An Aproximation to Solution of Space and Time Fractional Telegraph Equations by the Variational Iteration Method

Abstract

Sevimlican suggested an effective algorithm for space and time fractional telegraph equations by the variational iteration method. This paper shows that algorithm can be updated by either variational iteration algorithm-II or the fractional variational iteration method.

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As early as 1998, the variational iteration method was shown to be an effective tool for factional calculus [1]; afterwards, the method has been routinely used to solve various fractional differential equations for many years, see the review article in [2] for a detailed summarization. Sevimlican [3] also followed the solution given in [1]; however, the algorithm can be further improved.

Sevimlican considered the following one-dimensional space fractional telegraph equation:𝜕^𝛼𝑢𝜕𝑥^𝛼=𝜕²𝑢𝜕𝑡²+𝜕𝑢𝜕𝑡+𝑢,1<𝛼<2, (1) and obtained the following iteration formulation: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑥₀(𝜕𝑠−𝑥)^𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑠^𝛼−𝜕²𝑢_𝑛(𝑠,𝑡)𝜕𝑠²−𝜕𝑢_𝑛(𝑠,𝑡)𝜕𝑠−𝑢_𝑛(𝑠,𝑡)𝑑𝑠. (2) We can also construct a correction functional in the form 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑥₀𝜆𝜕^𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑠^𝛼−𝜕²𝑢_𝑛(𝑠,𝑡)𝜕𝑠²−𝜕𝑢_𝑛(𝑠,𝑡)−𝑢𝜕𝑠^𝑛(𝑠,𝑡)𝑑𝑠. (3) If the multiplier 𝜆 can be exactly identified, then one iteration results in the exact solution; however, the exact identification of the multiplier is impossible for most problems, and an approximate identification is always followed. To this end, 𝑢_𝑛 in (3) is assumed to be a known function, and it is generally called a restricted variable [4]. After identification of the multiplier, we obtain the following variational iteration Algorithm-II [5]: 𝑢_𝑛+1(𝑥,𝑡)=𝑢₀(𝑥,𝑡)−^𝑥₀(𝜕𝑠−𝑥)^𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑠^𝛼−𝜕𝑢_𝑛(𝑠,𝑡)𝜕𝑠−𝑢_𝑛(𝑠,𝑡)𝑑𝑠 (4) If we begin with 𝑢₀(𝑥,𝑡)=𝑢₀(0,𝑡)+𝑥𝑢_𝑥(0,𝑡), (4) leads to the same result as given in [3].

The fractional variational iteration method is also suitable for the present problem, see the solution process in [5–7].

Acknowledgment

The work is a project funded by PAPD (The Priority Academic Program Development of Jiangsu Higher Education Institutions).