The fractional analysis of thermo-elasticity coupled systems with non-linear and singular nature

: It is noted that the study of linear and non-linear thermo-elasticity systems has a key role in thermal conductivity, stresses, as well as elasticity, and temperature. The solution to these systems is one of the important parts of the present research. The fractional form of various thermo-elastic systems is discussed, and their solutions are attained in a sophisticated manner. The solutions of fractional and integer thermo-elastic systems are further discussed via graphs and tables. The graphs have shown the closed contact between the exact and LRPSM solutions. For appropriate modeling, the solutions of fractional problems are plotted and shown to be convergent towards integer-order problems solutions. From the tables, it is veriﬁed that as the terms of the obtained series solution increases, the higher accuracy is achieved quickly. The faster convergence and stability of the suggested method supports its modiﬁcation for other fractional higher non-linear complex systems in nature.


Introduction
Many scientists use the core concepts and theory of fractional calculus (FC) to investigate memory-related behaviours and dynamical aspects of scientific phenomena.The fundamental reason for the attraction with fractional operators comes from the fact that the usage of fractional differential and integral operators is related to the great application of different models in designing, chemical engineering, physical science, and mathematics.It has been rapidly growing and playing a key part in a variety of sectors, assisting in the modeling of innovative problems linked to the memory-based fractal framed repercussions and heredity-related procedures.The primary objective of combining an operator of fractional-order into the system and to examine the interactivity of longer-range, a higher degree of freedom, reduced imprecision due to the originality of the principle parameters of real nature, non-local effects that emphasize future representative and historical states, maximum information usage, and the systems having fractional-order are the particular cases of the system of conventional-order.By proposing new ideas, many researchers are laying the groundwork for the growth of FC [1,2,[4][5][6][7][8].The solutions of physical and technical importance fractional ordinary and fractional partial differential equations (FODEs,FPDEs) and integral equations (FIEs) [9] have gotten a lot of attention.Because most non-linear fractional-order problems do not have exact solutions, to investigate their approximation solutions, analytical and numerical approaches have been suggested and used.On other hand, many scholars have studied the mathematical characteristics of FPDEs and tried to solve it.
FC is related to real endeavours, and it is broadly used within chaos theory, optics, nanotechnology, human diseases and other fields [6,[10][11][12][13][14][15][16]].The analytical and numerical solutions for the above models play an essential part in depicting the aspect of nonlinear issues in related fields of study.Coupled one-dimensional nonlinear thermo-elasticity coupled systems can be found in a variety of scientific domains, including solid-state and plasma physics.Because of their importance and applications, thermo-elasticity problems have received a lot of attention.Coupled linear and nonlinear thermo-elasticity coupled system offer a broad field of study for studying the interaction of the mechanical and thermal domains.The study of associated stresses, thermal conductivity, and temperature, elasticity is known as thermo-elasticity.Recently, the investigation of these ideas have piqued the interest of numerous scholars working in many fields related to mathematics.Famous scientists, mathematicians, and engineers were influenced by the certainty of irrational physical behaviour, as depicted by elastic deformations obtained by temperatures stresses.For such types of systems exact solutions are difficult to obtain.As a result, some analytical and numerical approaches are developed for finding solutions to FPDEs, such as the VIM [17], ADM [18], q-HAM [19], the fast element-free Galerkin method [20], operational matrix method [21], fractional natural decomposition scheme [22], Fourier transform technique [23], Homotopy perturbation method [24], operational calculus method [25], Laplace-Sumudu transform method [26], multistage differential transformation method [27], iterative reproducing kernel method [28] have recently been developed.
El-Ajou [29] used Laplace residual power series method (LRPSM) for first time to study the solitary form exact solutions of time FPDEs.LRPSM [30,31] is made by using the hybride form of Laplace transform (LT) and RPSM [32][33][34][35][36][37].The LT is used firstly in the LRPSM to transfer the target problem into algebraic equations.The RPSM is then used to compute the series solution.Finally, inverse LT is used to get the approximate result.The LRPSM requires fewer calculations in less time and with more precision.LRPSM is an effective and simple strategy for creating a power series (PS) solution for FPDEs that does not need linearization, discretization and perturbation.A series of algebraic expressions are obtained to determine the PS coefficients.The methodology's key advantage is that it relies on simpler and more accurate derivation than other integration-based techniques.In theory, this method is an alternative way of solving FPDEs [38].The LRPSM is utilized to solve the non-linear systems that arise in thermo-elasticity in this article.The generalized LRPSM technique is provided, followed by the application of the LRPSM algorithm to a few numerical problems.Tables and graphs show the results and efficiency of the suggested method.The graphical representation is completed, and the results are extremely near to the actual solution of each target problem.The fractional-order LRPSM solutions are useful for analyzing the dynamics of the provided problems.
The current paper's summary is provided here.Section 2 discussed some necessary definitions and results from FC theory.and section 3 the basic technique is presented, certain test models are used to confirm the efficiency of LRPSM in section 4, the findings are reviewed in section 5, and the conclusion is provided in section 6.

