Abstract
In this article, natural transform iterative method has been used to find the approximate solution of fractional order parabolic partial differential equations of multidimensions together with initial and boundary conditions. The method is applicable without any discretization or linearization. Three problems have been taken as test examples and the results are summarized through plots and tables to show the efficiency and reliability of the method. By practice of a few iterations, we observe that the approximate solution of the parabolic equations converges to the exact solution. The fractional derivatives are considered in the Caputo’s sense.
1 Introduction
Fractional calculus was first studied in the seventeenth century and has recently received a lot of interest. Scientists have discovered that fractional calculus may describe memory and hereditary features of various problems in science and engineering due to fractionalorder derivatives. As a result, we observe the fractional calculus in many domains like signal processing, diffusion, physics, fluid mechanics, biology, chemistry, economics, polymer rheology etc. It has its significance almost in every field of science and technology. It models natural phenomena in more suitable way than classical calculus. Due to the rapid development in the daytoday life, fractional calculus has also played important role in engineering, biosciences, and finance [1,2,3,4]. Fractional calculus is the generalization of classical calculus. The differential equations arising in fractional calculus are termed as fractional differential equations (FDEs). FDEs have a noninteger order derivative and can be solved through derivative and integral operators related to fractional calculus. Different operators have been defined by mathematicians for the solution of FDEs [5,6,7].
In many instances exact solution to differential equations is not always possible and the approximate solutions to these equations are obtained by using different numerical and analytical methods. In order to obtain approximate solutions to differential equations, different numerical methods were developed over time. However, the numerical solutions were not enough to determine the overall properties of certain systems of differential equations which leads us to the development of some new analytical and semianalytical methods. These methodologies have revolutionized numerical analysis, allowing us to present difficult issues with both qualitative and quantitative analysis. For the solution of linear and nonlinear FDEs, several analytical, numerical, and homotopybased approaches have been used [8,9,10,11]. Some analytical techniques in combination with transformations have also been applied to handle FDEs more suitably. The homotopy analysis Sumudu transform Method is a combination of homotopy analysis method (HAM) and Sumudu transform used by Singh et al. for handling the fractional Caudrey–Dodd–Gibbon equations [12]. Dubey et al. used the local fractional natural HAM which is the combination of the HAM and local fractional natural transform for solving partial differential equations (PDEs) of fractional order [13]. Supriya Yadav et al. also used the qHASTM for solving the fractional reactiondiffusion equations [14]. Recently Jagdev Singh presented the composite fractional derivative to analyze the fractional blood alcohol model [15]. The derivative is considered in the Caputo’s sense due to its most applicability and popularity to the FDEs and can be handled easily by the proposed method.
One of the relevant algorithms we have used in this research is the natural transform iterative method (NTIM), which is based on new iterative method (NIM) and the natural transform [16,17]. The usage of natural transform and NIM make it easier and more appropriate to handle FDEs. For the investigation of PDEs of integer order and of fractional order, this method is free of discretization and linearization. NIM has been used by a number of other researchers to solve fractional order PDEs [18,19,20,21]. The proposed method’s convergence may be demonstrated, as demonstrated by Bhalekar et al. [22]. This work looks into the fractional order parabolic differential equations of fourthorder with variable coefficients. Fractional order parabolic PDEs have the general form as reported in ref. [23].
where
and boundary conditions as
where
The rest of this article is set out as follows. The first section contains some basic fractional calculus definitions. The basic concept of NTIM is explained in Section 2. The application of NTIM to parabolic equations is covered in Section 3. In Section 4, some numerical results are presented. Finally, Section 5 gives the conclusion.
2 Fractional calculus
Some definitions are presented from the fractional calculus.
2.1 Definition
The fractional integral in Riemann–Liouville’s (R–L) sense of a function f(ϕ) is defined as
where Γ is the gamma function defined as
2.2 Definition
The fractional order derivative of a function f (ϕ) in the Caputo’s sense is given as
for
2.3 Definition
Relationship of the Caputo’s derivative and R–L integral is given as
For
2.4 Definition
Natural transform of
u and s are the transformation variables.
2.5 Definition
The inverse of the natural transform
2.6 Definition
If
2.7 Theorem
If
H(s,u), respectively, then,
where [h * k] is convolution of h and k.
2.8 Remark
A few important natural transformations of some functions are given below.








