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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 703681, 10 pages
http://dx.doi.org/10.1155/2012/703681
Research Article

On Holomorphic Solution for Space- and Time-Fractional Telegraph Equations in Complex Domain

Institute of Mathematical Sciences, University Malaya, 50603 Kuala Lumpur, Malaysia

Received 19 February 2012; Revised 12 April 2012; Accepted 23 April 2012

Academic Editor: Josip E. Pecaric

Copyright © 2012 Rabha W. Ibrahim. 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.

Abstract

We consider some classes of space- and time-fractional telegraph equations in complex domain in sense of the Riemann-Liouville fractional operators for time and the Srivastava-Owa fractional operators for space. The existence and uniqueness of holomorphic solution are established. We illustrate our theoretical result by examples.

1. Introduction

Fractional differential equations have emerged as a new branch of applied mathematics which has been used for many mathematical models in science and engineering. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. Various types play important roles and tools not only in mathematics but also in physics, control systems, dynamical systems, and engineering to create the mathematical modeling of many physical phenomena. Naturally, such equations required to be solved. Many studies on fractional calculus and fractional differential equations, involving different operators such as Riemann-Liouville operators, Erdelyi-Kober operators, Weyl-Riesz operators, Caputo operators, and Grünwald-Letnikov operators, have appeared during the past three decades with its applications in other fields [16]. Recently, the existence of analytic solutions for fractional differential equations in complex domain is posed [710].

The study of nonlinear problems is of crucial importance in all areas of mathematics, mechanics, and physics. Some of the most interesting features of physical systems are hidden in their nonlinear behavior and can only be studied with appropriate methods designed to tackle and process nonlinear problems. One of the famous nonlinear problem is the telegraph equations used in signal analysis for transmission and propagation of electrical signals and also used in modeling reaction diffusion [11, 12]. Numerical solutions for these equations are posed; for example, the Adomian decomposition method (ADM) and homotopy perturbation method (HPM) were used to solve the space- and time-fractional telegraph equations (see [13]), Variational iteration method (VIM) was used to solve the linear and nonlinear telegraph equation (see [1416]). Note that in [17], the methods (HPM) and (VIM) were compared for linear fractional partial differential equations. Also time-periodic solutions of these equations are studied (see [18]). Recently, Herzallah and Baleanu [19] gave a new model of the abstract fractional-order telegraph equation, and they studied the existence and uniqueness theorems of the strong and mild solutions as well as the continuation of this solution.

In the present paper, we establish the existence and uniqueness of solutions for classes of the space- and time-fractional telegraph equations in complex domain. The fractional derivative for the space is taken in the sense of the Srivastava-Owa operators while the fractional time derivative is taken in the sense of the Riemann-Liouville operators because the Riemann-Liouville fractional derivative could hardly pose the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators.

Definition 1.1. The fractional-order (arbitrary) integral of the function 𝑓 of order 𝛼>0 is defined by 𝐼𝛼𝑎𝑓(𝑡)=𝑡𝑎(𝑡𝜏)𝛼1Γ(𝛼)𝑓(𝜏)𝑑𝜏.(1.1) When 𝑎=0, one writes 𝐼𝛼𝑎𝑓(𝑡)=𝑓(𝑡)𝜙𝛼(𝑡), where () denoted the convolution product, 𝜙𝛼𝑡(𝑡)=𝛼1Γ(𝛼),𝑡>0,(1.2) and 𝜙𝛼(𝑡)=0,𝑡0, and 𝜙𝛼𝛿(𝑡) as 𝛼0 where 𝛿(𝑡) is the delta function.

Definition 1.2. The fractional-order (arbitrary) derivative of the function 𝑓 of order 0<𝛼<1 is defined by 𝐷𝛼𝑎𝑑𝑓(𝑡)=𝑑𝑡𝑡𝑎(𝑡𝜏)𝛼𝑑Γ(1𝛼)𝑓(𝜏)𝑑𝜏=𝐼𝑑𝑡𝑎1𝛼𝑓(𝑡).(1.3) From Definitions 1.1 and 1.2, one has 𝐷𝛼𝑡𝜇=Γ(𝜇+1)𝑡Γ(𝜇𝛼+1)𝜇𝛼𝐼,𝜇>1,0<𝛼1,𝛼𝑡𝜇=Γ(𝜇+1)𝑡Γ(𝜇+𝛼+1)𝜇+𝛼,𝜇>1,𝛼>0.(1.4) We will use the notation 𝜕𝛼/𝜕𝑡𝛼 for the fractional-order derivative.

