A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions

1 Introduction

One of the most used mathematical functions for estimating processes is the spline functions. Spline functions are identified as piecewise polynomial functions that are used to solve many ordinary differential equations and partial differential equations. In recent work, these functions are also widely used to solve fractional partial differential equations. Compared to traditional differential equations, fractional order differential equations may be able to explain dynamic processes more effectively. The researchers of the twentieth century have done an astounding quantity of research on fractional calculus [1]. Caputo and Febrizio [2] presented a new definition of fractional derivative with a smooth kernel. The caputo fractional derivative is used by Diethelm and Ford [3] to solve multi-order fractional differential equations. El-Sayed et al. [4] investigated the fractional-order logistic equation’s numerical solution, existence, stability, and uniqueness. The relativistic Klein–Gordon equation (KGE) describes spinless particles and is a version of the Schrödinger equation. Numerous mathematicians have solved the KGE using various methods, such as Shakeri and Dehghan [5] used the variational iteration method to find the analytical and approximate solution of this equation. Onate et al. [6] analytically derived both the energy equation and the accompanying unnormalized wave function. Gorka [7] used the Galerkin technique, the logarithmic Sobolev inequality, and the compactness theorem to find out the existence of a unique solution. Kumbinarasaiah [8] calculate a nonlinear solution of KGE by transforming a system of nonlinear algebraic equations. Lou and Chen [9] defined some unique kinds of solutions to Kleine–Gordon reduction equations. Chatterjee [10] solved the KGE iteratively using an 1 N expansion approach to get the energy spectrum of a scalar particle in a spherically symmetric potential. Subasi and Araz [11] examined the phases involved in determining the best control and the methods for approximating this control, respectively. Atangana and Araz [12] used the recently presented numerical approach to integral and partial differential equations of both integer and noninteger order. Subasi et al. [13] investigated the optimality conditions in an optimal control problem governed by a hyperbolic problem. Dehghan and Shokri used collocation point and thin plate splines radial basis functions to approximate the solution in the study of Dehghan and Shokri [14]. Berat et al. [15] used a cubic B-spline (CBS) collocation finite element method and L 2 algorithm for the solution of the time fractional (TF) nonlinear KGE. Bratsos [16] solved the nonlinear KGE by using a predictor–corrector technique based on the application of rational approximates of second-order to the matrix-exponential term in a three-time level recurrence relation. Zhang et al. [17] introduced a simple direct space–time meshless approach based on the radial or nonradial basis function for the one-dimensional KGE. Shakeri and Dehghan [18] used He’s variational iteration method to give approximate and analytical solutions for this equation.

Ganji et al. [19] presented a novel method for solving the time fractional Klein–Gordon equation (TFKGE) by employing the clique polynomial as the basis function for the operational matrices in his article. Saifullah et al. [20] solved the nonlinear KGE with the Caputo fractional derivative. In order to solve a set of linear and nonlinear fractional KGEs that arise in quantum and classical relativistic physics, Liu et al. [21] offered an analytical and numerical method. Bentrcia and Mennouni [22] discussed behavior analysis of solutions to a TFKGE in a multidimensional a bounded domain. Chen et al. [23] considered the numerical approximation of a nonlinear TFKGE in a bounded domain. Mohebbi et al. [24] investigated three equations in his article which are the TF Cattaneo equation and the linear time-fractional KG plus dissipative KG equations. They solved the TF derivative described in Caputo’s sense by a scheme of order O ( τ 3 − α ) , 1 < α < 2 , and the space derivative is discretized with a fourth-order compact procedure. Hassani et al. [25] solved the variable order nonlinear TFKGE by using a set of basis functions, namely the generalized polynomials and the Caputo-type definition. As there are a number of equations with fractional derivative but this article is concerned with the following nonlinear TFKGE:

with ICs:

and boundary conditions (BCs)

where ∂ α U ( y , s ) ∂ s α is the Atangana–Baleanu time fractional derivative (ABTFD). Displacement of wave at point ( y , s ) is represented by U ( y , s ) , fractional order of the time derivative is represented by α , where α ∈ ( 1 , 2 ] , μ 1 , μ 2 , and μ 3 are real numbers, f ( y , s ) is the source term, and σ = 2 or 3.