3 NTIM [17]
Consider the FDE of the form
where
Taking the natural transform of Eq. (9), we have
by applying the differentiation property of natural transform (given in definition 2.6) to Eq. (11) we have
Using the initial conditions and rearranging Eq. (12) we obtain
Where the linear term
and
Using Eqs. (14) and (15) in Eq. (13) we obtain
The recursive relation of Eq. (16) by the use of natural transform is
Applying the inverse natural transform to Eq. (17) the solution component can be obtained as
The n terms’ approximate solution of Eqs. (9) and (10) by the proposed method is obtained by adding the components as
3.1 Convergence of NTIM [23]
3.1.1 Theorem
If N is analytic in a neighborhood of
for any m and for some real
To show the boundedness of
Sufficient condition for convergence is given in the following theorem.
3.1.2 Theorem
If
The above mentioned are the required conditions for the convergence of the series
4 Applications
4.1 Problem 1
Consider the (1 + 1) dimension parabolic equation of the form [24]
with initial conditions
and boundary conditions as
Rearranging Eq. (20) as
and applying the natural transform, we get
Using the differentiation property of natural transform, we have
which after rearranging and simplification we have
Applying the inverse natural transformation, we obtain
Using the initial conditions, we have
Using the recursive relation, as
Adding the solution components, we get the approximate solution as
For β = 2, Eq. (30) reduces to
which converges to the exact solution as
4.2 Problem 2
Consider the (2 + 1) dimension parabolic equation of the form [24]
with initial conditions
and boundary conditions as
Rearranging Eq. (33) as
Applying the basic procedure of NTIM and using the initial conditions, we obtain the solution components as
Adding the solution components, we get the approximate solution as
For β = 2, Eq. (33) reduces to
which converges to the exact solution given as
4.3 Problem 3
Consider the (3 + 1) dimension parabolic equation of the form [24]
with initial conditions
and boundary conditions as
Rearranging Eq. (41) as
Applying the basic procedure of NTIM and using the initial conditions, we obtain the solution components as
Adding the solution components, we get the approximate solution as
For β = 2, Eq. (41) reduces to
which converges to the exact solution given as
5 Numerical results and discussions
We have obtained the analytical approximate solution of the fourthorder parabolic time fractional PDEs by applying the natural transform in combination with the NIM. It is noted that the solution pattern converges to the exact solution after a few iterations which shows the reliability of our proposed method. All the solutions are approximated up to fourthorder for problems 1–3. Figures 1 and 2 show the approximate solution and exact solution, respectively, for problem 1. Figure 3 shows the absolute error for problem 1. Figures 4 and 5 show the absolute error and relative error by 2D plots. Figures 6 and 7 show the comparison of the fractional value of β as it converges to the exact solution when the value of β approaches to 2. Similarly, Figures 8 and 9 represent the approximate and exact solution in 3D plots for problem 2. The absolute error is shown by 3D plot in Figure 10 by keeping one parameter ղ constant. Figures 11 and 12 are the 2D graphs of the absolute and relative errors, respectively. The approximate solution is compared with the exact solution by giving different values to β in Figures 13 and 14. The approximate solution and exact solution for problem 3 have been shown through 3D plots by Figures 15 and 16, respectively. The absolute error is shown for the said problem in Figure 17. The absolute and relative errors are shown in Figures 18 and 19, respectively, with the help of 2D plots. The different fractional values are compared with exact solution by giving different values to β (Figures 20 and 21). The numerical values for different values of β are compared in Tables 1 and 2 for problem 1, Tables 3 and 4 for problem 2, and Tables 5 and 6 for problem 3. As the value of β approaches 2, which changes the FDE to a classical PDE, the approximate solution converges to the exact solution, as seen in the figures and tables. The parameters have been interpreted by keeping some parameters constant and others to expand through 2D and 3D plots. All the figures show that the method is convergent and has excellent degree of accuracy.
τ  β = 1.5  β = 1.7  β = 1.9  β = 2.0  Exact  Abs. error 