The Srivastava-Owa fractional operators are (see [20]) as follows.

Definition 1.3. The fractional derivative of order 𝛼 is defined, for a function 𝑓(𝑧), by 𝐷𝛼𝑧1𝑓(𝑧)=𝑑Γ(1𝛼)𝑑𝑧𝑧0𝑓(𝜁)(𝑧𝜁)𝛼𝑑𝜁,0𝛼<1,(1.5) where the function 𝑓(𝑧) is analytic in simply connected region of the complex 𝑧-plane containing the origin, and the multiplicity of (𝑧𝜁)𝛼 is removed by requiring log(𝑧𝜁) to be real when (𝑧𝜁)>0.

Definition 1.4. The fractional integral of order 𝛼 is defined, for a function 𝑓(𝑧), by 𝐼𝛼𝑧1𝑓(𝑧)=Γ(𝛼)𝑧0𝑓(𝜁)(𝑧𝜁)𝛼1𝑑𝜁,𝛼>0,(1.6) where the function 𝑓(𝑧) is analytic in simply-connected region of the complex 𝑧-plane () containing the origin, and the multiplicity of (𝑧𝜁)𝛼1 is removed by requiring log(𝑧𝜁) to be real when (𝑧𝜁)>0.

Remark 1.5. From Definitions 1.3 and 1.4, we have 𝐷𝛼𝑧𝑧𝜇=Γ(𝜇+1)𝑧Γ(𝜇𝛼+1)𝜇𝛼𝐼,𝜇>1,0<𝛼<1,𝛼𝑧𝑧𝜇=Γ(𝜇+1)𝑧Γ(𝜇+𝛼+1)𝜇+𝛼,𝜇>1,𝛼>0.(1.7) By using the majorant concept, we establish the existence and uniqueness of holomorphic solution.

Definition 1.6. The major are relations, described as follows: if 𝑎𝑎(𝑥)=𝑖𝑥𝑖 and 𝐴𝐴(𝑥)=𝑖𝑥𝑖, then one says that 𝑎(𝑥)𝐴(𝑥) if and only if |𝑎𝑖||𝐴𝑖| for each 𝑖.

2. Existence of Unique Solution

In the present paper, we consider the fractional differential equations 𝜃(𝑧)𝐷𝛼𝑧𝜕𝑢(𝑡,𝑧)+𝜆𝛽𝑢(𝑡,𝑧)𝜕𝑡𝛽=𝐹𝑡,𝑧,𝑢,𝑢𝑡,0<𝛼1,1<𝛽2,(2.1) subject to the initial condition 𝑢|𝑡=0=0 in a neighborhood of 𝑧=0, where 𝑡𝐽=[0,1],𝑧𝑈={𝑧|𝑧|<1},𝑢(𝑡,𝑧) is an unknown function, 𝜃(𝑧)=𝜗𝑧𝛼 such that 𝜗0,𝑧𝑈,𝜆0 is a constant, and 𝐹 is a holomorphic function on 𝐽×𝑈××. Thus, we can expand it as follows: 𝐹(𝑡,𝑧,𝑢,𝑣)=𝛾(𝑡)𝑧+𝛿(𝑡)𝑢+𝜂(𝑡)𝑣+𝑅2(𝑡,𝑧,𝑢,𝑣),(2.2) where 𝛾(𝑡)=𝜕𝐹𝜕𝑧(𝑡,0,0,0),𝛿(𝑡)=𝜕𝐹𝜕𝑢(𝑡,0,0,0),𝜂(𝑡)=𝜕𝐹𝜕𝑣(𝑡,0,0,0),(2.3) and the degree of 𝑅2(𝑡,𝑧,𝑢,𝑣) with respect to (𝑡,𝑧,𝑢,𝑣) is greater than or equal to 2.

