Invariant Solutions and Conservation Laws of the Time-Fractional Telegraph Equation

Najafi, Ramin; Çelik, Ercan; Uyanık, Neslihan

doi:https://doi.org/10.1155/2023/1294070

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2023 | Article ID 1294070 | https://doi.org/10.1155/2023/1294070

Invariant Solutions and Conservation Laws of the Time-Fractional Telegraph Equation

Ramin Najafi,¹Ercan Çelik,²and Neslihan Uyanık³

Academic Editor: Igor Freire

Received07 Nov 2022

Revised24 Dec 2022

Accepted29 Jul 2023

Published29 Aug 2023

Abstract

In this study, the Lie symmetry analysis is given for the time-fractional telegraph equation with the Riemann–Liouville derivative. This equation is useable to describe the physical processes of models possessing memory. By applying classical and nonclassical Lie symmetry analysis for the telegraph equation with time-fractional derivatives and some technical computations, new infinitesimal generators are obtained. The actual methods give some classical symmetries while the nonclassical approach will bring back other symmetries to these equations. The similarity reduction and conservation laws to the fractional telegraph equation are found.

1. Introduction

For decades, because fractional differential equations have a major role in many different fields of science, the theory of fractional differential equations has attracted broad interest in the different areas of applied sciences [1–5]. The fractional derivatives have also been of basic importance in the mathematical modeling of many systems in control [6], chemistry–biochemistry [7, 8], finance [9], disease transmission dynamics [10–12], and other disciplines. Fractional derivatives supply a perfect and elegant tool to describe diverse materials and processes with memory and hereditary characteristic of a variety of different phenomena. By Xu et al. [13], Zhang et al. [14], Xu and Zhang [15], and Zhang et al. [16], the analytical solutions of the fractional differential equations have been obtained. The telegraph equations describe the current and voltage of wave propagation of electric signals in a cable transmission line to find distance and time, and has many applications such as neutron transport [17], random walk of suspension flows [18], signal analysis for transmission, propagation of electrical signals [19] etc. Since the telegraph equation with classical derivatives can not wells match the abnormal diffusion phenomena during the finite long transmit progress, where the voltage or the current wave possibly exists, in modeling of this equation, from the point of view of memory effects, we obtain the fractional telegraph equations with fractional derivatives [20–22]. The time-fractional telegraph equation can be written as follows:where are arbitrary constants, is smooth function and represent the Riemann–Liouville fractional derivatives. For , the equation becomes the classical telegraph equation. Recently, the fractional telegraph equation has been studied comprehensively theoretically and numerically [23–26]; however, there are limited works using analytical methods. Lie group theory plays a fundamental role in the theory of differential equations and it is efficiently used to get exact solutions to differential equations and fractional differential equations in some disciplines [27–33]. To obtain new infinitesimal generators of a fractional differential equation and new solutions, we use the class of conditional symmetries or the so-called nonclassical symmetry method, which is proposed by Bluman and Cole [34] for the first time. A number of investigators have applied the method to find analytical solutions for the fractional differential equations [35–40]. Conservation laws have a crucial role in the mathematical physics. They describe the physically conserved quantities such as energy, mass, and other constants of the motion. Noether’s [41] theorem produces a link between symmetries and conservation laws of the differential equations.

In this paper, we intend to get the nonclassical Lie group method and conservation laws of the telegraph equation with fractional order. We obtain by using the nonclassical method, infinitesimal generators that are new when compared with obtained symmetries by the classical method. The continuation of the article is as given below: mathematical preliminaries of the Riemann–Liouville fractional derivative and a description of classical and nonclassical Lie symmetry analysis are presented in Section 2. In Section 3 classical, nonclassical Lie symmetries, and similarity reductions of the telegraph equation with fractional order are constructed. Utilizing the obtained symmetries nontrivial conservation laws are given in Section 4.

2. Preliminaries

This section deals with basic concepts, definitions, and lemmas of fractional derivatives and classical, and nonclassical Lie symmetry analysis.

2.1. Basic Definition on Fractional Derivatives

Definition 1. [5] Let . The Riemann–Liouville fractional derivative of order of a function is defined byfor all .

