On the Exact Solution of Wave Equations on Cantor Sets

Abstract

The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples.

1. Introduction

Fractional calculus (FC) has attracted the attention of many scientists in different scientific fields due to its numerous applications in our day life problems. In these applications, there are contributions of different parts of FC. Among the different parts, local FC has a wide range of applications in the fields of physics and engineering based on the fractals. The fractal curves [1] are everywhere continuous but nowhere differentiable and therefore, the classical calculus cannot be used to interpret the motions in Cantor time-space [2]. Calcagni [3] studied continuous geometries with some specific dimensions. Local FC [4,5,6,7] started to be considered as one of the useful ways to handle the fractals and other functions that are continuously but non-differentiable.

Mandelbrot [8] described fractal geometry as a workable geometric middle ground between the excessive geometric order of Euclid and the geometric chaos of general mathematics and extensively illustrated wide range of applications fractals in many scientific fields like in, physics, engineering, mathematics and geophysics. Zhang and Baleanu [9] studied local fractional wave equations under fixed entropy. Srivastava et al. [10] studied an initial value problem by the help of Sumudu Transform. Li et al. [11] studied local fractional Poisson and Laplace equations and provided its application in fractal domain. Christianto and Rahul [12] gave the derivation of Proca equations on Cantor sets. Hao et al. [13] investigated the Helmholtz and Diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora [14] studied diffusion problems in fractal media in Cantor sets. Yang et al. [15] and Su et al. [16] studied wave equations in Cantor sets.

Many techniques are utilized for handling the local fractional problems in both ordinary and partial derivatives. For instance, The Yang-Laplace Transform [4], the local fractional Laplace variation iteration method (LFLVIM) [7] and many others. These methods are widely used in different scientific fields [6,7,8,9,10]. This area of research is much popular in the community of scientists and we are continuously observing recent developments in it. These developments are useful in engineering and physics and we feel further attention of scientists for the exploration of its different aspects.

This paper is organized as follows: In the first section, we have pointed out the essential and related work to local fractional differential equations (LFDEs) and the techniques which have been produced for handling the LFDEs. In Section 2, we have presented the preliminary results, which we will utilize for the production of iterative scheme. In Section 3, we produce an iterative scheme based on the LFLVIM for the solution of wave equations. Section 4 demonstrates the efficiency of our scheme by the help of several examples.

2. Preliminaries

In this section we are presenting the basic and related definitions and relations from local fractional calculus [1,2,3,4,5].

A function $\text{g}\left(\text{t}\right)$ is said to be local fractional continuous function if $\text{f}\left(\text{t}\right)$ satisfies

$$\left|\text{g}\left(\text{t}\right)-\text{g}\left({\text{t}}_0\right)\right|<{\text{a}}^{\text{γ}},$$

(1)

where $\text{γ}\in \left(0,1\right]$, $\left|\text{t}-{\text{t}}_0\right|<b$, for $\text{a},\text{ b}>0$ and $\text{a},\text{ b}\in \text{R}.$

The local fractional derivative of a function $\text{g}\left(\text{t}\right)\in {\text{C}}_{\text{γ}}\left(\text{a},\text{b}\right)$ of order γ is defined as:

$$\frac{{\text{d}}^{\text{γ}}\text{f}\left(\text{t}\right)}{{\text{dt}}^{\text{γ}}}=\frac{{\Delta }^{\text{γ}}\left(\text{f}\left(\text{t}\right)-\text{f}\left({\text{t}}_0\right)\right)}{{\left(\text{t}-{\text{t}}_0\right)}^{\text{γ}}},$$

(2)

where

$${\Delta }^{\text{γ}}\left(\text{f}\left(\text{t}\right)-\text{f}\left({\text{t}}_0\right)\right)=\text{Г}\left(1+\text{γ}\right)\left(\text{f}\left(\text{t}\right)-\text{f}\left({\text{t}}_0\right)\right)$$

(3)

and

$$\frac{{\text{d}}^{\text{γ}}}{{\text{dt}}^{\text{γ}}}\left(\frac{{\text{t}}^{\text{mγ}}}{\text{Г}\left(\text{mγ}+1\right)}\right)=\frac{{\text{t}}^{\left(\text{m}-1\right)\text{γ}}}{\text{Г}\left(\left(\text{m}-1\right)\text{γ}+1\right)},\text{ m}\in \text{N}.$$

(4)