A generalization of the KGE that takes into account fractional time derivatives and nonlinearity is the nonlinear TFKGE. Because it can realistically describe memory effects, anomalous diffusion, and complicated wave processes, this equation is important in many areas of physics, applied mathematics, and engineering. Memory effects are introduced by the fractional time derivative, which means that the system’s current state is dependent on its complete past. In situations where traditional integer-order equations are ineffective, this is helpful for simulating complicated materials, viscoelasticity, and anomalous diffusion. Fractional KG models describe quantum fields having aberrant or fractal features in relativistic quantum mechanics. Nonlinear wave processes in complex and dispersive mediums are governed by the equation. Fractional derivatives make it possible to represent wave dispersion and attenuation in biological systems, optical fibers, and plasma waves more accurately. Fractional dynamics is used in inflationary cosmology to model cosmic inflation, dark energy, and early universe variations. The evolution of scalar fields in curved space–time is described by it. Kinks and domain walls are examples of soliton solutions that arise in condensed matter physics and cosmology as a result of the nonlinear KGE. It talks about localized oscillating waves called breather solutions.

Vivas-Cortez et al. [26] used the extended CBS functions and Caputo TF derivative for the numerical solutions of the generalized nonlinear TFKGE. Ahmad et al. [27] examined the Casson hybrid nanofluid’s magnetohydrodynamic flow along a vertical open channel while accounting for Newtonian heating and viscous dissipation. Ahmad et al. [28] considered a mathematical model for the disease dynamics in both prey and predator by considering the susceptible- infected-recovered-susceptible model with the prey–predator Lotka–Volterra differential equations. Ahmad et al. [29] introduced a model of pine wilt disease in trees that takes into account the relationship between nematodes and transmitting beetles with pine trees that are asymptomatic and those that are symptomatic. Ain et al. [30] solved the nonlinear TFKGE by using a redefined extended CBS and Caputo fractional derivative. Sarboland and Aminataei [31] applied the multi-quadratic quasi-interpolation scheme and the integrated radial basis function network scheme to solve TFKGE. Amin et al. [32] introduced the Haar collocation method to construct a numerical method for solving multi-term fractional differential equations up to fourth order. Amin et al. [33] developed the Haar collocation algorithm to solve the first-order HIV infection CD4+ T-cells model. Saeed and ur Rehman [34] suggested a technique for solving the fractional delay differential equations that make use of the Hermite wavelet approach and the method of steps. Mohammadi and Cattani [35] generalized the classical Legendre wavelet to a new fractional-order wavelet foundation. Amin et al. [36] created a novel collocation technique based on the Haar wavelet for the numerical solution of the fractional Volterra model for a species population expansion in a closed system. Yousif and Hamasalh [37] used a nonpolynomial spline fractional continuity method to calculate the Burgers-Fisher equation. Algehyne and Ibrahim [38] analyzed the severe acute respiratory syndrome corona virus-utilizing contemporary calculus and the Atangana–Bleanu Caputo-fractional derivative. Al-Raeei [39] simulate the spatial version of the fractional Schrödinger equation for the electrical screening potential using the Riemann–Liouville definition of fractional derivatives and numerical simulation methods. Zhang et al. [1] suggested a simple and direct space–time meshless approach based on the radial or nonradial basis function for the one-dimensional KGEs.

In this article, the CBS functions are used to obtain the approximate solution of TFKGE. For spatial integration, the CBS functions are used. The ABTFD is discretized by using usual a finite central difference approach. Stability and convergence are computed to demonstrate the scheme’s effectiveness. To demonstrate the correctness of the proposed approach, four test examples are provided along with comparisons to existing problems in the literature. Some results in the form of tables and graphs are provided for verification and correctness of the proposed scheme.

This article is organized as follows. Section 2 has several significant definitions. Section 3 provides an explanation of the method’s description, where Section 4 contains the initial vector. Stability and convergence are found in Sections 5 and 6, respectively. The numerical examples and closing remarks are given in Sections 7 and 8, respectively.

