Numerical solution of two-dimensional fractional differential equations using Laplace transform with residual power series method

1 Introduction

Fractional differential equations (FDEs) have drawn the interest of researchers in the study of differential calculus due to their broad implications and wide range of applications in various types of problems arising in signal processing systems, diffusion-reaction processes, electrical network systems, and some other technical issues [1].

The extended versions of the classical differential equations are the FDEs [2,3], which have been widely applied in a variety of scientific domains during the past few decades. There are various applications in science that describe the importance of the principles of fractional calculus. Despite the fact that there are many available numerical and analytical techniques for solving FDEs mathematically, researchers are putting their efforts in developing new technique that can lead to more accurate solution of the fractional equations.

There are many trustworthy and effective numerical and analytical procedures that can be used to address time-fractional problems with greater accuracy. Time fractional differential problems have been solved using a variety of methods, including the homotopy analysis method [4], Laplace transform [5], Adomian decomposition method [6], variational iteration method [7], promoted residual power series method [8], homotopy perturbation method [9], differential transform method [10], iterative method [11], and others. Local fractional integral transforms [12] are also applied to find the solutions of such equations in numerical forms. Recently, a well-known fractional relaxation oscillation equation is solved by using residual power series method [13] and obtained highly reliable and efficient results.

One of the well-known semi-analytical methods for solving various types of systems of ordinary, fractional, and partial differential equations is the residual power series method (RPSM) [14]. While this analytical approximation generates solution in the form of a polynomial, the approach offers the solution as convergent power series with easily calculable components. There are numerous ways in which the RPSM is different from the traditional higher order Taylor series technique. One of the ways that make this method different from the Taylor’s series approximation is that the RPSM can be easily hybrid with the transform. There are various linear and non-linear equations that are solved using RPSM such as a time-fractional order Schrödinger equation [15], fractional relaxation-oscillation equation [13], time-fractional foam drainage model [16], fractional Boussinesq equation [17], fractional coupled physical equations arising in fluid flow [18], and many others. There are many other well-known equations which are solved by using transform along with RPSM such as Laplace transform with residual power series method (LRPSM) is used to solve temporal-fractional NWS equation, fractional Burger’s equation, and fractional Drinfeld–Sokolov–Wilson system [8], fractional Riccati differential equation [19], and fractional reaction–diffusion Brusselator model [20]. The Elzaki transform with residual power series method is also successfully implemented to solve fractional order equations such as the two-dimensional fractional order diffusion equation [21] and time-fractional biological population diffusion equations [22].

In the last several decades, numerous fractional generalizations of the time-fractional diffusion equation have been presented and have been the subject of significant discussion in both the academic literature and various diffusion model applications [21]. The time-fractional diffusion equation is a partial differential equation that characterizes the temporal evolution of a quantity, such as heat, mass, or particles, by adding the principles of fractional calculus. In contrast to the conventional diffusion equation that employs integer-order derivatives, the time-fractional diffusion equation incorporates fractional derivatives in the temporal domain. This feature enables the model to effectively capture non-local and memory-dependent behaviours in diffusion processes, making it particularly advantageous for modelling phenomena characterized by long-range interactions or intricate temporal dependencies. Although the fractional diffusion equations have been solved by many numerical and analytical approaches. Here an attempt is made to implement the LPRSM to solve the equation for various values of the fractional power.

Another equation which is presented in this work is a time-fractional order biological population equation. The time-fractional order biological population equation is a mathematical framework employed to elucidate the temporal dynamics of a biological population, integrating the principles of fractional calculus [22]. Within the framework of a biological population equation, the fractional order is commonly utilized to denote a level of memory or a more intricate reliance on previous occurrences compared to conventional differential equations with integer orders. The utilization of these equations in the field of ecology and population dynamics serves to effectively capture many behaviours, including but not limited to population growth, competition, predation, and the influence of environmental factors, with enhanced precision.

Consider the general FDE in two dimensions of the form,

for t > 0 , 1 − 1 n < α ≤ 1 with initial condition,

and

where D t α = ∂ α ∂ t α , L [ x , y ] is the linear function in x and y , NL [ x , y ] is the general non-linear function in x and y , and φ ( x , y , t ) is taken as a continuous function.

For LRPSM, the solutions of Eqs (1) and (2) can be written as fractional power series form regarding the primary point t = 0 .

The order in which the manuscript is given is as follows: Introduction is presented in Section 1. Preliminary definitions are covered in Section 2. The research methodology used to solve two-dimensional fractional order differential equations using LRPSM is thoroughly justified in Section 3. The numerical solutions to these equations are provided in Section 4. The explanations and results are provided in Section 5. The conclusion for the research topic is presented in Section 6.