The local fractional integral of a function $\text{g}\left(\text{t}\right)$ on [a, b] is defined by

$${{\text{I}}^{\left(\text{γ}\right)}}_{\left[\text{a},\text{b}\right]}\text{f}\left(\text{t}\right)=\frac{1}{\text{Г}\left(\text{γ}+1\right)}{{\displaystyle \int }}_{\text{a}}^{\text{b}}\text{g}\left(\text{x}\right){\left(\text{dx}\right)}^{\text{γ}}=\frac{1}{\text{Г}\left(\text{γ}+1\right)}\underset{\Delta \text{x}\to 0}{\lim }{\displaystyle \sum }_{\text{i}=0}^{\text{i}=\text{M}-1}\text{g}\left({\text{x}}_{\text{i}}\right){\left(\Delta {\text{x}}_{\text{i}}\right)}^{\text{γ}},$$

(5)

where $\left[\text{a},\text{b}\right]$ is divided into $\text{M}-1$ sub-intervals $\left({\text{t}}_{\text{i}},{\text{t}}_{\text{i}+1}\right)$ and $\Delta {\text{t}}_{\text{i}}={\text{t}}_{\text{i}+1}-{\text{t}}_{\text{i}}$, with $\text{a}={\text{t}}_0$, $\text{b}={\text{t}}_{\text{M}}$ and

$${{\text{I}}^{\left(\text{γ}\right)}}_{0,\text{t}}\left(\frac{{\text{t}}^{\text{mγ}}}{\text{Г}\left(\text{mγ}+1\right)}\right)=\frac{{\text{t}}^{\left(\text{m}+1\right)\text{γ}}}{\text{Г}\left(\left(1+\text{m}\right)\text{γ}+1\right)}.$$

(6)

The Mittage-Leffler function in fractal space is defined by

$${\text{E}}_{\text{γ}}\left({\text{t}}^{\text{γ}}\right)={\displaystyle \sum }_{\text{j}=0}^{\infty }\frac{{\text{t}}^{\text{jγ}}}{\text{Г}\left(\text{jγ}+1\right)}.$$

(7)

Yang-Laplace Transforms (YLT)

In this subsection, we are giving definitions and some basic results related to the Yang-Laplace transforms (YLT). For a function$\text{g}\left(\text{t}\right)$ satisfying the following inequality,

$$\frac{1}{\text{Г}\left(\text{γ}+1\right)}{{\displaystyle \int }}_0^{\infty }\left|\text{g}\left(\text{t}\right)\right|{\left(\text{dt}\right)}^{\text{γ}}<\text{m}<\infty ,$$

(8)

the YLT is defined by

$${\mathrm{Ɫ}}_{\text{γ}}\left\{\text{f}\left(\text{t}\right)\right\}=\frac{1}{\text{Г}\left(\text{γ}+1\right)}{{\displaystyle \int }}_0^{\infty }{\text{E}}_{\text{γ}}\left(-{\text{s}}^{\text{γ}}{\text{t}}^{\text{γ}}\right)\text{g}\left(\text{t}\right){\left(\text{dt}\right)}^{\text{γ}},$$

(9)

where $\text{γ}\in \left(0,1\right]$ and ${\text{s}}^{\text{γ}}={\text{β}}^{\text{γ}}+{\text{i}}^{\text{γ}}{\text{ω}}^{\text{γ}}$ for ${\text{i}}^{\text{γ}}$, the fractal imaginary unit and $\text{Re}\left({\text{s}}^{\text{γ}}\right)={\text{β}}^{\text{γ}}>0$. The YLT has the following properties:

$${\mathrm{Ɫ}}_{\text{γ}}\left\{\text{ah}\left(\text{t}\right)+\text{bg}\left(\text{t}\right)\right\}=\text{a}{\mathrm{Ɫ}}_{\text{γ}}\left\{\text{h}\left(\text{t}\right)\right\}+\text{b}{\mathrm{Ɫ}}_{\text{γ}}\left\{\text{f}\left(\text{t}\right)\right\},$$

(10)

$${\mathrm{Ɫ}}_{\text{γ}}{\left\{\text{h}\left(\text{t},\text{x}\right)\right\}}^{\text{mγ}}={\text{s}}^{\text{mγ}}{\mathrm{Ɫ}}_{\text{γ}}\left\{\text{h}\left(\text{t},\text{x}\right)\right\}-{\text{s}}^{\left(\text{m}-1\right)\text{γ}}\text{h}\left(0,\text{x}\right)-{\text{ s}}^{\left(\text{m}-2\right)\text{γ}}\text{h}{\left(0,\text{x}\right)}^{\left(\text{γ}\right)}$$

$$-{\text{s}}^{\text{γ}\left(\text{m}-3\right)}\text{h}{\left(0,\text{x}\right)}^{\left(2\text{γ}\right)}-\dots -\text{h}{\left(0,\text{x}\right)}^{\left(\left(\text{m}-1\right)\text{γ}\right),}$$