Proposition 1. [5] Let be analytical in for some , and let . Thenfor , where denotes the -th derivative of at .

Proposition 2. [5] (Leibniz’s formula for Riemann–Liouville fractional derivative) Let and assume that and are analytical on with some . Then,for .

Definition 2. [5] Let . The function defined bywhenever the series converges is called the two-parameter Mittag–Leffler function with parameters and .

2.2. Description of Classical and Nonclassical Lie Symmetry Analysis

Let us consider a one-parameter Lie group of infinitesimal transformations on an open subset with coordinate where is the group parameter. Then its associated Lie algebra is spanned by the following infinitesimal generators

We first recall the details of the classical symmetries method for the fractional differential equation. Let us consider the following time-fractional differential equationdefined over . A one-parameter group of transformations as in Equation (7), is a classical Lie symmetry group of Equation (10) if and only if for every infinitesimal generator of we havewherewith

Using Leibniz’s formula for Riemann–Liouville fractional operator and the chain rule, we can writewhere

Here , , denotes the total derivative operator defined byand , denote the total fractional derivative and the total derivative of , w.r.t. , respectively.

According to the group of transformations in nonclassical methods Equation (7), we consider the invariant surface conditionand we investigate the invariance of both the original equations together with this equation. Because for every , and the invariant surface condition is always invariant under the group of transformations Equation (7), then a one-parameter group Equation (7), is a nonclassical Lie symmetry group of Equation (10) if and only if for every infinitesimal generator of , we have

3. Lie Symmetries and Similarity Reductions of the Time-Fractional Telegraph Equation

In this section, we study the invariance properties of the time-fractional telegraph equation. Let us consider the time-fractional telegraph equationwhere are arbitrary constants and is given smooth function. Now we wish to obtain infinitesimals , and so that the one-parameter group Equation (7) be a symmetry group admitted by Equation (19). According to Equation (11) and by applyingto Equation (19), we have the classical Lie’s invariance criterionwhenever satisfies Equation (19). Substituting into this equation, replacing

by whenever it occurs, and equating the coefficients of the various monomials in the partial derivatives of , we obtain the classical determining equations for the symmetry group of the time-fractional telegraph equation. Notice that is defined similarly to . To analyze theses, the coefficients of arerespectively, hence is constant. On the other hand, due to the lower limit of integral in Definition 2.1 be invariant under the group of transformations Equation (7), we have to suppose that

Similarly, the coefficients of (or ) shows that and the coefficients of and (or ) reveals thatwhere are arbitrary constants, is an arbitrary function and satisfies the following fractional equation

Thereforeand the classical symmetry algebra of Equation (19) is spanned by the three vector fields

Example 1. Let where is arbitrary constant. To obtain a classical invariant solution of Equation (19), we consider the one-parameter group generated by with invariant solution Substituting this solution into Equation (19), the reduced fractional ordinary differential equation (ODE) isIf, for example, the solution of this equation can be written in terms of Mittag–Leffler functionswhere are arbitrary constants and is the Riemann–Liouville fractional integral operator of order , thusis the solution of time-fractional telegraph equation.
If where thenChoosing suitable values, the figures for Equation (31) have been displayed in Figures 1 and 2.

(a)

(b)

(a)

(b)

