Numerical Simulation of Time Fractional BBM-Burger Equation Using Cubic B-Spline Functions

The unidirectional propagation of long waves in certain nonlinear dispersive system is explained by the Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. The purpose of this study is to investigate the BBM-Burger equation numerically using Caputo derivative and B-spline basis functions. The fractional derivative is considered in Caputo form, and L 1 formula is used for discretization of temporal derivative. The interpolation of space derivative is done with the help of B-spline functions. The e ﬀ ect of α and time on solution pro ﬁ le of travelling wave for di ﬀ erent domain of x is discussed in this paper. The numerical results have been presented to show that the cubic B-spline method is e ﬀ ective and e ﬃ cient in solving the time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation. Moreover, the convergence and stability of the proposed scheme are analyzed. The error norms are also calculated to check the accuracy of the proposed scheme. The numerical results re ﬂ ect that the proposed scheme can be used for linear and highly nonlinear models.


Introduction
The fractional calculus has a long history, beginning on 30 September, 1695 [1]. Fractional derivatives are an excellent method for explaining the memory and genetic characteristics of various materials and processes. Today, fractional differential equation (FDE) plays an important role in number of fields such as mathematics and dynamic systems [2]. Noninteger-order derivatives and integrals have proved to be more useful in formulating certain electrochemical problems than conventional models [3]. Fractional partial differential equations (FPDEs) provide a powerful tool for modeling a variety of potentially emerging phenomena with a wide range of applications across applied sciences, engineering, and physics. It is the classical generalization of the calculation that involves operators of integration of noninteger orders and differentiation. They are considered to be valuable and efficient tools in both theoretical and techno-logical fields to tackle the complexity and nonlinearity of specific problems, including chemical science, fluid flow, fiber optics, signaling system, polymers, device recognition, and elastic materials [4,5]. A great deal of attention has been paid to managing fractional differential equations (FDEs). Specifically, fractional-order partial differential equations (PDEs) are increasingly described in viscoelasticity, mathematical biology, finance [6][7][8], airplane designing, traffic, population, and particle chemistry [9][10][11]. Elasticity, plasma, fluid mechanics, optical fiber, and other disciplines of mathematics and physics involve wave phenomena. The Benjamin-Bona-Mahony-Burger (BBM-Burger) equation, which addresses dissipation as well as nonlinearity and dispersion in the simulation of unidirectional plane waves, is defined as follows [12]: subject to IC: ϕðx, 0Þ = f ðxÞ, x ε ½a, b and BCs as follows: ϕ ða, tÞ = g 1 ðtÞ, ϕðb, tÞ = g 2 ðtÞ, where α and β are positive constants and domain of Ω = ða, bÞ × ð0, T. Nonlinearity, dispersive, and dissipative effects are all included in Equation (1). The terms ϕ xx and ϕ xxt indicate the effects of dissipation and dispersion, respectively. Cracked rock, thermodynamics, acoustic waves in aharmonic crystals [13], acoustic-gravity waves in fluids, and other applications of the Benjamin-Bona-Mahony-Burger (BBM-Burger) equation and their special cases can be found in engineering and science. Benjamin-Bona-Mahony-Burger (BBM-Burger) equation is used in the analysis of the surface waves of long wavelength in liquids, hydromagnetic waves in cold plasma, and acoustic-gravity waves in compressible fluids.
The water wave model is notoriously difficult to solve analytically. So, several investigations have been done in recent years to determine the numerical solution of water wave model (see, for example, [14,15] and the references therein). Researchers introduced many different methods to develop an approximate analytical solution for the fractional differential equation and systems, such as the cubic B-spline method (CBS) by [16,17] and homotopy analysis method (HAM) by [18], unified method (UM) [19], Runge-Kutta method [20], Hermite wavelet technique together with Newton Raphson iteration method [21], and Lie symmetry method [22]. The time fractional Benjamin-Bona-Mahony-Burger (BBM-Burger) equation was proposed to discuss the dynamic physical system. The time fractional operator form of BBM-Burger equation used in [23] is where α is the fractional order of the equation. The initial and boundary conditions are respectively. The exact solution of Equation (2) Many scholars contributed by developing new analytical and numerical schemes to solve fractional model of BBM-Burger. Zhang et al. [23] implemented modified residual power series method (RPSM) for the simulations of BBM-Burger equation. G′/G expansion method has been implemented to investigate the exact solution of time fractional model of BBM-Burger equation in [24]. The