We need the following result.

Lemma 2.1 (see [21]). Let 𝑅>0, and let 𝑓(𝑥) be a holomorphic function on 𝐷𝑅={𝑥|𝑥|<𝑅}. If for any 𝑟>0,0<𝑟<𝑅,𝑓(𝑥) satisfies max|𝑥|𝑟||𝑓||𝜌(𝑥)(𝑅𝑟)𝜇(2.4) for some 𝜌>0 and 𝜇0, then one has max|𝑥|𝑟|||𝜕𝑓(𝑥)|||𝜕𝑥(𝜇+1)𝑒𝜌(𝑅𝑟)𝜇+1.(2.5)

Theorem 2.2. Assume the problem (2.1). If 𝛼(0,1], 𝜗0, and Γ(𝑛+1)Γ(𝑛+1𝛼)𝜗𝛿(𝑡)0,𝑛,(2.6) then (2.1) has a unique holomorphic solution 𝑢(𝑡,𝑧) near (0,0)𝐽×𝑈.

Proof . We realize that (2.1) has a formal solution 𝑢(𝑡,𝑧)=𝑘=1𝑢𝑘(𝑡)𝑧𝑘,(𝑧𝑈).(2.7) Then, substitut the series (2.7) into (2.1) and comparing the coefficients of 𝑧𝑘 in two sides of the equation yield Γ(2)𝑢Γ(2𝛼)𝜗𝛿(𝑡)1(𝑡)+𝜆𝑢1(𝛽)(𝑡)𝜂(𝑡)𝑢1(𝑡)=𝛾(𝑡)𝑢1(𝛽)(𝑡)=𝑓1𝑡,𝑢1Γ(3)𝑢Γ(3𝛼)𝜗𝛿(𝑡)2(𝑡)+𝜆𝑢2(𝛽)(𝑡)𝜂(𝑡)𝑢2(𝑡)=𝜙1𝑢1,𝑢1𝑢2(𝛽)(𝑡)=𝑓2𝑡,𝑢2Γ(𝑛+1)𝑢Γ(𝑛+1𝛼)𝜗𝛿(𝑡)𝑛(𝑡)+𝜆𝑢𝑛(𝛽)(𝑡)𝜂(𝑡)𝑢𝑛(𝑡)+𝜙𝑛1𝑢1,,𝑢𝑛1,𝑢1,,𝑢𝑛1𝑢𝑛(𝛽)(𝑡)=𝑓𝑛𝑡,𝑢𝑛.(2.8) Thus, we obtain the following formula: 𝑢𝑛(𝛽)[](𝑡)+(Γ(𝑛+1)/Γ(𝑛+1𝛼))𝜗𝛿(𝑡)𝜆𝑢𝑛(𝑡)=Φ(𝑡),(2.9) where 𝜙0(𝑡)=𝛾(𝑡) and 𝜙Φ(𝑡)=𝑛1𝑢1,,𝑢𝑛1,𝑢1,,𝑢𝑛1+𝜂(𝑡)𝑢𝑛(𝑡)𝜆.(2.10) Hence, from the idea given in [22], solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem, problem (2.9), and consequently problem (2.1) has a unique solution in a neighborhood of 𝑧=0.
Now, we proceed to prove that the formal series solution (2.7) is convergent near (0,0)(𝐽,𝑈). We expand the remainder term 𝑅2(𝑡,𝑧,𝑢,𝑣) into Taylor series with respect to (𝑡,𝑢,𝑣), that is, 𝑅2(𝑡,𝑧,𝑢,𝑣)=𝑚+𝑛+𝑝2𝑎𝑚,𝑛,𝑝(𝑡)𝑧𝑚𝑢𝑛𝑣𝑝,(2.11) such that(i)𝑎𝑚,𝑛,𝑝(𝑡) is holomorphic in 𝐽,(ii)|𝑎𝑚,𝑛,𝑝(𝑡)|𝐴𝑚,𝑛,𝑝,𝐴𝑚,𝑛,𝑝>0 on 𝐽,(iii)𝑚+𝑛+𝑝2𝐴𝑚,𝑛,𝑝𝑧𝑚𝑉𝑛+𝑝 converges in (𝑧,𝑉) where 𝑉>0 satisfies |𝑢|𝑉 and |𝑣|𝑉.From (2.9), we observe that [](Γ(2)/Γ(2𝛼))𝜗𝛿(𝑡)𝜆+𝜕𝛽𝜕𝑡𝛽𝑢1(𝑡)=𝛾(𝑡)+𝜂(𝑡)𝑢1[](Γ(𝑘)/Γ(𝑘𝛼))𝜗𝛿(𝑡)𝜆+𝜕𝛽𝜕𝑡𝛽𝑢𝑘=(𝑡)𝑚+𝑛+𝑝2𝑚+𝑘1++𝑘𝑛+𝑙1++𝑙𝑝=𝑘𝑎𝑚,𝑛,𝑝(𝑡)×𝑢𝑘1××𝑢𝑘𝑛×𝜕𝑢𝑙1𝜕𝑡××𝜕𝑢𝑙𝑝.