Example 2. If there exists a class of solutions to the Equation (19) admitting group Equation (7), defines a solution too [39], then we haveNow we consider the one-parameter group generated by of the Equation (19), where satisfies the Equation (25). According to Equation (32), we can writeThe solution of this equation can be written in the formsoIf we assume, for instance, thereforeSubstituting this solution into Equation (19), thus according to Equation (25) and assuming thatthe reduced fractional ODE isThe solution of this equation can be written in terms of Mittag–Leffler functionswhere are arbitrary constants. According to Equation (37), if we assume that then we determine the solution of Equation (19) in the following cases: If , then we havewhere denote the real roots to the auxiliary equation of Equation (40). If , then we havewhere denote the repeated real root to the auxiliary equation of Equation (37). If , then we havewhere denote the complex roots to the auxiliary equation of Equation (37) and are arbitrary constants.
Our next goal is now to obtain nonclassical determining equations, employing to Equation (19) and according to Equation (18), the nonclassical infinitesimal invariance criterion is given bywhich must be satisfied whenever satisfies Equation (19) and the invariant surface condition Equation (17). According to invariant surface condition, we consider two different cases: and . In the case , without loss of generality, we can choose , then we haveAfter substituting into Equation (43) with , replacing by similar to classical case, and also resubstituting the expressions wherever it occurs, we attain the nonclassical determining equations for the symmetry group. Now we analyze the order of the derivatives which appear. The coefficient of is , thus we conclude or .
If according to coefficient of (or ), we havewhere are arbitrary functions. After substituting into the remaining term giveswhere and satisfy the following systemTo find the function , we solve equation Equation (48) thuswhere are arbitrary constants. A comparison with Equation (24) shows that the nonclassical method is more general than the classical method for the symmetry group.
If and the set of determining equations for the nonclassical symmetry group of Equation (19) are incompatible.
In the case and , we haveand nonclassical determining equations unfortunately is impossible.

Example 3. According to Equation (50), let . Assume thatand. Then by considering invariat surface condition Equation (20), we havewhose general solution is given bySubstituting of this solution into Equation (19) reduces it to following fractional ODEAfter solving this fractional ODE, taking into account Equation (53) we find the exact solutions of the Equation (19). For example, whenever and let denote the roots of the equation , assuming we havesoIf , thenwhere are arbitrary constants.

Example 4. Suppose that, andThen in view of the invariant surface condition, we have . Differentiating this equation with respect to leads toThus, the general solution of the equation iswhere and are arbitrary functions. Substituting into Equation (19), after some simplifications, we find the reduced fractional ODEsWe can solve these equations similar to previous examples and obtain the exact solutions to Equation (19).

4. Conservation Laws

In the current section, we will focus on the conservation laws of Equation (19). A vector is called a conserved vector for Equation (19), if it fulfills the conservation equationwhere the conserved density and the spatial flux are functions of , integer-order derivatives, fractional integrals and fractional derivatives of . Equation (63) is called a conservation law for Equation (19). The new conservation theorem proposed by Ibragimov [42] provides a method to construct conservation laws for differential equations. Based on this theorem, the Lagrangian formal for Equation (19) can be introduced as follows:where is a new dependent variable. Thus the functional of Equation (19) through the formal Lagrangian Equation (64) is presented as follows:

A necessary condition for functional to have an extremum is

Here the Euler–Lagrange operator can be represented in the formwhere and are the adjoint operators of and are defined by

Thus according to Equation (66), the adjoint equation to Equation (19) is given by

We also use fundamental identity [42–44]where and represent the Noether’s operators, is the identity operator, is given by Equation (20) and we can rewrite in the following formwhere and the same as Equation (16), are defined by

Therefore according to Equation (70), we havewhereand the integral operator is defined by

This integral satisfies the property

So for Equation (19), the operator and can be written by the following formula

Since for all infinitesimal generator admitted by Equation (19) and its any solutiontherefore, in view of Equation (70), the conservation law of Equation (19) can be written as follows:

Now, we present the components of conserved vectors for Equation (19). Using Equations (81) and (82), we havewhere is an arbitrary nontrivial solution of Equation (69) and for different infinitesimal generator , of Equation (19), are defined as follows:

5. Conclusion

In this paper, the methods of classical and nonclassical Lie symmetry analysis are applied to the time-fractional telegraph equation. The Lie symmetry method has had an enormous success when applied to a diverse range of differential equations. The idea of looking for the exact solutions and conservation law for the more derivatives of fractional and integer-order will motivate the nonclassical Lie symmetry method to study such types of fractional partial differential equations (PDEs). The results shown in this work demonstrate how the nonclassical approaches grab the new exact solutions to differential equations with fractional order. Moreover, we constructed conservation laws for the time-fractional telegraph equation by Ibragimovâ€s conservation theorem adapted to the time-fractional PDEs.

Data Availability

Data supporting this research article are available on request.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Copyright © 2023 Ramin Najafi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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