Preliminaries
This section discusses the fundamental definitions and theorems of fractional derivatives in the Caputo meaning.

Definition
The derivative of a function P (ω, ξ) of order µ is expressed as in Caputo sense [3] (1)

Definition
Suppose that P (ω, ξ) is continuous piecewise and having µ as exponential order, LT can be explained as [29]: where the inverse LT is given as The properties of the LT and its inverse are summarised in the following Lemma [39].

Lemma
Assume that P (ω, ξ) is of exponential order ξ and continuous piecewise, and

Remarks
The inverse LT of the Eq. ( 4) represented as: It is equal to the illustration of the fractional order Taylor's formula in [40].
The following Theorem explains and establishes the FPS convergence in the 2.5 Theorem.

Theorem
Let the function P (ω, ζ) is piecewise continuous on interval I × [0, ∞) and of exponential order ϱ can be presented as the fractional expansion in Theorem where 0 < ζ ≤ 1, then the remainder R n (θ, s) satisfies the below inequality Theorem which satisfy the following [29]:

LRPS Methodology
In this section, we will discuss the methodology of LRPSM for nonlinear one dimensional thermo-elasticity coupled system with initial conditions (IC's) where P and R are displacement and temperature difference respectively, Using Eq. ( 9) in Eq. ( 7), we get Using LT to Eq. ( 10) and using IC's from Eq. ( 8), we get Let the approximate solution of Eq. ( 11) has the following form The j th -truncated term series are Laplace residual functions (LRFs) [29] are And the j th -LRFs as: The following list includes some key facts regarding the Laplace residual function that are critical to determining the approximation of the solution [29].
To find the coefficients f i (ω) and g i (ω), we recursively solve the following system In the last, we apply inverse LT to Eq. ( 13), to get the j th approximate solutions of P j (ω, ξ) and R j (ω, ξ).

Problem
Nonlinear thermo-elasticity coupled system in one-dimensional is given as: [42] D Subject to the ICs Exact solution for the Eq. ( 17) is Using LT to Eq. ( 17) and using IC's from Eq. ( 18), we get The j th -truncated term series are And the j th -LRFs as: Putting the j th truncated term series of Eq. ( 21) into the j th truncated Laplace residual function of Eq. ( 22), multiplying the resulting expression by s jζ+1 and then solve the systems lim j→∞ s jζ+1 LRes P,j (ω, s) = 0 and lim j→∞ s jζ+1 LRes R,j (ω, s) = 0 to find the unknown coefficients κ j (ω) and ϱ j (ω) for j = 1, 2, 3, • • • , the following are the first few terms of the approximate solutions Substituting κ j (ω) and ϱ j (ω) for j = 1, 2, 3, • • • in Eq. ( 21), we have Utilizing inverse LT on Eq. ( 24) Putting ζ = 1 in Eq. ( 25), we get the exact solution given in Eq. ( 19).

