Solution of the time-fractional generalized Burger-Fisher equation using the fractional re duce d differential transform method

Vhisht K.Tmboli , , PritiV.Tndel b ,

aDepartment of Mathematics, B.P.Baria Science Institute, Veer Narmad South Gujarat University, Surat, Gujarat, India

b Department of Mathematics, Veer Narmad South Gujarat University, Surat, Gujarat, India

ABSTRACT‘‘The time-fractional generalized Burger-Fisher equation (TF-GBFE)”is used in various applied sciences and physical applications, including simulation of gas dynamics, financial mathematics, fluid mechanics, and ocean engineering.This equation represents a concept for the coordination of reaction systems,as well as advection, and conveys the understanding of dissipation.The Fractional Reduced Differential Transform Method (FRDTM) is used to evaluate “the time-fractional generalized Burger-Fisher equation(TF-GBFE).”Todeterminethemethod’s validity, whenthesolutionsareobtained, theyarecorrelatedto exact solutions of α= 1 order, and even for various values of α.Three-dimensional graphs are used to depict the solutions.Additionally, the analysis of exact and FRDTM solutions indicates that the proposed approach is very accurate.

Keywords:Fractional reduced differential transform method (FRDTM)Time fractional generalized Burger-Fisher equation (TF-GBFE)Fractional calculus Caputo sense

1.Introduction

The fractional calculus was firstly introduced in the year 1695 by Gottfried Wilhelm Leibniz.Numerous studies on fractional phenomena have been conducted in the fields of science and engineering, including plasma physics, viscoelastic compounds, medical sciences, diffusion, fluid flow, genetics, electric networks, electrochemistry of corrosion, finance, probability, statistics, polymers,and optical fibers.In recent years, “fractional differential equations(FDEs)”have grown in common amongst scientists and engineers,and the research field of non-linear FDEs (NLFDEs) is limitless[1-6] .

Numerous mathematical methods have been established and studied in order to determine the exact solution of NLFDEs.For example, “the sine-cosine method [7,8] , the tanh method[9] , the Adomian decomposition method [10] , the sub-equation method [11,12] , the variational iteration method [13] , the first integral method [14,15] , Sumudu transforms [16] , Homotopy perturbation method [17,18] , q-homotopy analysis transform method(q-HATM) [19] , Fractional Reduced Differential Transformation Method (FRDTM) [20] .”

Numerous researchers have made important contributions to fractional models in recent years and investigated mathematical simulations of the human liver and HIV using Caputo-Fabrizio fractional derivatives [21,22] .The Picard-Lindelöfand approach to fixed-point theory is used to explore the presence of unique solutions to all of these models.Baleanu et al.[23-27] developed solutions to fractional hybrid equations with hybrid boundary conditions that include variants for the thermostat model using the famous Dhage fixed point theorems for maps with a single value and others with a set value and also addressed fractional integrodifferential equation modeling.

Many phenomena of integer-order differential equations cannot be well-described.That makes the research significance of fractional non-linear differential equations.Recently, academics have developed an interest in Non-linear Fractional Partial Differential Equations (NFPDEs).NFPDEs have gained considerable popularity in recent years due to their applications in various areas of science and engineering.The generalized Burger-Fisher equation”hasalreadybeen equated using the Reduced Differential Transform Method (RDTM) [28] .We conducted this research on“the time-fractional generalized Burger-Fisher equation (TFGBFE)”for the better understanding of mechanism governed by Nonlinear Fractional Partial Differential Equations (NFPDEs).“The timefractional generalized Burger-Fisher equation (TF-GBFE)”is a nonlinear equation that combines the reaction, advection, and dissipation mechanisms.This non-linear equation makes use of the Burger and Fisher diffusion transfer properties, as well as reaction form characteristics.

“The time-fractional generalized Burger-Fisher equation (TFGBFE)”is a critical fluid dynamic model, and several researchers have considered studying it to aid in the mathematical analysis of physical flows and the analysis of the different numerical methods.“The time-fractional generalized Burger-Fisher equation (TF-GBFE)”is highly non-linear because it incorporates reaction, convection,and diffusion mechanisms.It is named Burger-Fisher because it combines convective and diffusion properties from Burger’s equation with diffusion and reaction characteristics from Fisher’s equation.