2 Preliminaries

Laplace transform is a well-known transform or map which is helpful to obtain coefficients in the series solution of considered equations. Some of the well-known concepts for the Laplace transform are as follows:

Suppose f ( t ) be a function defined for all t > 0 , then the Laplace transform of f ( t ) denoted by L { f ( t ) } is defined as, L { f ( t ) } = ∫ 0 ∞ e − pt f ( t ) d t , provided that the integral exists with parameter t . Laplace transform is a map of p and is denoted by f ̅ (p).

Therefore,

Again, the function f ( t ) is called the inverse Laplace transform of f ̅ (p). It is useful for finding inverse Laplace transform on numerical solution of FDEs considered in this study.

Another important concept related to the series solution is its convergence. A series of the form ∑ n = 0 ∞ c n ( x − x 0 ) n = c 0 + c 1 ( x − x 0 ) + c 2 ( x − x 0 ) 2 + … is known as a general power series in x − x 0 . In particular, an infinite series ∑ n = 0 ∞ c n x n = c 0 + c 1 x + c 2 x 2 + … is known as power series in x . The power series ∑ n = 0 ∞ c n ( x − x 0 ) n converges (absolutely) for | x | < R where R = lim n → ∞ c n c n + 1 , provided that the limit exists.

3 Methodology

The following steps are used in the methodology of the LRPSM [23] for solving two-dimensional FDEs in fractional power series [24]:

Step 1: Performing Laplace transform on diffusion equation,

By property of Laplace transform,

Hence, Eq. (4) can be written as

Step 2: The solutions in LRPS form can be written as

Again, the k ^th-Laplace residual function of (6) is,

Step 3: Using adopting properties of LRPS,

h 0 = Ψ ( x , y , 0 ) and h k − ∇ 2 h k − 1 = 0 or h k = ∇ 2 h k − 1 .

Following are some useful relations which are used in LRPSM:

i ) L Res ( x , y , s ) = 0 and lim k → ∞ L Res k ( x , y , s ) = L Res ( x , y , s ) , for s > 0 .

Step 4: At last, by performing inverse Laplace transform [25] to Ψ ′ ( x , y , s ) , the k ^th-approximate solution Ψ ( x , y , t ) can be obtained.

4 Numerical examples

The numerical solutions of two-dimensional FDEs by using LRPSM are calculated in this section. In order to present the efficiency of the method, two well-known equations are considered for the numerical solution.

Example 1 The two-dimensional fractional order diffusion equation is given by

with initial condition,

and exact solution is,

Also, the exact solution for 0 < α < 1 is Ψ ( x , y , t ) = E α ( z ) sin x sin y ,

where E α ( z ) = ∑ k = 0 ∞ z k ᴦ ( 1 + k α ) , and z = − 4 i t α .

To obtain the solution, applying Laplace transform on both sides of (1),

By property of Laplace transform, L [ D t α Ψ ( x , y , t ) ] = s α L [ Ψ ( x , y , t ) ] − s α − 1 Ψ ( x , y , 0 ) , then Eq. (14) can be written as

The solutions in LRPS form can be written as

Then, we obtain

Therefore, h 0 = Ψ ( x , y , 0 ) , h 1 = ∇ 2 h 0 , and h 2 = ∇ 2 h 1 .

Again, the k th - Laplace residual function of (15) is

By adopting properties of LRPSM

h 0 = Ψ ( x , y , 0 ) and h k − ∇ 2 h k − 1 = 0 or h k = ∇ 2 h k − 1 .

Hence,

Therefore, by LRPSM, the solution of the given equation in infinite form is

At last, by taking inverse Laplace transform in (18), we get the required solution of given equation by using LRPSM as

Example 2 The two-dimensional time-fractional order biological population equation is given by

with the initial condition

and the exact solution when

To obtain the solution, performing Laplace transform on Eq. (20) results in

By using the property, L [ D t α Ψ ( x , y , t ) ] = s α L [ Ψ ( x , y , t ) ] − s α − 1 Ψ ( x , y , 0 ) , and after simplification it becomes,

or,

where

Now the transformed function Ψ ′ ( x , y , s ) can be written in the following expansion as

The kth-truncated series of (25) can be written as

Then, by the Laplace residual function of (24) is

Again the k ^th-Laplace residual function of (27) is

To find the values of f k ( x , y ) , k = 1,2,3 , … … , substituting the k ^th-truncated series (26) into the k ^th-Laplace residual function (28), we obtain:

For k = 1 , from (29), the first Laplace residual function is

On solving Eq. (30) and using the relation lim s → ∞ ( s α + 1 L Res 1 ( x , y , s ) ) = 0 for k = 1 , it is obtained that

For k = 2 , from (29), the second Laplace residual function is

On solving Eq. (31) and using the relation lim s → ∞ ( s 2 α + 1 L Res 2 ( x , y , s ) ) = 0 for k = 2 , it is obtained that

Hence, by LRPSM, the solution of given equation in infinite form is

At last, by taking inverse Laplace transform in (32), we obtain the required solution of given equation by LRPSM as

5 Explanation and results

The reliability and efficiency of the LRPSM are discussed in this section from the obtained results for the two-dimensional fractional order diffusion equation and time-fractional order two-dimensional biological population equation.