2 Preliminaries

The ABTFD is a modern concept in fractional calculus, which provides advantages in modeling complex systems with memory and hereditary properties. Some of its advantages include generalization of classical models, inclusion of nonsingular kernels, better representation of memory effects, physical interpretability, flexibility in applications, and preservation of initial conditions (ICs). ABTFD is based on the generalized Mittag–Leffler kernel, which gives it a nonlocal and memory-preserving nature, while the conformable derivative is a straightforward extension of the classical derivative, designed to maintain some of the intuitive properties of derivatives for fractional orders. With memory effects, the Atangana–Baleanu fractional derivative offers a strong framework for simulating complex structures. However, for broader applicability, its few shortcomings in terms of BC formulation and limited analytical solutions must be addressed.

3 Description of the numerical method

The ABTFD has been utilized to discretize the TF term of presented problem. Assume that K equal subintervals of length Δ s = S K split the time domain [ 0 , S ] , using the knots 0 = s 0 , s 1 , … , s K = S , where s k = k Δ s , k = 0 , 1 , … , K . Now, ABTFD can be discretized as

Theoretically robust and easy to implement, finite difference schemes are a popular tool in scientific computing, particularly for problems with regular domains and simple BCs. Through forward finite difference formulation, Eq. (3.1) is modified as

Hence

where E ˜ l = E α , 2 [ − α 2 − α ( l Δ s ) α ] and q l = ( ( l + 1 ) E ˜ l + 1 − l E ˜ l ) . Furthermore, the truncation error ℏ Δ s k + 1 is indicated as

where ϒ is a constant.

Using the θ -weighted scheme and Eqs. (3.2) and (1.1) takes the form

A numerical technique for solving differential equations that offer a modifiable trade-off between explicit, Crank–Nicolson, and implicit time-stepping is the θ -weighted scheme. The method is totally implicit when θ = 1 , fully explicit when θ = 0 , and the Crank–Nicolson approach when θ = 0.5 . We adopt θ = 1 for the direction of space throughout the work, Eq. (3.4) yields

where β = A B ( α ) Δ s ( 2 − α ) , U i k = U ( y i , s k ) , and f i k + 1 = f ( y i , s k + 1 ) . After rearrangement, Eq. (3.5) yields

It is clear that when l = k or k = 0 , the term U − 1 is obtained. To handle this term, the ICs is used by the help of central difference approximation to obtain

Using the CBS approximations and their derivatives at the particular knot y i in Eq. (3.6), then

Using (2.4) in (3.7), we obtain

where T 1 = ( ψ i − 1 k + 4 ψ i k + ψ i + 1 k ) σ . Eq. (3.8) can be written as

where

There are N + 1 linear equations in N + 3 unknowns in system (3.9). In order to obtain a numerical solution of the system, the remaining two equations are extracted from the BCs, Eq. (1.3) yields

After using BCs, an ( N + 3 ) × ( N + 3 ) matrix system is obtained that can be uniquely solved using any suitable numerical technique. The above-mentioned system of linear equations given in Eq. (3.9) can be written in matrix form as

where

4 Initial vector

The initial vector ψ 0 = [ ψ − 1 0 , ψ 0 0 , ψ 1 0 , ⋅ , … , ψ N + 1 0 ] is achieved by utilizing the ICs as

System (4.1) having ( N + 3 ) × ( N + 3 ) equations and unknown, respectively. It can be written in matrix form as

where

5 Stability analysis

The stability of the suggested scheme is examined in this section using the von Neumann stability method. The increase of error in a single Fourier mode is taken into account in the von Neumann analysis of stability [45]. It is well known that the stability of linear schemes can be examined using this method. It is assumed that every nonlinear term in Eq. (1.1) is zero [45]. For the sake of simplicity, we will discuss the suggested scheme’s stability for the linear form of Eq. (1.1) as

Use the technique described in Section 3 for θ = 1 , we have

Let us use the Fourier approach now [46,47] and assume that the growth factor is represented by the symbol ∅ i k . Its estimation can be shown as ∅ ˜ i k . The error ϱ i k can be given as

From Eq. (5.1), we obtain

From initial and BCs, it can be written as

The mesh function is described as

In terms of Fourier series, ϱ k ( y ) can be written as

where

Ω k ( r ˆ ) = 1 b − a ∫ a b ϱ k ( y ) e − 2 π j r ˆ y b − a d y , k = 0 , 1 , … , K and ϱ k = [ ϱ 1 k , ϱ 2 k , … ϱ N − 1 k ] T .

By applying norm, we obtain

By Parseval’s identity (2.1), we obtain