(11)

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{t }}^{\text{mγ}}\}=\frac{\text{Г}\left(\text{mγ}+1\right)}{{\text{s}}^{\left(\text{m}+1\right)\text{γ}}},$$

(12)

$${\text{cos h}}_{\text{γ}}\left({\text{t}}^{\text{γ}}\right)={\displaystyle \sum }_{\text{j}=0}^{\infty }\frac{{\text{t}}^{2\text{jγ}}}{\text{Г}\left(2\text{jγ}+1\right)}.$$

(13)

3. Iteration Scheme

In this section, we produce an iterative scheme for the solution of LFWEs based on LFLVIM. For this we consider the following LFDE

$${\text{v}}_{\text{t}}^{\mathrm{m\gamma }}-\text{p}\left(\text{x}\right){\text{v}}_{\text{x}}^{\mathrm{n\gamma }}=0,$$

(14)

where $\text{m},\text{n}$ are orders of local fractional partial derivatives with respect to $t and x$ respectively. Applying the local fractional variation iteration method for the correction local fractional operator for (14), we have

$${\text{v}}_{\text{m}+1}\left(\text{t},\text{x}\right)={\mathrm{v}}_{\text{m}}\left(\text{t},\text{x}\right)+{{\text{I}}^{\left(\text{γ}\right)}}_{0,\text{t}}\left(\frac{\mathfrak{X}{\left(\text{x}\right)}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}\right)\left({\text{v}}_{\text{t}}^{\text{mγ}}-\text{p}\left(\text{x}\right){\text{v}}_{\text{x}}^{\text{nγ}}\left(\text{t},\text{x}\right)\right),$$

(15)

where $\frac{\mathfrak{X}{\left(\text{x}\right)}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}$ is the Lagrange multiplier and (15) leads to

$${\text{v}}_{\text{m}+1}\left(\text{t},\text{x}\right)={\text{ v}}_{\text{m}}\left(\text{t},\text{x}\right)+{{\text{I}}^{\left(\text{γ}\right)}}_{0,\text{t}}\left(\frac{\mathfrak{X}{\left(\text{t}-\text{x}\right)}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}\right)\left({\text{v}}_{\text{t}}^{\text{mγ}}\left(\text{t},\text{x}\right)-\text{p}\left(\text{x}\right){\text{v}}_{\text{x}}^{\text{nγ}}\left(\text{t},\text{x}\right)\right).$$

(16)

Applying the operator YLT, of order $\gamma$ that is ${\mathrm{Ɫ}}_{\text{γ}}$ on (16), we have

$${\mathrm{Ɫ}}_{\text{γ}}\left\{{\text{v}}_{\text{m}+1}\left(\text{t},\text{x}\right)\right\}={\mathrm{Ɫ}}_{\text{γ}}\left\{{\text{v}}_{\text{m}}\left(\text{t},\text{x}\right)\right\}+{\mathrm{Ɫ}}_{\text{γ}}\left\{\frac{\mathfrak{X}{\left(\text{x}\right)}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}\right\}{\mathrm{Ɫ}}_{\text{γ}}\left\{{\text{v}}_{\text{t}}^{\text{mγ}}-\text{p}\left(\text{x}\right){\text{v}}_{\text{x}}^{\text{nγ}}\left(\text{t},\text{x}\right)\right\}.$$

(17)

Taking the $\gamma$ order local fractional variation of (17) with respect to$t$, and assuming that the term $\text{p}\left(\text{x}\right){\text{v}}_{\text{x}}^{\mathrm{n\gamma }}\left(\text{t},\text{x}\right)$ be invariant, we have

(18)

From (18) we obtain the Lagrange-Multiplier as follows

$${\mathrm{Ɫ}}_{\text{γ}}\left\{\frac{\mathfrak{X}{\left(\text{x}\right)}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}\right\}=\frac{-1}{{\text{s}}^{\text{mγ}}},$$

(19)

and by the help of (18) and (19), we have the following relation

(20)