Problem
Nonlinear one-dimensional coupled system thermo-elasticity given by [42] Subject to the ICs The exact solution of Eq. ( 26) is Using LT to Eq. ( 26) and using IC's from Eq. ( 27), we get The j th -truncated term series are And the j th -LRFs as: Putting the j th truncated term series of Eq. ( 39) into the j th truncated Laplace residual function of Eq. ( 40), multiplying the resulting expression by s jζ+1 and then solve the systems lim j→∞ s jζ+1 LRes P,j (ω, s) = 0 and lim j→∞ s jζ+1 LRes R,j (ω, s) = 0 to find the unknown coefficients κ j (ω) and ϱ j (ω) for j = 1, 2, 3, • • • , the following are the first few terms of the approximate solutions Substituting κ j (ω) and ϱ j (ω) for j = 1, 2, 3, • • • in Eq. ( 39), we have Utilizing inverse LT on Eq. ( 42), we get the approximate solution as Putting ζ = 1, we get the exact solution which is given in Eq. ( 26).
Subject to the ICs P (ω, Eq. ( 35) exact solution is as follows: Using LT to Eq. ( 35) and using IC's from Eq. ( 36), we get The j th -truncated term series are And the j th -LRFs as: Putting the j th truncated term series of Eq. ( 39) into the j th truncated Laplace residual function of Eq. ( 40), multiplying the resulting expression by s jζ+1 and then solve the systems lim j→∞ s jζ+1 LRes P,j (ω, s) = 0 and lim j→∞ s jζ+1 LRes R,j (ω, s) = 0 to find the unknown coefficients κ j (ω) and ϱ j (ω) for j = 1, 2, 3, • • • , the following are the first few terms of the approximate solutions Substituting κ j (ω) and ϱ j (ω) for j = 1, 2, 3, • • • in Eq. ( 39), we have Utilizing inverse LT on Eq. ( 42), we get the approximate solution as Putting ζ = 1, we get the exact solution which is given in Eq. ( 37).

Results and disscusions
In this section, the numerical solutions of the coupled system considered in Problem 4.1 are discussed, which is given in Tables 1 and 2. Further, in Tables 1 and 2. The effectiveness of the technique is indicated, as the number of iterations increases, the solution approaches to the exact solution.We can see from the tables, the present technique has the higher accuracy.The 2D and 3D plots are presented to highlight the LRPSM results at different values of parameters.The 2D plots of problem 4.1 at various fractional-order for P (ω, ξ) is shown in Fig. 1 and for R(ω, ξ) is shown in Fig. 5 17, 19, 20, 18, 21, and 22.With the help of FC, we can study and analyze the physical behavior of non-linear problem by simulating and displaying its physical properties.The suggested technique is more suitable and efficient in analyzing complex coupled fractional-order problems.All the numerical calculations are done by Maple 2020.

Conclusion
The present study is the broader and useful concept in the fractional analogue of some thermo-elasticity systems.In this work, the thermo-elasticity systems are first represented in their Caputo fractional definition and then tried successfully to investigate their useful dynamics by using an accurate procedure.The non-linearity is directly handled by the current technique, which is very uncommon in other existing technique.The Laplace transformation is applied to convert the given problems into simple form and then implemented residual power series method to achieve the completed solution.It is observed that the results are fully compatible with the actual dynamics of the suggested problems.In particular, fractional solutions provide many choices to describe the useful dynamics of problems.The present study can makes a valuable contribution toward the analysis of other higher non-linear complex phenomena.

Figure 1 .
Figure 1.2D graph shows the comparison of P (ω, ξ) LRPSM solution at various fractional order of Example 4.1.

Figure 2 .
Figure 2. 2D graph shows the comparison of LRPSM and exact solutions for P (ω, ξ) at various fractional order of Example 4.1.

Figure 6 .
Figure 6.2D graph shows the comparison of LRPSM and exact solutions for R(ω, ξ) at various fractional order of Example 4.1.
and the solution in 3D surfaces are shown in Figs. 3 and 7.The comparison of LRPSM and exact solutions are plotted in Figs. 2, 4, 6, 8, 10, 12, 14 and 16 as 2D and 3D representations for problems 4.1, 4.2 and 4.3 respectively.Similarly, 2D and 3D plots of Problems 4.2 and 4.3 at various fractional-order are discussed in Figs. 9, 13, 11 and 15, respectively.Meanwhile, the response of the LRPSM solution in term of AE for various arbitrary orders are shown in Tables.3, 4, 5 and 6 respectively for Problem 4.2 and Problem 4.3.Also for Problem 4.3, the solutions are plotted in 2D and 3D graphs in Figs.