𝜕𝑡(2.12) Without loss of generality, we may assume that there exists a constant 𝐾>0 such that ||𝑢1||||||(𝑡)𝐾,𝜕𝑢1(𝑡)||||𝜕𝑡𝐾.(2.13) Denoting 1𝐶=,(Γ(2)/Γ(2𝛼))𝜗𝛿(2.14) then we pose the following formula: 𝐶𝑉(𝑧)=𝐾𝑧+1𝑟𝑚+𝑛+𝑝2𝐴𝑚,𝑛,𝑝(1𝑟)𝑚+𝑛+𝑝2𝑧𝑚𝑉𝑛𝑉𝑝,(2.15) where 𝑟 is a parameter with 0<𝑟<1. Since (2.15) is an analytic functional equation in 𝑉 then in view of the implicit function theorem, (2.15) has a unique holomorphic solution 𝑉(𝑧) in a neighborhood of 𝑧=0 with 𝑉(0)=0. Expanding 𝑉(𝑧) into Taylor series in 𝑧, we have 𝑉(𝑧)=𝑘1𝑉𝑘𝑧𝑘,(2.16) where 𝑉𝑘=𝐶1𝑟𝑚+𝑛+𝑝2𝑚+𝑘1++𝑘𝑛+𝑙1++𝑙𝑝=𝑘𝐴𝑚,𝑛,𝑝(1𝑟)𝑚+𝑛+𝑝2×𝑉𝑘1××𝑉𝑘𝑛×0!𝑒𝑉𝑙1××(𝑝1)!𝑒𝑉𝑙𝑝𝐶=𝑘(1𝑟)𝑘1,𝑘>0,(2.17) with 𝐶1=𝐾.
Next, our aim is to show that the series 𝑘1𝑉𝑘𝑧𝑘 is a major series for the formal series solution 𝑘1𝑢𝑘𝑧𝑘 near 𝑧=0. For this purpose, we will show that ||𝑢𝑘||𝑉𝑘on𝐽𝑟[]||||=0,𝑟,𝑟<1,(2.18)𝜕𝑢𝑘||||𝜕𝑡(𝑘1)!𝑒𝑉𝑘on𝐽𝑟.(2.19) Since (1𝑟)<1 implies 1(1𝑟)𝑚+𝑛+𝑝21,𝑟<1,(2.20) then we have ||𝑢𝑘||𝐶𝑚+𝑛+𝑝2𝑚+𝑘1++𝑘𝑛+𝑙1++𝑙𝑝=𝑘𝐴𝑚,𝑛,𝑝×||𝑢𝑘1||×||𝑢(𝑡)×𝑘𝑛||×||||(𝑡)𝜕𝑢𝑙1(𝑡)||||||||𝜕𝑡××𝜕𝑢𝑙𝑝(𝑡)||||𝜕𝑡𝐶𝑚+𝑛+𝑝2𝑚+𝑘1++𝑘𝑛+𝑙1++𝑙𝑝=𝑘𝐴𝑚,𝑛,𝑝×𝑉𝑘1××𝑉𝑘𝑛×0!𝑒𝑉𝑙1××(𝑝1)!𝑒𝑉𝑙𝑝𝐶𝑚+𝑛+𝑝2𝑚+𝑘1++𝑘𝑛+𝑙1++𝑙𝑝=𝑘𝐴𝑚,𝑛,𝑝(1𝑟)𝑚+𝑛+𝑝2×𝑉𝑘1××𝑉𝑘𝑛×0!𝑒𝑉𝑙1××(𝑝1)!𝑒𝑉𝑙𝑝𝐶𝑘(1𝑟)𝑘2𝐶𝑘(1𝑟)𝑘1=𝑉𝑘.(2.21) Hence, we obtain the inequality (2.19). Next, by using Lemma 2.1, we pose that ||||𝜕𝑢𝑘||||𝜕𝑡(𝑘1)𝑒𝐶𝑘(1𝑟)𝑘1(𝑘1)!e𝐶𝑘(1𝑟)𝑘1=(𝑘1)!𝑒𝑉𝑘.(2.22) This completes the proof of Theorem 2.2.