Hence, we obtain the following iterative scheme

$${\text{v}}_{\text{m}+1}\left(\text{t},\text{x}\right)={{\mathrm{Ɫ}}_{\text{γ}}}^{-1}\left\{\frac{1}{{\text{s}}^{\text{mγ}}}\left\{-{\text{s}}^{\left(\text{m}-1\right)\text{γ}}{\mathrm{v}}_{\text{m}}\left(0,\text{x}\right)-{\text{ s}}^{\left(\text{m}-2\right)\text{γ}}{\mathrm{v}}_{\text{m}}{\left(0,\text{x}\right)}^{\left(\text{γ}\right)}-{\text{s}}^{\text{γ}\left(\text{m}-3\right)}{\mathrm{v}}_{\text{m}}{\left(0,\text{x}\right)}^{\left(2\text{γ}\right)}-\dots -{\mathrm{v}}_{\text{m}}{\left(0,\text{x}\right)}^{\left(\left(\text{m}-1\right)\text{γ}\right)}\right\}\right\}.$$

(21)

Consequently, we have the solution of (14) as

$$\text{v}\left(\text{t},\text{x}\right)=\underset{\text{m}\to \infty }{\lim }{{\mathrm{Ɫ}}_{\text{γ}}}^{-1}\left({\mathrm{Ɫ}}_{\text{γ}}\left\{{\mathrm{v}}_{\text{m}}\left(\text{s},\text{x}\right)\right\}\right).$$

(22)

4. Interpretation of the Iterative Scheme

This section is reserved for the interpretation of the iterative scheme (22). The iterative scheme is applied on some examples of wave equations for their solutions.

Example 1. Consider the following wave equation on Cantor sets,

$$\frac{{\partial }^{2\text{γ}}}{\partial {\text{t}}^{2\text{γ}}}\text{v}\left(\text{t},\text{x}\right)-\text{C}\frac{{\partial }^{2\text{γ}}}{\partial {\text{x}}^{2\text{γ}}}\text{v}\left(\text{t},\text{x}\right)=0,$$

(23)

and the initial-boundary conditions read as

$$\frac{{\partial }^{\text{γ}}\text{v}\left(0,\text{x}\right)}{\partial {\text{t}}^{\text{γ}}}=0,\text{ v}\left(0,\text{x}\right)={\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right).$$

(24)

By the use of (20), we have

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_{\text{m}+1}\left(\text{t}\right)\}={\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_{\text{m}}\left(\text{t},\text{x}\right)\}-\frac{1}{{\text{s}}^{2\text{γ}}}\left\{{\text{s}}^{2\text{γ}}{\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_{\text{m}}\left(\text{t},\text{x}\right)\right\}-{\text{s}}^{\text{γ}}{\text{v}}_{\text{m}}\left(0,\text{x}\right)$$

$$-{\text{ v}}_{\text{m}}{\left(0,\text{x}\right)}^{\left(\text{γ}\right)}-\text{C}\frac{{\partial }^{2\text{γ}}}{\partial {\text{x}}^{2\text{γ}}}{\text{v}}_{\text{m}}\left(\text{s},\text{x}\right)\}.$$

(25)

Using the initial conditions (24), we get

$${\text{v}}_0\left(\text{s},\text{x}\right)={\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_0\left(0,\text{x}\right)\}=\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{\text{γ}}},$$

(26)

From (24)–(26), we proceed to

(27)

For the second iteration, we utilize (24)–(27), as under

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_2\left(\text{t},\text{x}\right)\}={\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_1\left(\text{t},\text{x}\right)\}-\frac{1}{{\text{s}}^{2\text{γ}}}\left\{{\text{s}}^{2\text{γ}}{\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_1\left(\text{t},\text{x}\right)\right\}-{\text{s}}^{\text{γ}}{\text{v}}_1\left(0,\text{x}\right)$$

$$-{\text{ v}}_1{\left(0,\text{x}\right)}^{\left(\text{γ}\right)}-\text{C}\frac{{\partial }^{2\text{γ}}}{\partial {\text{x}}^{2\text{γ}}}{\text{v}}_1\left(\text{s},\text{x}\right)\}$$

$$=\frac{1}{{\text{s}}^{2\text{γ}}}\left\{{\text{s}}^{\text{γ}}{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)+\text{C}\left(\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{\text{γ}}}+\text{C}\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{3\text{γ}}}\right)\right\}$$

(28)

$$=\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{\text{γ}}}+\text{C}\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{3\text{γ}}}+{\text{C}}^2\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{5\text{γ}}}={\text{v}}_2\left(\text{s},\text{x}\right)$$

