Comment on “An Approximation to Solution of Space and Time Fractional Telegraph Equations by He's Variational Iteration Method”

Abstract

The variational iteration method was applied to the time fractional telegraph equation and some variational iteration formulae were suggested in (Sevimlican, 2010). Those formulae are improved by Laplace transform from which the approximate solutions of higher accuracies can be obtained.

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Sevimlican [1] considered the application of the variational iteration method [2, 3] to find approximate solutions of space and time fractional telegraph equations. The author suggested the following variational iteration formula for (5.1) in [1] 𝑢_𝑛+1=𝑢_𝑛+^𝑥₀𝜆()𝜕𝑠,𝑥^𝛼𝑢(𝑠,𝑡)𝜕𝑠^𝛼−𝜕²𝑢(𝑠,𝑡)𝜕𝑡²−𝜕𝑢(𝑠,𝑡)()𝜆()𝜕𝑡−𝑢𝑠,𝑡𝑑𝑠,1<𝛼<2,𝑠,𝑥=𝑠−𝑥. (1) However, in this comment, it is pointed out that the identification of the Lagrange multiplier 𝜆(𝑠,𝑥)=𝑠−𝑥 from (4.1) to (4.9) can be improved.

According to the technique of determination of the Lagrange multipliers [4, 5], firstly, construct a correctional functional as 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛()+𝑥,𝑡_𝑥0𝜆(𝜕𝑠,𝑥)^𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑠^𝛼−𝜕²𝑢_𝑛(𝑠,𝑡)𝜕𝑡²−𝜕𝑢_𝑛(𝑠,𝑡)𝜕𝑡−𝑢_𝑛(𝑠,𝑡)𝑑𝑠,𝛼>0. (2)

Assuming the Lagrange multiplier 𝜆(𝑠,𝑥)=𝜆(𝑥−𝑠), take the Laplace transform to both sides of (2) 𝑢_𝑛+1(𝑆,𝑡)=𝑢_𝑛(𝑆,𝑡)+𝐿^𝑥₀𝜆()𝜕𝑠,𝑥^𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑠^𝛼−𝜕²𝑢_𝑛(𝑠,𝑡)𝜕𝑡²−𝜕𝑢_𝑛(𝑠,𝑡)𝜕𝑡−𝑢_𝑛(𝑠,𝑡)𝑑𝑠, (3) where 𝑢_𝑛(𝑆,𝑡) is the Laplace transform of 𝑢_𝑛(𝑥,𝑡).

Taking the variation 𝛿 with respect to 𝑢_𝑛(𝑆,𝑡), one can obtain 𝛿𝑢_𝑛+1(𝑆,𝑡)=𝛿𝑢_𝑛(𝑆,𝑡)+𝛿𝐿^𝑥₀𝜆()𝜕𝑠,𝑥^𝛼𝑢_𝑛(𝑠,𝑡)𝜕𝑠^𝛼−𝜕²𝑢_𝑛(𝑠,𝑡)𝜕𝑡²−𝜕𝑢_𝑛(𝑠,𝑡)𝜕𝑡−𝑢_𝑛()=𝑠,𝑡𝑑𝑠1+𝜆(𝑆)𝑆^𝛼𝛿𝑢_𝑛(𝑆,𝑡),𝜆(𝑆)1=−𝑆^𝛼. (4) Then, the Lagrange multiplier can be determined as 𝜆()=()𝑠,𝑥−1^𝛼(𝑠−𝑥)^𝛼−1Γ(𝛼). (5) Instead 𝜆(𝑠,𝑥)=𝑠−𝑥 (see (4.9) in [1]).

As a result, the variational iteration formula is obtained as 𝑢_𝑛+1=𝑢_𝑛+^𝑥₀(−1)^𝛼(𝑠−𝑥)^𝛼−1Γ(𝛼)𝜕^𝛼𝑢(𝑠,𝑡)𝜕𝑠^𝛼−𝜕²𝑢(𝑠,𝑡)𝜕𝑡²−𝜕𝑢(𝑠,𝑡)()𝜕𝑡−𝑢𝑠,𝑡𝑑𝑠,0<𝛼. (6)

The variational iteration formulae (5.10) and (5.17) are not right which also should be corrected, respectively.

Equation (5.10) in [1] should be 𝑢_𝑛+1=𝑢_𝑛+^𝑥₀(𝑠−𝑥)^2𝛼−1Γ()𝜕2𝛼^2𝛼𝑢(𝑠,𝑡)𝜕𝑠^2𝛼−𝜕²𝑢(𝑠,𝑡)𝜕𝑡²−𝜕𝑢(𝑠,𝑡)()𝜕𝑡−𝑢𝑠,𝑡+𝑠²+𝑡−1𝑑𝑠,0<𝛼. (7)

Equation (5.17) in [1] should be 𝑢_𝑛+1=𝑢_𝑛+^𝑡₀(𝑠−𝑡)^2𝛼−1Γ()𝜕2𝛼^2𝛼𝑢(𝑥,𝑠)𝜕𝑠^2𝛼𝜕+𝜆²𝑢(𝑥,𝑠)𝜕𝑠²()−𝜐𝜕𝑢𝑥,𝑠𝜕𝑠𝑑𝑠,0<𝛼. (8)

Conclusions

As is well known, the VIM became an efficient analytical tool in nonlinear science since it was proposed and the method was often used in fractional differential equations. This study illustrates the method in fractional calculus can be improved by the Laplace tranform method with which the Lagrange mutipliers can be identified explicitly.

Recently, there are also other new applications of the variation iteration method to various nonlinear problems, that is, fuzzy equations [6, 7] and 𝑞-fractional differential equations [8]. Readers are referred to the recent review article [9].

Acknowledgment

This work is financially supported by the Zhejiang Natural Science Foundation (Grant no. LQ12A01010).