authors in [25] utilized Lie symmetry approach to develop solution of fractional BBM equation. New soliton solutions for ð3 + 1Þ -dimensional Extended Date-Jimbo-Kashiwara-Miwa Equation has been established by means of Hirota method together with quadratic test functions in [26]. In [27], the authors studied numerical solution of ð3 + 1Þ-dimensional shallow water wave equation and explicit traveling wave solutions for Zoomeron evolution equation using Lie vector technique. Ali et al. [28,29] developed new method based on hybrid orthonormal Bernstein and block-pulse function wavelet method (HOBW) for the solution of fractional voltera-Fredholm differential equation and for the solution of 2D Bratu problem. Runge-Kutta (RK4) method and Keller-Box scheme have been implemented for the analysis of the melting heat transport of blood with time-dependent cross nanofluid model by [30]. The authors in [31] developed the Gudermannian neural networks to solve the nonlinear fifth-order induction motor model (FO-IMM). Ali [32,33] analyzed heat transfer of Ostwald-de Waele fluid using method of lines and penalty-artificial compressibility technique (PACT) for studying free convection of magnetohydrodynamic unsteady incompressible flow in a square cavity, respectively. New homotopy analysis transform method has been established in [34] to constitute analytical solution of BBM-Burger equation. The exact solution of BBM-Burger equation of fractional order has been developed by HAM in [12]. The author in [35] investigated the solution of Benjamin-Bona-Mahony equation using linearized difference scheme. The linear difference scheme with Crank-Nicolson has been implemented in [36] for the treatment of Benjamin-Bona-Mahony equation with a time fractional nonlocal viscous term. Majeed et al. [37] employed cubic B-spline scheme for the approximate solution of time fractional nonhomogeneous BBM-Burger equation. [38] used Pohozhaev nonlinear capacity method to investigate the blowing-up solutions of different nonlinear fractional partial differential equations like KdV, time fractional modified KdV-Burger, BBM-Burger, Rosenau, Ostrovsky, and Degasperis-Procesi. [39] has investigated the analytical solution of time fractional Benjamin-Bona-Mahony equation using method based on Laplace Transformation and Adomian decomposition method. Two problems have been solved with different initial conditions. In [40], the authors have derived the Benjamin-Bona-Mahony and KdV equations from water wave model. Further travelling wave equations of these models have been derived by two different methods. Space-time fractional BBM-Burger models have been solved analytically using Adomian decomposition method by [41]. The energy method is used by [42] to prove the uniqueness and existence of BBM-Burger model. The authors also used the Fourier Spectral method to get numerical solution of this model. In [43], the authors used EMAEM method to get different solutions of 3D fractional WBBM models. The sine-cosine function method is used in [44] to investigate the analytical solution of Benjamin-Bona-Mahony and MRLW models. The modified simple method and G/G expansion method have been used in [45] to find the exact solution of TFBBM equations.
In this study, cubic B-spline method along with Caputo derivative will be used to solve time fractional BBM-Burger equation numerically. L1 formula will be used to tackle the noninteger-order derivative, while B-spline basis functions will be used to discretize the space derivative. The 2 Journal of Function Spaces advantages of B-spline collocation method over the existing schemes are that it efficiently delivers a piecewise-continuous, closed form solution and it is simpler and can be used to a wide range of problems involving partial and fractional partial differential equations. The advantage of this method over others is that once the solution has been computed, the information needed for interpolation at different locations in the interval is available. The key benefit of this method is that it can approximate the analytical curve to a given degree of smoothness. As a result, the spline approach, when compared to previous methods, has the flexibility to obtain an estimate at any point in the domain with more accurate results. The behavior of solution profile for different domains of x will be studied, and the effect of parameter α will be discussed for various values. The objective of this study is to solve nonlinear fractional-order partial differential equation with mixed partial derivative numerically using scheme based on cubic B-spline bases. We proved that the proposed scheme is unconditionally stable. The organization of the paper is as follows: Section 2 gives brief introduction and preliminaries. Section 3 presents the algorithm and implementation of the suggested scheme to solve the fractional model of BBM-Burger. The stability and convergence of the proposed scheme are analyzed in Sections 4 and 5, respectively. Discussion on results is presented in Section 6, and Section 7 reviews the concluding remarks.
Here, ρ is for positive integer.

