Fractional Klein Gordon Equation Research Articles

Existence of nontrivial solutions for fractional Klein–Gordon equation coupled with Born–Infeld theory with critical exponents

In this paper, we investigate the existence of solutions for a fractional Klein–Gordon equation coupled with Born–Infeld theory when the nonlinearity exhibits critical growth. By overcoming the lack of compactness, we obtain that there is a radially symmetric solution by using variational methods.

A numerical approach for solving nonlinear fractional Klein–Gordon equation with applications in quantum mechanics

Abstract In this paper, we propose a numerical approach for solving the nonlinear fractional Klein–Gordon equation (FKGE), a model of significant importance in simulating nonlinear waves in quantum mechanics. Our method combines the Bernoulli wavelet collocation scheme with a functional integration matrix to obtain approximate solutions for the proposed model. Initially, we transform the main problem into a system of algebraic equations, which we solve using the Newton–Raphson method to extract the unknown coefficients and achieve the desired approximate solution. To theoretically validate our method, we conduct a comprehensive convergence analysis, demonstrating its uniform convergence. We perform numerical experiments on various examples with different parameters, presenting the results through tables and figures. Our findings indicate that employing more terms in the utilized techniques enhances accuracy. Furthermore, we compare our approach with existing methods from the literature, showcasing its performance in terms of computational cost, convergence rate, and solution accuracy. These examples illustrate how our techniques yield better approximate solutions for the nonlinear model at a low computational cost, as evidenced by the calculated CPU time and absolute error. Additionally, our method consistently provides better accuracy than other methods from the literature, suggesting its potential for solving more complex problems in physics and other scientific disciplines.

Relativistic energy spectra of diatomic molecules through the solutions of general conformable fractional (GCF) Klein-Gordon equation for Hulthen potential with position-dependent mass (PDM) and the thermodynamic properties

In this study, we investigate the thermodynamic properties of some diatomic molecules modelled by Hulthen potential with PDM. The GCF Klein-Gordon equation in the GCF cylindrical coordinate for the Hulthen potential and the PDM function is solved by using the GCF Laplace method to obtain the GCF energy equation and GCF wave function. The effect of the fractional order, quantum number, potential parameter, and PDM parameter on the energy eigenvalue of the H2, LiH, and HF diatomic molecules is discussed through the corresponding graphs computed using MATLAB software. The results show that the fractional order, quantum number, and potential parameter cause the energy spectrum to increase. In addition, the thermodynamic properties are examined by calculating the partition function. Based on the results, internal energy tends to be constant at lower temperatures, and the specific heat capacity decreases with increasing β . Whereas free energy and entropy increase monotonically with increasing β . To validate our findings, we compare our results with those obtained in the literature.

An efficient technique for time fractional Klein-Gordon equation based on modified Laplace Adomian decomposition technique via hybridized Newton-Raphson Scheme arises in relativistic fractional quantum mechanics

An efficient technique for time fractional Klein-Gordon equation based on modified Laplace Adomian decomposition technique via hybridized Newton-Raphson Scheme arises in relativistic fractional quantum mechanics

On the construction of various soliton solutions of two space-time fractional nonlinear models

Abstract In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein–Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters.

Fast Two-Grid Finite Element Algorithm for a Fractional Klein- Gordon Equation

In this article, we propose a spatial two-grid finite element algorithm combined with a shifted convolution quadrature (SCQ) formula for solving the fractional Klein-Gordon equation. The time direction at tn − θ is approximated utilizing a second-order SCQ formula, where θ is an arbitrary constant. The spatial discretization is performed using a two-grid finite element method involving three steps: calculating the numerical solution by solving a nonlinear system iteratively on the coarse grid, obtaining the interpolation solution based on the computed solutions in the first step, and solving a linear finite element system on the fine grid. We present a numerical algorithm, validate the two-grid finite element method’s effectiveness, and demonstrate the computational efficiency for our method by the comparison of the computing results between the two-grid finite element method and the standard finite element method.

Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation

<abstract><p>In this study, we investigate the fundamental properties of ($ 3+1 $)-$ D $ Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both $ 2D $ and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields.</p></abstract>

Traveling Wave Solutions of the Conformable Fractional Klein–Gordon Equation With Power Law Nonlinearity

This article investigates the construction of new traveling wave solutions for the conformable fractional Klein–Gordon equation, which is a well‐known mathematical and physical model that can be used to explain spinless pion and de Broglie waves. In order to accomplish this task, a classic and effective analysis method, namely, the extended tanh–coth method, was utilized. By introducing appropriate transformations, the conformable fractional Klein–Gordon equations are reduced to ordinary differential equations, and then the solutions with hyperbolic function form are obtained. In addition, the effect of fractional parameters on waveform is analyzed by drawing two‐ and three‐dimensional graphics. These results contribute to a deeper understanding of the dynamics of the conformable fractional Klein–Gordon equation. The research in this article also indicates that the extended tanh–coth method is a straightforward and concise technique that has the potential to be applicable to many other conformable fractional partial differential equations that appear in mathematical physics.

Analysis of the n-Term Klein-Gordon Equations in Cantor Sets

The effectiveness of the local fractional reduced differential transformation method (LFRDTM) for the approximation of the solution related to the extended n-term local fractional Klein-Gordon equation is the main aim of this paper in which fractional complex transform and local fractional derivative have been employed to analyze the n-term Klein-Gordon equations, and Cantor sets. The proposed method, along with the existence of the solutions demonstrated through some examples, provides a powerful mathematical means in solving fractional linear differential equations. Considering these points, the paper also provides an accurate and effective method to solve complex physical systems that display fractal or self-similar behavior across various scales. In conclusion, the fractional complex transform with the local fractional differential transform method has been proven to be a robust and flexible approach towards obtaining effective approximate solutions of local fractional partial differential equations.

A numerical study on the nonlinear fractional Klein–Gordon equation

This article helps to develop a numerical approach based on Fibonacci wavelets for solving fractional Klein-Gordan equations (FKGEs). The FKGEs are solved with Caputo-type fractional derivative. Using the definition of Fibonacci wavelets, we constructed the operational matrices of integration. These operational matrices of integration led to the development of the collocation method called the Fibonacci wavelet collocation method (FWCM). This method transforms the given nonlinear partial differential equation into a system of nonlinear algebraic equations using collocation points to determine the unknown coefficients. By substituting the unknown coefficients in the method, we obtained the numerical solution of the present approach. We furnish the different error norms for the present technique. The obtained results are compared with the Clique polynomial method. These findings demonstrate the computational attractiveness, efficiency, effectiveness, reliability, and robustness of the proposed method for addressing a variety of physical models in science and engineering.

Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

Spectral Element Method for Fractional Klein–Gordon Equations Using Interpolating Scaling Functions

On the standing wave in coupled fractional Klein–Gordon equation

Abstract The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave.

Improved Uniform Error Bounds of Exponential Wave Integrator Method for Long-Time Dynamics of the Space Fractional Klein-Gordon Equation with Weak Nonlinearity

Improved Uniform Error Bounds of Exponential Wave Integrator Method for Long-Time Dynamics of the Space Fractional Klein-Gordon Equation with Weak Nonlinearity

A study on fractional tumor-immune interaction model related to lung cancer via generalized Laguerre polynomials

BackgroundCancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies.MethodsIn this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data.ResultsWe propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results.ConclusionThe findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems.