0.2  0.196306  0.198551  0.199884  0.200325  0.200325  2.77556 × 10^{−17} 
0.4  0.373684  0.383362  0.390131  0.392663  0.392663  1.16573 × 10^{−15} 
0.6  0.525633  0.546467  0.562749  0.569348  0.569348  2.11164 × 10^{−13} 
0.8  0.649208  0.68249  0.711002  0.723334  0.723334  8.87512 × 10^{−12} 
1.0  0.743628  0.788051  0.829477  0.848483  0.848483  1.61161 × 10^{−10} 
1.2  0.809641  0.861515  0.914196  0.939806  0.939806  1.72071 × 10^{−9} 
1.4  0.849104  0.902788  0.962692  0.993662  0.993662  1.27334 × 10^{−8} 
1.6  0.864641  0.913114  0.974042  1.0079  1.0079  7.20455 × 10^{−8} 
1.8  0.859358  0.894854  0.948843  0.981963  0.981963  3.32043 × 10^{−7} 
2.0  0.836591  0.851263  0.889127  0.916874  0.916875  1.30162 × 10^{−6} 

β = 1.5  β = 1.7  β = 1.9  β = 2.0  Exact  Abs. error 

0.2  0.737484  0.78154  0.822624  0.841473  0.841473  1.59829 × 10^{−10} 
0.4  0.737545  0.781604  0.822692  0.841543  0.841543  1.59842 × 10^{−10} 
0.6  0.73796  0.782044  0.823155  0.842016  0.842016  1.59932 × 10^{−10} 
0.8  0.739496  0.783672  0.824868  0.843769  0.843769  1.60265 × 10^{−10} 
1.0  0.743628  0.788051  0.829477  0.848483  0.848483  1.61160 × 10^{−10} 
1.2  0.752774  0.797744  0.83968  0.85892  0.85892  1.63143 × 10^{−10} 
1.4  0.770535  0.816565  0.859491  0.879185  0.879185  1.66992 × 10^{−10} 
1.6  0.801924  0.84983  0.894504  0.915000  0.915000  1.73795 × 10^{−10} 
1.8  0.853609  0.904602  0.952155  0.973972  0.973972  1.84996 × 10^{−10} 
2.0  0.934144  0.989948  1.04199  1.06586  1.06586  2.02449 × 10^{−10} 
τ  β = 1.5  β = 1.7  β = 1.9  β = 2.0  Exact  Abs. error 

0.2  0.389908  0.394367  0.397014  0.397891  0.397891  9.99201 × 10^{−16} 
0.4  0.74222  0.761443  0.774889  0.779918  0.779918  2.10221 × 10^{−12} 
0.6  1.04403  1.08541  1.11775  1.13085  1.13085  1.81609 × 10^{−10} 
0.8  1.28948  1.35558  1.41221  1.4367  1.4367  4.29227 × 10^{−9} 
1.0  1.47703  1.56525  1.64753  1.68528  1.68528  4.98537 × 10^{−8} 
1.2  1.60821  1.71117  1.8158  1.86667  1.86667  3.69378 × 10^{−7} 
1.4  1.68681  1.79318  1.91213  1.97364  1.97364  2.00653 × 10^{−6} 
1.6  1.71828  1.8138  1.93469  2.00193  2.00192  8.68357 × 10^{−6} 
1.8  1.70936  1.77785  1.8847  1.95043  1.9504  3.15864 × 10^{−5} 
2.0  1.66774  1.69208  1.76625  1.82122  1.82112  1.00171 × 10^{−4} 

