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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 703681, 10 pages
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.
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.
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 [1–6]. Recently, the existence of analytic solutions for fractional differential equations in complex domain is posed [7–10].
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 ), Variational iteration method (VIM) was used to solve the linear and nonlinear telegraph equation (see [14–16]). Note that in , the methods (HPM) and (VIM) were compared for linear fractional partial differential equations. Also time-periodic solutions of these equations are studied (see ). Recently, Herzallah and Baleanu  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 of order is defined by When , one writes , where denoted the convolution product, and , and as where is the delta function.
The Srivastava-Owa fractional operators are (see ) as follows.
Definition 1.3. The fractional derivative of order is defined, for a function , by where the function is analytic in simply connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .
Definition 1.4. The fractional integral of order is defined, for a function , by where the function is analytic in simply-connected region of the complex -plane containing the origin, and the multiplicity of is removed by requiring to be real when .
Definition 1.6. The major are relations, described as follows: if and , then one says that if and only if for each .
2. Existence of Unique Solution
In the present paper, we consider the fractional differential equations subject to the initial condition in a neighborhood of , where is an unknown function, such that is a constant, and is a holomorphic function on . Thus, we can expand it as follows: where and the degree of with respect to is greater than or equal to 2.
We need the following result.
Lemma 2.1 (see ). Let , and let be a holomorphic function on . If for any satisfies for some and , then one has
Proof . We realize that (2.1) has a formal solution
Then, substitut the series (2.7) into (2.1) and comparing the coefficients of in two sides of the equation yield
Thus, we obtain the following formula:
Hence, from the idea given in , 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 .
Now, we proceed to prove that the formal series solution (2.7) is convergent near . We expand the remainder term into Taylor series with respect to , that is, such that(i) is holomorphic in ,(ii) on ,(iii) converges in where satisfies and .From (2.9), we observe that Without loss of generality, we may assume that there exists a constant such that Denoting then we pose the following formula: where is a parameter with . Since (2.15) is an analytic functional equation in then in view of the implicit function theorem, (2.15) has a unique holomorphic solution in a neighborhood of with . Expanding into Taylor series in , we have where with .
Next, our aim is to show that the series is a major series for the formal series solution near . For this purpose, we will show that Since implies then we have Hence, we obtain the inequality (2.19). Next, by using Lemma 2.1, we pose that This completes the proof of Theorem 2.2.
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: where is the unknown function. Since the condition is satisfied, then in view of Theorem 2.2, the problem (3.1) has a unique holomorphic solution near . To verify this, we put as a formal solution. Therefore, satisfies The last equation is equivalent to the equation of the form where is a contraction function whenever ; thus the holomorphic solution exists uniquely and converges in a neighborhood of the origin (the Banach fixed-point theorem).
Example 3.2. Assume the following equation: where is the unknown function. The condition of Theorem 2.2 is valid; hence, the problem (3.5) has a unique holomorphic solution near . To realize this, (3.5) can be reduced to the following problem: Thus, the problem (3.6) has a unique solution.
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 . 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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