Modified Modelling for Heat Like Equations within Caputo Operator

The present paper is related to the analytical solutions of some heat like equations, using a novel approach with Caputo operator. The work is carried out mainly with the use of an effective and straight procedure of the Iterative Laplace transform method. The proposed method provides the series form solution that has the desired rate of convergence towards the exact solution of the problems. It is observed that the suggested method provides closed-form solutions. The reliability of the method is confirmed with the help of some illustrative examples. The graphical representation has been made for both fractional and integer-order solutions. Numerical solutions that are in close contact with the exact solutions to the problems are investigated. Moreover, the sample implementation of the present method supports the importance of the method to solve other fractional-order problems in sciences and engineering.


Introduction
Heat is the source of energy that can be transferred from one device to another due to differing temperatures. A thermodynamic analysis refers to the amount of heat transferred as a scheme that undergoes a transition from one state of equilibrium to another [1,2]. Heat transfer is the science that deals with determining the rate of such energy transfers. Transformation of energy as heat is usually from the higher to the lower temperature, and the heat transfer ceases when both mediums exceed the same temperature [3,4]. Heat can be transmitted in three differing classes-radiation, conduction and convection. The problem of fractal heat conduction describes heat transport in inhomogeneous materials as fibrous materials, coal deposits, textiles and other discontinuous media where homogenization is not acceptable because this approach neglects the important physical characteristics of transport processes. Non-smoothness raises problems and avoids the implementation of classical calculus a fractional and integer-order [5][6][7].
Definition 3. The Laplace transform is described as [44] L Definition 5. The Mittag-Leffler function is expressed as [44]

The Presentation of the Method
In this section, we will briefly discuss ILTM, to solve fractional-order nonlinear PDEs [44].
Using Laplace transform of Equation (6) we get Applying the property of Laplace differentiation [44] L[ϑ( , , By using the inverse Laplace transform of Equation (9), we obtain From the iterative technique, Since R is a linear operator and the non-linear operator N is split as [44] Putting Equations (11)- (13) in Equation (10), we obtain Using Equation (14), we defined the following iterative formula [44] ϑ 0 ( , , The approximate m-term solution of Equations (6) and (7) in form of series as [44] ϑ( , ,

Implementation of the Method
In this section, ILTM and ISTM are applied to examine the solution of fractional-order heat-like equations. It has been shown that the ILTM and ISTM are an accurate and appropriate analytical technique to solve non-linear FPDEs. Example 1. Consider the following one-dimensional fractional-order heat equation [8,14,48]: with initial condition The Sumudu transformation to Equation (19) is expressed as Using the inverse Sumudu transformation of Equation (22),we get Using the iterative technique described in Equations (11)-(13), we obtain the following solution components of Example 1 The analytical solution of the series form is given as Therefore, the approximate solution of Equation (19) is given as Now that in the case ρ = 1 This is the exact solution for this case. Figure 1 represents the exact and ISTM solutions in graphs (a) and (b), respectively, for Example 1 at ρ = 1. The best contact is observed between the exact and ISTM solutions of Example 1. In Figure 2, the solutions at different fractional orders of the derivatives are calculated for Example 1. In Figure 2, the fractional order solutions of Example 1 at γ = 1, 0.8, 0.6 and 0.5 are expressed in three and two dimensions by sub-figures (a) and (b) respectively. The graphical representation in Figure 2 confirmed the convergence phenomena of fractional order solutions towards the integer-order solution of Example 1.
The Laplace transformation to Equation (32) is expressed as Using inverse Laplace transformation of Equation (35),we get Using iterative technique describe in Equations (11)-(13), we obtain the following solution components of Example 2 ϑ 0 ( , , τ 1 ) = sin sin , The analytical solution of series form is given as Therefore the approximate solution of the equation is given as when ρ = 1 we get, the exact solution is The comparison between ILTM and the exact solution has been done in Figure 3 for Example 2. The comparison has shown the closed resemblance between the actual and ILTM solutions. In Figure 4, the error analysis of ILTM has been done for Example 2 at ρ = 1. It is observed that the error associated with ILTM is consistent throughout the collocation points.  Example 3. Consider the following two-dimensional fractional-order heat equation [8,14]: with initial condition The Laplace transformation to Equation (45) is expressed as Using the inverse Laplace transformation of Equation (48), we get Using the iterative technique described in Equations (11)-(13), we obtain the following solution components of Example 3 The series form of the analytical solution is given as Therefore the approximate solution of the equation is given as when ρ = 1 we get, the exact solution is ϑ( , , τ 1 ) = 2 sinh τ 1 + 2 cosh τ 1 . (58) In Figure 5, the graphical representation of the ILTM and exact solutions of Example 3 are presented. The closed contact between the exact and ILTM solutions in graphs (a) and (b) of Figure 5 for Example 3 is observed. In Figure 6, the ILTM solutions at different fractional orders for Example 3 is shown. The analysis shows that there is a strong convergence of the fractional-order solutions towards the integer order solution of Example 3. The solution convergence can be seen in both one and two-dimensional graphs represented by (a) and (b), respectively.
In Figure 7, the fractional order solutions of Example 4 are given in both dimensional graphs (a) and (b), respectively. In both cases, the convergence phenomena of fractional-order solutions to the integer-order solution can be seen for Example 4. The implementation of the suggested method for various numerical examples has supported the validity of the proposed method. The present technique has the greater capacity to solve partial differential equations of fractional-order and integer-order as well.

Conclusions
The fractional view analysis of the heat like equations, using an efficient analytical approach, was the focus of the present research work. The approximate analytical solutions for both fractional and integer-order heat like equations are obtained in a sophisticated manner. The graphical analysis of the obtained solutions has been done successfully. The analysis has confirmed the strong agreement between the proposed and exact solutions to the problems. The present method have proved to be an effective and straightforward procedure as compared with other analytical and numerical techniques. Moreover, the suggested method required fewer calculations and therefore can be extended for the solutions of other fractional-order problems.

Conflicts of Interest:
The authors declare no conflict of interest.