“The time-fractional generalized Burger-Fisher equation (TFGBFE)”is a significant non-linear diffusion equation in ocean engineering because it represents the distant field of wave propagation in the ocean.The effects of convection, turbulent diffusion,and non-linear radiation on the irregularity of Sea Surface Temperature (SST) results in strong turbulent diffusion to cause the travelling waves.The velocity and the moving direction of the travelling wavefronts are dependent on the currents induced by winds [29] .

“The time-fractional generalized Burger-Fisher equation (TFGBFE)”is accountable for the convection-diffusion model that may simulate underwater landslide, which in consequence might create the most hazardous tsunamis in the coastal region [30] .

The globe has seen an explosion of essential facts, such as tsunami wave fields caused by Japan’s 2011 earthquakes.With the proper techniques, these tremendous abilities of ocean waves may be twisted into a diverse energy source for various purposes.To mitigate the tremendous force of such massive crises or harness them as beneficial energy sources, we must examine the mathematical frameworks of such natural issues.If we approach these issues differently, we may choose the optimal technique for analyzing such potential disasters and take necessary precautions.The current study may be expanded to include oceanic modeling, nonlinear mechanical models, thermal systems [31,32] .

The TF-GBFE is described as follows with the fractional order α and any real constants a, b, and ψ:

where a, b, ψare non-zero parameters, and the initial condition is given below:

Eq.(1) has an exact solution:

Numerous scholars have examined NFPDEs to get precise answers through a variety of techniques.FRDTM is one of the leading analytical methods among various available methods to solve NFPDEs.

This paper aims to solve the above TF-GBFE using the Fractional Reduced Differential Transform Method (FRDTM) [20] and utilize 3D graphs to equate with the outcomes of the exact solution.The results are collected in accordance with the orders α= 0 .25 , α= 0 .50 , α= 0 .75 and α= 1 at different x and time t values.For the proposed procedure’s efficacy, the errors among exact and the FRDTM solutions for α= 1 are also compared, which provides a reference for the accuracy and stability of the solution.

As a pioneer in analytical techniques, FRDTM may also be used to solve other NFPDEs since it is a very powerful and flexible technique as non-linear terms are easily handled without discretization and linearization.

2.Preliminaries and representations

This section explains some of the fundamental concepts and properties of the theory of fractional calculus.

Definition 2.1.In Caputo’s terminology, the fractional derivative( Dα) ofu ( x ) isasfollows[33] :

Definition 2.2.Caputo’s fractional derivative for order α＞ 0 encompassing the whole domain, as shown byis described as follows [34] :

Property.The modified Riemann-Liouville derivative has some helpful formulas and significant properties as follows [35] :

3.A brief overview of the fractional reduced differential transform method

Take the analytical function ω ( x, t ) , the multiplication of two independent variable functions, denoted by the symbol ω ( x, t ) =φ ( x ) ·η( t ) .As a consequence, the function may be expressed in the following manner:

Definition 3.1.For analytic and continuously differentiable functions, the t -dimensional spectrum function ω ( x, t ) is provided in the domain with respect to x and tas follows:

where the parameter αdenotes the time-functional derivative order.

The term ω ( x, t ) refers to the original function, and in this whole paper, the word Ωk( x ) refers to the reduced transformed function.The inverse differential transformation of Ωk( x ) can be considered as:

Uniting Eqs.(7) and (8) , we get

When t 0 = 0 , Eq.(8) turns into

The reduced differential transformation process has many properties, which are [36-39] mentioned in Table 1 below.

Table 1 The fractional reduced differential transform method’s axioms.

Table 2 ω ( x, t ) of “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”in case of a = -1 , b = ψ = 1 , and various values of α.

Table 3 ω ( x, t ) of “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”in case of a = -1 , b = 1 , ψ = 2 , and various values of α.

Table 4 ω ( x, t ) of “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”in case of a = b = 0 .001 and ψ = 1 , and various values of α.