A Fast θ Scheme Combined with the Legendre Spectral Method for Solving a Fractional Klein–Gordon Equation

In the current work, a fast θ scheme combined with the Legendre spectral method was developed for solving a fractional Klein–Gordon equation (FKGE). The numerical scheme was provided by the Legendre spectral method in the spatial direction, and for the temporal direction, a θ scheme of order O(τ2) with a fast algorithm was taken into account. The fast algorithm could decrease the computational cost from O(M2) to O(MlogM), where M denotes the number of time levels. In addition, correction terms could be employed to improve the convergence rate when the solutions have weak regularity. We proved theoretically that the scheme is unconditionally stable and obtained an error estimate. The numerical experiments demonstrated that our numerical scheme is accurate and efficient.

Analytical solution for the dynamics and optimization of fractional Klein-Gordon equation: an application to quantum particle.

Klein-Gordon equation characterizes spin-particles through neutral charge field within quantum particle. In this context, fractionalized Klein-Gordon equation is investigated for the comparative analysis of the newly presented fractional differential techniques with non-singularity among kernels. The non-singular and non-local kernels of fractional differentiations have been employed on Klein-Gordon equation for the development of governing equation. The analytical solutions of Klein-Gordon equation have been traced out by fractional techniques by means of Laplace transforms and expressed in terms of series form and gamma function. The data analysis of fractionalized Klein-Gordon equation is observed for Pearson's correlation coefficient, probable error and regression analysis. For the sake of comparative analysis of fractional techniques, 2D sketch, 3D pie chart, contour surface with projection and 3D bar sketch have been depicted on the basis of embedded parameters. Our results suggest that varying frequency has reversal trends for quantum wave and de Broglie wave.

NEW FRACTAL SOLITON SOLUTIONS FOR THE COUPLED FRACTIONAL KLEIN–GORDON EQUATION WITH β-FRACTIONAL DERIVATIVE

In this paper, we derive some novel fractal soliton solutions of the coupled fractional Klein–Gordon equation with the [Formula: see text]-fractional derivative via two efficient methods, which are fractal functional variable method and fractal sech-function method. The two new mathematical schemes are quite concise and effective, and then numerous new exact fractal soliton solutions of other nonlinear fractal evolution equations can be obtained. Finally, some 3D figures are sketched to describe these new fractal soliton solutions.

Numerical simulation for generalized space-time fractional Klein–Gordon equations via Gegenbauer wavelet

Abstract This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

Fractal–fractional Klein–Gordon equation: A numerical study

In this work, we solve a new kind of the fractional Klein–Gordon problem numerically. In fact, we study the mentioned problem under fractal–fractional operator with the Riemann–Liouville frame with Mittag-Leffler kernel. We use an efficient operational matrix (OM) technique employing the shifted Chebyshev cardinal functions (CCFs) to get the approximate solutions of the considered equation. Moreover, an OM for the considered derivative is gained using the basic functions. To get the approximate solutions of the presented equation we change the principal model into an algebraic system. To see the numerical results of the problem, we provide the related graphs of the exact and approximate solutions along with the absolute errors of each example. The accuracy and reliability of the numerical solutions can be found form the figures. Also, for each example Tables displaying the values of solutions and errors are reported.

Analysis of local fractional Klein-Gordon equations arising in relativistic fractal quantum mechanics

In this paper, we present the implementation of a local fractional homotopy perturbation method pertaining to the local fractional natural transform (LFNT) operator for local fractional Klein-Gordon equations (LFKGEs) under distinct fractal initial conditions. The Klein-Gordon equation is a relativistic wave equation which estimates the nature and movement of zero spin particles at high velocities and energies resembling the lightspeed. This equation admits the laws of special relativity and describes zero spin particles with relativistic energy. This work also examines uniqueness and convergence analyses of the obtained solution for a general local fractional partial differential equation via applied method. The numerical simulations are presented for each LFKG equation on the Cantor set. The computational process ensures that the copulation of local fractional homotopy perturbation method and LFNT operator provides series solutions in closed form with fast convergence for local fractional Klein-Gordon equation in a very lucid way. Furthermore, the results have also been compared with solutions obtained by other methods.