β = 1.5  β = 1.7  β = 1.9  β = 2.0  Exact  Abs. error 

0.2  1.47601  1.56416  1.64639  1.68411  1.68411  4.98191 × 10^{−8} 
0.4  1.47601  1.56417  1.64639  1.68412  1.68412  4.98193 × 10^{−8} 
0.6  1.47605  1.56421  1.64644  1.68417  1.68417  4.98207 × 10^{−8} 
0.8  1.47627  1.56445  1.64669  1.68442  1.68442  4.98282 × 10^{−8} 
1.0  1.47703  1.56525  1.64753  1.68528  1.68528  4.98537 × 10^{−8} 
1.2  1.47906  1.5674  1.6498  1.6876  1.6876  4.99224 × 10^{−8} 
1.4  1.48372  1.57234  1.65499  1.69291  1.69291  5.00794 × 10^{−8} 
1.6  1.49319  1.58237  1.66555  1.70372  1.70372  5.03992 × 10^{−8} 
1.8  1.51084  1.60108  1.68525  1.72386  1.72386  5.09950 × 10^{−8} 
2.0  1.54156  1.63363  1.71951  1.75891  1.75891  5.20318 × 10^{−8} 
τ  β = 1.5  β = 1.7  β = 1.9  β = 2.0  Exact  Abs. error 

0.2  1.19041  1.15741  1.13645  1.12911  1.12911  3.86358 × 10^{−14} 
0.4  1.06156  0.993212  0.943736  0.924434  0.924434  3.84487 × 10^{−11} 
0.6  0.962711  0.864368  0.788285  0.756862  0.756862  2.17861 × 10^{−9} 
0.8  0.883437  0.760652  0.661903  0.619666  0.619666  3.80252 × 10^{−8} 
1.0  0.818041  0.675718  0.558569  0.50734  0.50734  3.48164 × 10^{−7} 
1.2  0.762877  0.605227  0.473684  0.415373  0.415375  2.11990 × 10^{−6} 
1.4  0.715275  0.546021  0.40366  0.34007  0.34008  9.74103 × 10^{−6} 
1.6  0.672942  0.495657  0.345645  0.278398  0.278434  3.64289 × 10^{−5} 
1.8  0.633511  0.452083  0.29732  0.227846  0.227963  1.16411 × 10^{−4} 
2.0  0.594103  0.413349  0.25673  0.186311  0.18664  3.28612 × 10^{−4} 

β = 1.5  β = 1.7  β = 1.9  β = 2.0  Exact  Abs. error 

0.2  0.0826461  0.0682673  0.0564318  0.0512562  0.0512562  1.75874 × 10^{−8} 
0.4  0.236281  0.195173  0.161336  0.146539  0.146539  5.02815 × 10^{−8} 
0.6  0.411698  0.34007  0.281112  0.25533  0.25533  8.76108 × 10^{−8} 
0.8  0.606632  0.501089  0.414216  0.376226  0.376226  1.29093 × 10^{−7} 
1.0  0.818041  0.675718  0.558569  0.50734  0.50734  1.74082 × 10^{−7} 
1.2  1.04223  0.8609  0.711646  0.646378  0.646378  2.21790 × 10^{−7} 
1.4  1.27498  1.05316  0.870575  0.79073  0.79073  2.71321 × 10^{−7} 
1.6  1.51176  1.24874  1.03225  0.937575  0.937576  3.21708 × 10^{−7} 
1.8  1.74784  1.44375  1.19345  1.08399  1.08399  3.71947 × 10^{−7} 
2.0  1.97855  1.63432  1.35098  1.22708  1.22708  4.21044 × 10^{−7} 
6 Conclusion
The proposed method is tested upon the time fractional parabolic equations of fourth order. The method shows its convergence as the solution pattern after a few iterations converges to the exact solution. The algorithm of the proposed method is easy to apply without any discretization. The approximate solutions are found to be in excellent agreement with the exact solution. The numerical values of approximate solution for different fractional values have been compared in tables, and the results have been shown in 3D and 2D graphs. We found that for β = 2, the suggested method’s approximate solution converges to the precise solution of the problems, ensuring NTIM’s dependability and correctness. The results show that the NTIM successfully delivers accurate, faster converging solutions while using fewer computer resources than other approaches in the literature.

Funding information: The authors state no funding involved.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.
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