The nonlinearity in many problems occurs because of the complexity of the given physical phenomena. The present paper investigates the nonlinear fractional partial differential equations’ solutions using the Caputo operator with Laplace residual power series method. It is found that the present technique has a direct and simple implementation to solve the targeted problems. The comparison of the obtained solutions has been done with actual solutions to the problems. The fractionalorder solutions are presented and considered to be the focal point of this research article. The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem. Because of the simple implementation, the present technique can be extended to solve other important fractional order problems.
The integral and derivative of fractional order are considered to be common topics of fractional calculus (FC) due to their numerous applications in applied sciences. This topic has gained much popularity among researchers because of its popularity and significance while modelling various procedures in nature. For example, these problems are increasingly being applied to equations in fluid flow, diffusion, polymer physics, electric network rheology, relaxation, reactiondiffusion, diffusive transport akin to diffusion, turbulence, anomalous diffusion, porous structures, and dynamical processes in complex systems, as well as a variety of other physical phenomena [
It is very rare to calculate the exact solution of nonlinear FPDEs in the literature. Using linearization, successive, or perturbation methods, only approximate solutions can be obtained. Iterative Laplace transform method [
In many cases, analytical or exact solutions are very difficult to investigate. Therefore, mathematicians have tried to develop and use several numerical techniques with fractional derivative and integral operators [
Without linearization, perturbation, and discretization, the residual power series method (RPSM) is an effective and uncomplicated technique for constructing a power series (PS) solution for FPDEs. Unlike the traditional PS method, the RPS method does not require a recursion relation or comparison of the coefficients of the related terms. A series of algebraic expressions are obtained to calculate the PS coefficients. The methodology’s main advantage is that it relies on simpler and more accurate derivation as compared to other techniques that are based on integration. This method is a different way of solving FPDEs in theory [
In this article, the LRPSM is used to solve the time and space FPDEs. The aim of the study is to use LRPSM with spacetime fractional derivatives of the form to obtain numerical solutions to nonlinear fractional partial differential equations.
The generalized LRPSM procedure is presented and then LRPSM algorithm is applied to solve few numerical problems. The results and the accuracy of the suggested technique is shown by tables and graphs. The graphical representation is done and the obtained solutions are vary closed to the actual solutions of each target problem. The fractional order LRPSM solutions provide the analysis of some useful dynamics of the given FPDE’s. The tables have shown that LRPSM has the higher degree accuracy. LRPSM is comparatively a very simple and direct procedure to evaluate the solutions of nonlinear FPDEs and their systems. The proposed required fewer calculations to compute the nonlinear terms in each problem.
The article layout as follows. The fundamental concepts regarding FC are described in
The Caputo’s derivative of
Let
In [
Let
(Fractional Taylor’s formula [
If
Suppose that
If
A PS representation of the form
The extension form of PS is
Consider that
If
Here, we will go through the procedure that LRPSM takes to solve time and space FPDEs.
We apply the LT to
By the fact that
We write the
The
As stated in [
Therefore, to determine the coefficient functions
We apply the inverse LT to
Consider the time and space FPDE [
We apply the LT to
The
Now, to determine
Putting the values of
Applying inverse LT to
Consider the time and space FPDE [
We apply the LT to
The
Now, to determine
Putting the values of
Applying inverse LT to
Putting
Consider the time and space FPDE [
We apply the LT to
The
Now, to determine
Putting the values of
Applying inverse LT, we get
AE at 
AE at 
AE at 
AE at 


0.2  1.547471725 
5.929622809 
1.211474741 
1.66667 

0.4  3.377538834 
1.248095530 
2.497361834 
2.33333 

0.01  0.6  5.251026905 
1.909389588 
3.786907904 
2.00000 
0.8  7.145473987 
2.573550349 
5.078019823 
1.66667 

1  9.052369674 
3.239411093 
6.370013095 
2.33333 

0.2  1.540758992 
9.767460497 
2.832920738 
1.309333 

0.4  4.345131868 
2.462522892 
6.668790065 
1.317667 

0.05  0.6  7.334389807 
4.004032035 
1.056982054 
1.326000 
0.8  0.1040238760  5.570672050 
1.449846305 
1.334333 

