLEIN–GORDON EQUATION

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with [Formula: see text]-weighted scheme is utilized to obtain the solution. The method is shown to be unconditionally stable and convergent. Numerical examples indicate that the obtained results compare well with other numerical results available in the literature.

A linearized high-order Galerkin finite element approach for two-dimensional nonlinear time fractional Klein-Gordon equations

In this paper, we propose a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations with a cubic nonlinear term. The employed time discretization is a weighted combination of the L2 − 1σ formula introduced recently by Lyu and Vong (Numer. Algorithms 78(2):485–511, 2018), Galerkin finite element method is used for the spatial discretization, and the cubic nonlinear term is handled explicitly. Using mathematical induction, we prove that the numerical solution is bounded and the fully discrete scheme is convergent with second-order accuracy in time. In numerical experiments, some problems with both smooth and non-smooth exact solutions are considered.

A Meshless Solution for the Variable-Order Time Fractional Nonlinear Klein–Gordon Equation

In this paper, we study the variable-order (V-O) time fractional Klein–Gordon equation which widely appears in the various fields of engineering and mathematical physics. The numerical method which we have used for solving this equation is based on a combination of the radial basis functions (RBFs) method and finite difference scheme. In the first stage the V-O time-dependent derivative is discreticized, and then we approximate the solution by the RBFs. The main goal is to show that the collocation method based on RBFs is suitable for solving V-O fractional differential equations. The applicability of the proposed method is investigated by solving some numerical examples. The obtained results show that the proposed approach is very efficient and accurate. Also, the effect of replacing V-O fractional derivative of order $$\alpha (x,t)$$ with its approximations on the behavior of approximate solutions relative to the exact solution is investigated numerically.

A new analysis for Klein-Gordon model with local fractional derivative

This work adopts Yang’s local fractional derivative to define the fractional Klein-Gordon equation in a fractal space or microgravity space. The variational principle of local fractional Klein-Gordon equation is successfully established via fractional semi-inverse transform method and the classical He’s variational iteration method (HVIM) is used to obtain its approximate analytical solution.

An efficient numerical technique for variable order time fractional nonlinear Klein-Gordon equation

This paper studies the variable order time fractional nonlinear Klein-Gordon equation (V-TFNKGE). An optimization method using the Caputo type definition and a set of basis functions, namely the generalized polynomials (GP), is proposed. The solution is expanded in terms of the GP with unknown free coefficients and control parameters. First, a new variable order time fractional operational matrix of the Caputo type for the GP is derived. Then, based on a operational matrix and the Lagrange multipliers, an optimization process achieves the approximate solution. Additionally, the convergence of the proposed method is analyzed and two numerical examples illustrate its good performance.