Proposed Scheme
To implement the numerical schemes, let ½0, T be the domain for time and is splitted up into N uniformly sized subintervals ½t n , t n+1 : t n = nΔt, n = 0, ⋯, N where Δt = T/N: The time fractional derivative in Equation (5) for l = 1 can be evaluated at t = t n+1 by the formula [46]. Using Equation (6) together with forward difference scheme and Crank-Nicolson for time derivative, Equation (2) becomes the following: where ϕ n = ϕðx, tÞ and γ = 1/ðΔtÞ α Γð2 − αÞ. Replacing Equation (7) to Equation (8), we get To apply the proposed scheme, the domain of spatial variable ½0, L is partitioned into N subdomain by ½x r+1 , x r , : x r = rh, where h = L/N and r = 0, ⋯, N: The solution of Equation (9) at ðn + 1Þ th time step is given by where Ψ j ðt n+1 Þ′s are unknowns to be determined and Λ j ðxÞ are spline function basis of third degree defined as follows [16]: Different derivatives like ϕ, ϕ ′ , and ϕ ′ ′ of Equation (10) in terms of Ψ j at knot points can be expressed as follows: The variation of ϕ N ðx, tÞ over the typical component ½x j , x j+1 is given by On substituting Equation (10) and its necessary derivatives of Equation (12) in Equation (9), we obtain the following set of equations: where Equation (14) will be solved for ℘ = 0 first, and then, loop will be executed for ℘≥1, which gives N + 1 system of linear equations with N + 3 unknowns ðΨ 1 , ⋯, Ψ N+1 Þ T . To get consistent solution of this system, we need two more equations, for this derivative equations will be used.
Equations (14) and (16) have the matrix form as follows: : The approximate solution for ðn + 1Þ th time level can be achieved for n = 0, ⋯, N by using vector Ψ in Equation (10).

Stability Analysis
The most frequently used formula to determine the stability is von Neumann Fourier formula [47]. Solution will be con-sidered in single Fourier mode.
Using Equation (21) in Equation (14), we have By putting values of χ 0 , χ 1 , χ 2 and G 1 , G 2 , G 3 and after simplification, we have where Z ⋆ , Z ⋆⋆ , and γ were defined above. If augment factor jΨ ℘+1 j ≤ 1, then the suggested scheme will be stable. Otherwise, it will be unstable.
The above result indicates that the proposed scheme is stable unconditionally.

Convergence Analysis
Let ϕðx, tÞ be the exact solution of Equation (2). Assume that G > 0 which takes different values at various locations as in [49] and is independent of j, n, h, and χ with the smoothness conditions that We first linearize the term ϕ x by taking ϕ x as a constant "b." The linearized form of the proposed scheme is then where λ 0 = ðΔtÞ α Γð2 − αÞ.
Theorem 5. If ϕðx, tÞ and ϕ n j are the solutions of given equation and Equation (27), respectively, and if ϕðx, tÞ satisfies smoothness condition Equation (25), then for sufficiently small h and χ, Proof. Subtracting Equation (27) from Equation (28), we obtain where s n+l j = Oðχ 2 + χh 2 Þ. Multiplying the above equation by g n+1 j and summing over j from 1 to M, we obtain  Journal of Function Spaces Using the inequality <r, t > ≤krkktk, we obtain Since κλ 0 kðg n+1 Þ x k 2 ≥ 0, therefore Dividing Equation (33) throughout by kg n+1 k, we get where G = max ðmaxÞkg n−l k. Note that we have used s n+l j = Oðχ 2 + χh 2 Þ. This completes the proof.

