An efficient technique for nonlinear time-fractional Klein–Fock–Gordon equation
Introduction
The concept of fractional calculus (FC) is debuted over 324 years ago, but recently it magnetized the attention of many scholars working in diverse areas of science and technology. From the last few decades, FC becomes the most powerful tool to analyse and describe the nonlinear complex phenomena, due to its favourable properties such as analyticity, hereditary, nonlocality and memory effect. The concept of derivative with arbitrary order has been developed due to the complexities connected to the phenomenon with heterogeneities. The differential operators with non-integer order are capable to capture complex media having diffusion mechanism. Due to the rapid growth of computer and mathematical techniques and software's, many authors initiated to work on the fractional calculus with fundamentals and its applications.
There have been numerous pioneering orientations available for various definitions of fractional calculus, and which prescribed the foundation for fractional calculus [1], [2], [3], [4], [5], [6]. FC has been related to practical projects and it has been applied to chaos theory [7], biomathematics [8], optics [9], financial models [10], and other fields [11], [12], [13], [14], [15]. The solution for fractional differential equations describing these phenomena play a vibrant role in unfolding the behaviour of complex problems arises in real life.
On the other hand, the physicists Klein, Fock and Gordon derived the equation which describes relativistic electrons, called Klein–Fock–Gordon (KFG) equation. This equation is also known as relativistic wave equation and which is related to Schrodinger equation and it is a quantized version of the relativistic energy–momentum relation. The theoretical relevance of KFG equation is similar to that of the Dirac equation. In the present framework, we consider fractional Klein–Fock–Gordon (FKFG) equation. In order to incorporate the effect of hereditary, nonlocality and memory effect, we modified the time-derivative with the Caputo fractional derivative. Since, the small variation in the physical behaviour leads to great flexibility in phenomena, and it helps us to understand and capture the important nature of proposed model. The extent to which a fractional order model will span multiple scales is based on an underlying presumption that fractional derivatives can limn or capture salient features of complex phenomena. Moreover, the classical derivatives are local in nature, i.e., using classical derivatives we can describe changes in a neighbourhood of a point but using fractional derivatives we can describe changes in an interval. This property makes fractional order derivatives suitable to simulate more physical phenomena. Further, the integer order derivatives are narrow subset of fractional derivatives. Therefore, in order to introduce the above said futures in the considered problem and demonstrate the efficiency of the proposed technique, we consider the FKFG equation of the formsubjected to initial conditionswhere a and b are real constants and n is a positive integer. The KFG equation emerged in distinct physical phenomenon such as nonlinear optics, quantum field theory, the interaction of solitons in collision less plasma, condensed matter physics [16], dispersive wave-phenomena [17] and others.
Recently, large numbers of advanced techniques are developed to study the classical and fractional order differential system. In connection with this, Chinese Mathematician Liao Shi-Jun proposed homotopy analysis method (HAM) [18]. HAM has effectively and profitably employed to derive the solution for nonlinear complex problems exist in science and engineering without linearization and perturbation. But, HAM necessitates more time for computational work and huge computer memory. Hence, there is a necessity of the combination of this technique with previously existing transform algorithms.
In this paper, we analyse and find the solution for the time-fractional KFG equation by employing q-HATM. The proposed technique is recently introduced by Singh et al. [19] by using the concept of q-HAMwith Laplace transform. Since q-HATM is a modified technique of HAM, it does not necessitate linearization, perturbation or discretization, and in addition, it will reduce vast mathematical computations and is free from obtaining difficult physical parameters and polynomials. Recently, due to its efficacy and consistency, q-HATM is extremely employed by many authors to interpret results for numerous classes of nonlinear problems [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. The future technique provides us extremely large freedom to choose equation type of linear sub-problems, initial guess, and base function of the solution; due to this, the complicated nonlinear differential equations can often be solved in a simple way. The novelty of the proposed technique is that, it offers a simple procedure to find the solution, the huge convergence region and non-local effect in the obtained solution. The proposed scheme manipulates and controls the obtained solution, which quickly converges to the analytical solution in a small acceptable region, but which is not possible by any other traditional techniques. Moreover, it logically contains results of some traditional methods such as Adomian decomposition method (ADM), the δ-expansion method, HPM, RDTM and q-HAM, so that it has the prodigious generality. The future algorithm can reduce the computation of the work and time as weigh against the other classical methods while preserving the great accuracy.