1  0.1351013616  7.151740735 
1.844251126 
1.342667 

0.2  2.89111730 
6.18076440 
2.610104023 
2.1014667 

0.4  2.783094332 
2.397812731 
8.071310653 
2.1280333 

0.1  0.6  5.59856207 
4.317538941 
1.375500821 
2.1546000 
0.8  8.53246215 
6.298330555 
1.953186031 
2.1811667 

1  0.1151215528  8.312924610 
2.536006207 
2.2077333 
AE at 
AE at 
AE at 
AE at 


0.2  9.783254129 
3.041738403 
5.585918942 
3.0000 

0.4  7.816277292 
2.332577660 
4.178733371 
1.0000 

0.01  0.6  4.936203814 
1.448539702 
2.579866758 
1.0000 
0.8  1.270700930 
4.343826185 
8.515810999 
3.0000 

1  3.114377418 
6.863807506 
9.756686665 
5.0000 

0.2  0.2101487567  8.252714158 
1.844959799 
1.97370000 

0.4  0.1693089322  6.272596400 
1.357378055 
6.5790000 

0.05  0.6  9.619426598 
3.540019718 
7.710973462 
6.5790000 
0.8  6.88088047 
2.384783989 
1.191825756 
1.97370000 

1  0.1391892618  3.538120260 
5.826427340 
3.29000000 

0.2  0.2886918116  0.1249493949  3.018437865 
3.33333000 

0.4  0.2347545351  9.404629866 
2.176028113 
1.11111000 

1  0.6  0.1189245304  4.687155107 
1.106710623 
1.11112000 
0.8  5.693954892 
1.282208411 
1.115755320 
3.33334000 

1  0.2933045613  8.314396606 
1.442633714 
5.55560000 
LRPSM  ADM  

0.3  2.0 
2.9798 

0.01  0.6  1.0 
6.0101 
0.9  4.0 
9.04041 

0.3  1.31580 
1.80921 

0.05  0.6  6.5790 
3.78289 
0.9  2.63160 
5.75658 

0.3  2.222220 
2.77778 

0.1  0.6  1.111120 
6.11111 
0.9  4.444450 
9.44444 
AE at 
AE at 
AE at 
AE at 


0.2  4.979224126 
2.269449293 
5.514676025 
1.1880000 

0.4  8.334590765 
3.527902196 
7.863510967 
1.3860000 

0.01  0.6  0.1153348541  4.711902873 
9.994458235 
1.5840000 
0.8  0.1472311811  5.894558247 
1.207522865 
1.7820000 

1  0.1793811380  7.095505279 
1.415128281 
1.9800000 

0.2  6.236701377 
3.941705892 
1.274588648 
1.42857000 

0.4  0.1077711942  6.705855054 
2.023437703 
1.66667000 

0.05  0.6  0.1435445188  9.198944737 
2.687576822 
1.90476000 
0.8  0.1729787028  0.1162794147  3.331413454 
2.14286000 

1  0.1968479851  0.1404973461  3.972410448 
2.38095000 

0.2  4.042725784 
3.259267367 
1.253809850 
1.09090900 

0.4  7.39329309 
6.312390645 
2.311313478 
1.27272700 

0.1  0.6  8.73204114 
8.86309024 
3.227722400 
1.45454500 
0.8  8.52448893 
0.1121969429  4.108477061 
1.63636400 

1  6.88013028 
0.1346907447  4.982174851 
1.81818200 
IC  Initial conditions 
LT  Laplace transform 
LR  Laplace residual 
LRPSM  Laplace residual power series method 
FPDEs  Fractional partial deferential equation 
AE  Absolute error 
ADM  Adomian decomposition method 
The present article is related to the approximate analytical solutions of some nonlinear fractional partial differential equations using the Laplace residual power series method. The fractional derivatives in each targeted problem are represented by the Caputo operator. First, the proposed scheme is discussed for the general nonlinear problem and the few nonlinear problems related to nonlinear fractional partial differential equations are solved by using the proposed method. The obtained results are compared with the exact solution of each problem. The fractional order solutions are analyzed by using the suggested method successfully. It is observed that the present technique is the most suitable tool for the solutions of nonlinear fractional partial differential equations and possesses a higher degree of accuracy. In conclusion, this new hybrid technique is straightforward to solve the nonlinear fractional problems and can be used effectively in other branches of applied sciences.