Discussion
The numerical results obtained by the suggested scheme for various parameters are reported in this part in the form of tables and figures. L 2 and L ∞ norms are computed and defined as The approximate solution (10) can be written in piecewise form as follows: The numerical solution ϕðx, tÞ of travelling wave models is discussed graphically and numerically in our computational work, the values of the parameter α = 0:95, step length h = 0:01, and Δt = 0:00943 are chosen, and for t = 0:25 to t = 5 computation, it was carried out over different finite space range of x ε½−5, 5, x ε½−10, 10, and x ε½−20, 20. Figure 1 shows the wave evolution for different time with α = 0:95. Figure 1(a) depicts the behavior of travelling wave at t = 0:25, while Figure 1(b) demonstrates the identical patron of travelling waves for different time t = 0:25, 0.75, 2.5, and 5. The domain of x varies from -5 to 5 to observe the behavior of wave, and it is clearly shown that both  The graphical illustration displayed in Figure 2 expresses the behavior of solution profile for different time steps t = 0:75, 2.5, and 5 in broader domain of x ε½−10, 10 and keeping α, h, and Δt = 0:011 fixed. Figure 2 express the behavior of wave evolution at t = 0:75, 2.5, and 5 in wide domain of x ε½−10, 10, while keeping h, α, and Δt = 0:011 fixed in this case. As the number of iterations is raised, the solution profile produces better results, and the traveling wave gets smoother in a larger domain. The wave evolution is also observed at different values of α εð0, 1. It is observed that as the value of α approaches to 1, the numerical solution tends toward the exact ⋆ solution, and if we use the alpha value near 0, numerical value moves far away from theexact ⋆ solution as shown in Figure 3. The 3D form of numerical solution with different time level is shown in Figure 4 effect of α εð0, 1 is observed. The efficiency of the proposed scheme in tabular form for this model can be observed by Table 1. In this table, numerical and exact ⋆ solutions are compared. Table 2 presents the L 2 and L ∞ norms.
To study the effect of solution profile on a larger domain, we extended the x space domain from -20 to 20 in Figure 5. The parameters for this simulation were chosen α = 0:95, h = 0:01, and Δt = 0:01 for different times t = 0:25, 0.75, 2.5, and t = 5. Figure 4 displays the solution profile for time t = 0:25, while the effect of different time on solution profile of travelling wave is shown in Figure 4. The graphics illustrate the smoothness of the solution profile. Figure 6 shows how different fractional parameters, such as αε ð0, 1Þ, affect the solution profile.
The wave swings for small values of α in the broader domain x ε½−20, 20 and then smooths out and approaches the classical solution for α ⟶ 1. The numerical solution of wave model in 3D form can be seen in Figure 7. Both solutions are extremely similar to each other for different parameters, as seen in the graphics. The exact ⋆ and approximate outcomes are compared in Table 3. Both solutions are com-parable up to four decimal places, as may be shown from tabular data. The error norms obtained at x ε½−20, 20 are shown in Table 4. The behavior of solution profile for broader domain has been studied and illustrated in the figures. These graphics demonstrate that solution curves become smooth and swing as we extend the domain, and both results express the identical behavior. So, it is observed

Concluding Remarks
In this work, an algorithm based on B-spline functions and Caputo derivative has been successfully implemented on fractional BBM-Burger equation. The temporal derivative is considered in Caputo's form, and discretization has been done by L1 formula. The space derivative has been interpolated by means of Crank-Nicolson and B-spline function. Applicability of the scheme is examined on test problem, and the following points are extracted.
(i) The behavior of travelling wave is studied for different domains of x, and it is observed that as we extend the domain of x, the solution profile improves and wave becomes more smooth (ii) The effect of the fractional parameter α on various spatial ranges has been studied, and it has been discovered that the wave varies for smaller values of α before becoming smooth for larger values of α (iii) The approximation solution tends to overlap the classical solution, as α ⟶ 1 may be seen in visuals and tabular data (iv) The study's findings on the stability of the suggested scheme reveal that it is unconditionally stable

Data Availability
The experimental data used to support the findings of this study are available within this paper.