Figure 1 compares the behaviour of the solution of the fractional diffusion equation at different values of t when α = 0.5 , 0.7 , and 1.0 , with the exact solution. This shows that solutions are reliable for α ≤ 1.

Figure 1

Comparison of solution behaviour of fractional diffusion equation when α = 0.5, 0.7, and 1.

The absolute errors of different number of terms of 6, 8, 10, and 12 of numerical solutions of the diffusion equation at various values of t = 0.2, 0.4, 0.6, 0.8, and 1.0, and α = 0.7 with exact solution are presented in Figure 2. This shows that when the number of terms is increased, then the approximate solution approaches the numerical solution and hence errors are reduced.

Figure 2

Absolute error of fractional diffusion equation when value of α = 0.7.

The absolute errors of different number of terms of 6, 8, 10, and 12 of numerical solutions of the diffusion equation at various values of t = 0.2, 0.4, 0.6, 0.8, and 1.0 and α = 0.9 with exact solution are presented in Figure 3. This shows that when the number of terms is increased, errors are decreased.

Figure 3

Absolute error of fractional diffusion equation when α = 0.9.

In Figure 4, the absolute errors of different number of terms of 6, 8, 10, and 12 of numerical solutions of the diffusion equation at various values of t = 0.2, 0.4, 0.6, 0.8, and 1.0 and α = 1.0 with exact solution are presented. It is evident from the results that the errors are reducing with the addition of more terms in the solution.

Figure 4

Absolute error of fractional diffusion equation when α = 1.0.

Figure 5(a) and (b), respectively, show the errors for various values of t in two dimensions. The first graph is drawn for t = 0.5 and α as 0.5 while the second graph is presenting the errors in solution for α as 1 at t = 1 . This shows that when the value of α is approaching 1, the errors are improved even at higher time levels.

Figure 5

(a) The absolute errors for α = 0.5 at t = 0.5 and (b) absolute errors for α = 1 for t = 1.

Table 1 displays the numerical values of ( L ∞ ) , the maximum errors for prescribed values of t as 0.25, 0.50, 0.75, and 1.0 at various α values. This shows that maximum errors are decreasing as α is approaching 1.

Tables 2 and 3 display the numerical and exact solution of the diffusion equation at predetermined points in two dimensions where x varies from 0.0 to 1.0 as well as y varies from 0.1 to 0.2 and 0.3 to 0.4, respectively. From the table it is evident that both the solutions are close enough.

Tables 4 and 5 demonstrate the numerical solution and exact solution of the fractional order diffusion equation at predetermined sites in two dimensions where x ranges from 0.0 to 1.0 as well as y changes from 0.6 to 0.7 and from 0.8 to 0.9, respectively, which also verifies that both the solutions are close enough and reliable. This demonstrates the precision of the fractional order diffusion equation’s mathematical solutions using the LRPSM.

In Figure 6, the solution behaviour of two-dimensional time-fractional order biological population equation at various values of t = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 is presented when α = 0.5 , 0.7 , and 1.0 . From the figure, it can be seen that the lines representing the numerical and the exact solution are almost overlapping; hence, there is no significant difference between the solutions at different values of α .

Figure 6

Comparison of solution behaviour of fractional biological population equation when α = 0.5, 0.7, and 1.0.

Figure 7 displays the absolute errors of the two-dimensional time-fractional biological population equation at various values of t = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 for a fixed value of α taken as 0.7. This shows that when the number of terms is increased, then the errors of both solutions decreased.

Figure 7

Absolute error of fractional biological population equation when α = 0.7.

This demonstrates how effective and innovative the LRPSM is for solving two-dimensional time-FDEs and obtaining analytical approximate of solutions.

6 Conclusion

In this study, a semi-analytical method is applied to two important equations for analytically approximating solutions of these fractional order differential equations in two-dimensional by using LRPSM. The advantage of this method is that it reduces the amount of computational work necessary to find numerical solutions in residual power series form after applying Laplace transform in a way that the coefficients of those solutions are established in the aforementioned sequential algebraic steps. Using the LRPSM, the time-fractional order two-dimensional differential equations can therefore be solved accurately and effectively. It is demonstrated that the method is capable of solving time fractional order two-dimensional differential equations with adequate accuracy. So, this method provides trustworthy solutions with fewer errors.