The numerical and analytical solutions for the non-linear differential equations with integer and fractional order have essential importance. Since the majority of nonlinear phenomena are modelled by the aid of differential and integral equations of fractional order. There is a number of techniques are available in the literature to study these phenomena. The KFG equation is studied through distinct techniques like, variational iteration method [33], homotopy analysis method (HAM) [34], differential transform method (DTM) [35], modified differential transform method [36], and many others [37], [38], [39]. In the above cited research papers, the authors present only the numerical solution and they do not conduct a numerical simulation for all cases and they do not present the behaviour of the solution obtained for Klein–Fock–Gordon equation with the aid of fractional order. Hence in the present framework, we conduct the numerical simulation for each case and present the behaviour of obtained solution for diverse fractional order. This analysis may help the researcher to understand the nature of complex phenomena by the help of fractional order.
The rest of the paper is arranged as: In Section 2, the preliminaries of fractional order integrals and derivatives and Laplace transform is presented. In Section 3, concerned with the fundamental procedure of the proposed algorithm for fractional Klein–Fock–Gordon equation. In Section 4, the convergence analysis of the technique is presented. In Section 5, the solution for fractional KFG equation is investigated. In Sections 6 and 7, the numerical simulation and discussions, and concluding remarks are cited respectively.
Section snippets
Preliminaries
In this segment, we present the fundamental notion of FC and Laplace transform, which are essential in the present framework.
Definition 1 The Riemann–Liouville integral of a function f(t) ∈ Cδ(δ ≥ −1) having fractional order (μ > 0) is presented as follows
Definition 2 The Caputo fractional order derivative of is defined as follows:
Definition 3 Let be a Caputo fractional derivative, then the Laplace transform (LT
Proposed algorithm for time-fractional Klein–Fock–Gordon equation
In this segment, we applied the fundamental solution procedure of the proposed algorithm [40], [41], [42] for FKFG equitation. First, we consider a nonlinear FKFG equationsubjected to the initial conditionswhere denotes the Caputo fractional derivative of the function v(x, t). Here, v(x, t) is a bounded function (i.e., for a number ε > 0 we have ‖v‖ ≤ P), Now, by employing the Laplace transform on Eq. (5) and
Convergence analysis of the technique
Here, the convergence analysis has been present for FKFG equation using the proposed algorithm.
Theorem 1 (Uniqueness theorem) The obtained solution for the FKFG Eq. (5) with the help of q-HATM is unique wherever 0 < λ < 1, where .
Proof The solution for FKFG equation defined in Eq. (5) is presented aswhere If possible, let v and v* be the two distinct solutions for the FKKP
Solution for fractional KFG equation
In this section, we consider three different examples in ordered to validate the efficiency and applicability of the considered technique.
Example 5.1 Consider the nonlinear FKFG equation at a = 0, b = −1 and n = 2, then Eq. (1) reduced tosubjected to the initial conditions By using the proposed algorithm, we have
On solving Eq. (26) we get the iterative terms of vm(
Numerical results and discussion
In this section, we present the numerical simulation for time-fractional Klein–Fock–Gordon equation with the help of the proposed technique. In Tables 1, the comparison has been conducted for the obtained solution of FKFG equation considered in Ex. 5.1 for diverse values of xwith ADM, VIM and DTM, and it shows that the obtained solution has a great agreement with the solution available in the literature. We can also conform that, the result obtained by proposed technique is the special case (n
Conclusion
We beneficially applied q-HATM to find the solution for Klein-Fock-Gordon equation of fractional order in the present work. We consider three examples with distinct initial conditions to elucidate the efficiency and applicability of the future technique. The convergence analysis for the FKFG equation is presented with the assist of Banach's fixed point theory. The approximated analytical solution for the proposed problem is presented with in series form, which converges quickly. In the present
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