Table 5 ω ( x, t ) of “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”in case of a = b = 0 .0 0 01 and ψ = 2 , and various values of α.

4.Applications

To demonstrate the reliability and precision of FRDTM for “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”, let us observe the following two examples:

Example 1.Consider a = -1 , b = ψ = 1 in Eq.(1)

concerning the initial condition

Then, using the FRDTM’s fundamental properties, we may determine the transformed form of the Eq.(12) as follows:

We obtain the following by using the initial condition (13) ,

Substituting the Eq.(15) into Eq.(14) , we obtain the values of Ωk( x ) successively

After applying the inverse transformation to the Eq.(16) , obtains four terms approximations

Fig.1.Surface of FRDTM for α= 1 .

Fig.2.Exact solution.

Additionally, it is stated here that the approach’s convergence can be improved by using additional terms in the series solution.As a result, the precise solution to the problem can be provided by

Above Eq.(18) is the approximate solution of the taken Example 1 .The exact solution of Example 1 turns out to be

The Fig.1 shows a surface of FRDTM for α= 1 , and the Fig.2 depicts the exact solution’s surface for “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”of Example 1 , andthe Fig.3 shows the corresponding values of ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

Fig.3.ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

Fig.4.Surface of FRDTM for α= 1 .

Fig.5.Exact solution.

Example 2.Consider a = -1 , b = 1 and ψ = 2 in Eq.(1) ,

concerning the initial condition

Then, using the FRDTM’s fundamental properties, we may determine the transformed form of the Eq.(21) as follows:

We obtain the following by using the initial condition (22) ,

Substituting the Eq.(24) into Eq.(23) , we obtain the values of Ωk( x ) successively

After applying the inverse transformation to the Eq.(25) , obtains four terms approximations

Additionally, it is stated here that the approach’s convergence can be improved by using additional terms in the series solution.As a result,the precise solution to the problem can be provided by

Above Eq.(27) is the approximate solution of the taken Example 2 .The exact solution of Example 2 turns out to be

The Fig.4 shows a surface of FRDTM for α= 1 , the Fig.5 depicts the exact solution’s surface for “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”of Example 2 , and Fig.6 shows the corresponding values of ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

Fig.6.ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

Fig.7.Surface of FRDTM for α= 1 .

Fig.8.Exact solution.

5.Results and discussion

Case-1 Let us consider “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”for the case of a = b = 0 .001 and ψ =1 .The corresponding values of ω ( x, t ) for different values of x and t, the following table lists the solutions for various values of α, along with their exact solutions and associated absolute errors.

Fig.9.ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

Fig.10.Surface of FRDTM for α= 1 .

Fig.11.Exact solution.

Fig.7 shows the surface of FRDTM for α= 1 and the Fig.8 depicts the exact solution’s surface for “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”for the case of a = b =0 .001 and ψ = 1 , and Fig.9 shows the corresponding values of ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

Case-2 Let us see “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”for the case of a = b = 0 .0 0 01 and ψ = 2 .The corresponding values of ω ( x, t ) for different values of x and t, the following table lists the solutions for various values of α, along with their exact solutions and associated absolute errors.

Fig.10 shows the surface of FRDTM for α= 1 and the Fig.11 depicts the exact solution’s surface for “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”for the case of a = b =0 .0 0 01 and ψ = 2 , and Fig.12 shows the corresponding values of ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

Fig.12.ω ( x, t ) combined for FRDTM for α= 1 and the exact solution.

6.Conclusion

Time fractional generalized Burger-Fisher equation (TF-GBFE) is successfully solved using FRDTM in this analysis.As firmly developed, FRDTM can be effectively used to achieve an approximate series solution to time-dependent equations.As a consequence of this analysis, it is clear that solving “the time-fractional generalized Burger-Fisher equation (TF-GBFE)”using the FRDTM produces superior results as compared to current numerical approaches.As a result, the proposed methodology is a robust, reliable, and productive method for algorithmic simplicity in the computer world.Additionally, it is worth noting that the FRDTM findings agree very well with the exact solutions of illustrative cases selected for comparative purposes.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions to improve the final version of the paper.