3. Applications

In this section, we pose two examples that demonstrate the performance and efficiency for solving time- and space-fractional telegraph equations.

Example 3.1. Assume the following equation: 𝑧0.5𝐷1.128𝑧0.5𝜕𝑢(𝑡,𝑧)+1.5𝑢(𝑡,𝑧)𝜕𝑡1.5=𝑢𝑡[],(𝑡,𝑧)+1.5𝑢(𝑡,𝑧),𝑡𝐽=0,1𝑢(0,𝑧)=0,(3.1) where 𝑢(𝑡,𝑧) is the unknown function. Since the condition (Γ(𝑛+1)/Γ(𝑛+1𝛼))𝜗𝛿(𝑡)0,𝑛 is satisfied, then in view of Theorem 2.2, the problem (3.1) has a unique holomorphic solution near (0,0)𝐽×𝑈. To verify this, we put 𝑧𝑢(𝑡,𝑧)=𝜇(𝑡)𝑧+𝑣(𝑡,𝑧)𝑣(𝑡,𝑧)=𝑂2(3.2) as a formal solution. Therefore, 𝜇(𝑡) satisfies 𝜇(𝑡)(1.5)=𝜇(𝑡)+0.5𝜇(𝑡).(3.3) The last equation is equivalent to the equation of the form 𝜕𝛾𝜇(𝑡)𝜕𝑡𝛾=𝑔𝑡,𝜇(𝑡),𝜇],(𝑡),𝛾(1,2(3.4) where 𝑔 is a contraction function whenever |𝜇(𝑡)𝜌(𝑡)|𝑎|𝜇(𝑡)𝜌(𝑡)|,𝑎(0,0.5); thus the holomorphic solution 𝜇(𝑡) exists uniquely and converges in a neighborhood of the origin (the Banach fixed-point theorem).

Example 3.2. Assume the following equation: 𝑧0.5𝐷1.128𝑧0.5𝑢(𝑡,𝑧)+𝑢𝑡𝑡=𝑢𝑡[],𝑢(𝑡,𝑧)𝑢(𝑡,𝑧),𝑡𝐽=0,1(0,𝑧)=0,(3.5) where 𝑢(𝑡,𝑧) is the unknown function. The condition of Theorem 2.2 is valid; hence, the problem (3.5) has a unique holomorphic solution near (0,0)𝐽×𝑈. To realize this, (3.5) can be reduced to the following problem: 𝜓(𝑡)=𝜓(𝑡)2𝜓(𝑡),𝜓(0)=0.(3.6) Thus, the problem (3.6) has a unique solution.

4. Conclusion

The telegraph equation appears when we look for a mathematical model for the electrical flow in a metallic cable. In virtue of the laws of electricity, we posed generalized fractional partial differential equations (2.1) in a complex domain where the unknown variable is the voltage 𝑢. The conventional power series method (tool from classical functional analysis) has been successfully employed for finding the solution of space- and time-fractional telegraph equations. The space- and time-fractional derivatives are considered in the Srivastava-Owa and Riemann-Liouville sense, respectively. The results of the examples show that conventional power series method is a reliable and efficient method for solving space- and time-fractional telegraph equations and also other equations in complex domain.

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