For the third iteration, using (24)–(26) and (28), we have

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_3\left(\text{t},\text{x}\right)\}={\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_2\left(\text{t},\text{x}\right)\}-\frac{1}{{\text{s}}^{2\text{γ}}}\left\{{\text{s}}^{2\text{γ}}{\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_2\left(\text{t},\text{x}\right)\right\}-{\text{s}}^{\text{γ}}{\text{v}}_2\left(0,\text{x}\right)-{\text{ v}}_2{\left(0,\text{x}\right)}^{\left(\text{γ}\right)}$$

$$-\text{C}\frac{{\partial }^{2\text{γ}}}{\partial {\text{x}}^{2\text{γ}}}{\text{v}}_2\left(\text{s},\text{x}\right)\}$$

$$=\frac{1}{{\text{s}}^{2\text{γ}}}\left\{{\text{s}}^{\text{γ}}{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)+\text{C}\left(\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{\text{γ}}}+\text{C}\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{3\text{γ}}}+{\text{C}}^2\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{5\text{γ}}}\right)\right\}$$

(29)

$$=\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{\text{γ}}}+\text{C}\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{3\text{γ}}}+{\text{C}}^2\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{5\text{γ}}}+{\text{C}}^3\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{7\text{γ}}}={\text{v}}_2\left(\text{s},\text{x}\right).$$

Continuing this process up to the nth approximation, we deduce

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{ v}}_{\text{n}}\left(\text{t},\text{x}\right)\}=\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{\text{γ}}}+\text{C}\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{3\text{γ}}}+{\text{C}}^2\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{5\text{γ}}}+{\text{C}}^3\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{7\text{γ}}}+\dots +{\text{C}}^{\text{n}}\frac{{\text{E}}_{\text{γ}}\left({\text{x}}^{\text{γ}}\right)}{{\text{s}}^{\left(2\text{n}+1\right)\text{γ}}}.$$

(30)

Applying ${{Ɫ}_{\gamma }}^{-1}$ on (30), we obtain

(31)

Example 2. Consider the following wave equation on Cantor sets,

$$\frac{{\partial }^{2\text{γ}}\text{v}\left(\text{t},\text{x}\right)}{\partial {\text{t}}^{2\text{γ}}}-\frac{{\text{x}}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}\frac{{\partial }^{2\text{γ}}\text{v}\left(\text{t},\text{x}\right)}{\partial {\text{x}}^{2\text{γ}}}=0,$$

(32)

and the initial-boundary conditions read as

$$\frac{{\partial }^{\text{γ}}\text{v}\left(0,\text{x}\right)}{\partial {\text{t}}^{\text{γ}}}=0,\text{ v}\left(0,\text{x}\right)=\frac{{\text{x}}^{2\text{γ}}}{\text{Г}\left(2\text{γ}+1\right)}.$$

(33)

From (33), we form

$${\text{v}}_0\left(\text{s},\text{x}\right)={\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_0\left(0,\text{x}\right)\}=\frac{{\text{x}}^{2\text{γ}}}{{\text{s}}^{\text{γ}}\text{Г}\left(2\text{γ}+1\right)}.$$

(34)

From (20) and (32)–(34), we get

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_1\left(\text{t},\text{x}\right)={\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_0\left(\text{t},\text{x}\right)-\frac{1}{{\text{s}}^{2\text{γ}}}\left\{{\text{s}}^{2\text{γ}}{\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_0\left(\text{t},\text{x}\right)\right\}-{\text{s}}^{\text{γ}}{\text{v}}_0\left(0,\text{x}\right)$$

$$-{\text{v}}_0{\left(0,\text{x}\right)}^{\left(\text{γ}\right)}-\frac{{\text{x}}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}\frac{{\partial }^{2\text{γ}}\text{v}\left(\text{s},\text{x}\right)}{\partial {\text{x}}^{2\text{γ}}}\}$$

(35)

$$=\frac{{\text{x}}^{2\text{γ}}}{{\text{s}}^{\text{γ}}\text{Г}\left(2\text{γ}+1\right)}-\frac{{\text{x}}^{\text{γ}}}{{\text{s}}^{3\text{γ}}\text{Г}\left(\text{γ}+1\right)}={\text{v}}_1\left(\text{s},\text{x}\right)$$

For the second iteration, using (33)–(35), we proceed to

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_2\left(\text{t},\text{x}\right)={\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_1\left(\text{t},\text{x}\right)-\frac{1}{{\text{s}}^{2\text{γ}}}\left\{{\text{s}}^{2\text{γ}}{\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_1\left(\text{t},\text{x}\right)\right\}-{\text{s}}^{\text{γ}}{\text{v}}_1\left(0,\text{x}\right)$$

$$-{\text{v}}_1{\left(0,\text{x}\right)}^{\left(\text{γ}\right)}-\frac{{\text{x}}^{\text{γ}}}{\text{Г}\left(\text{γ}+1\right)}\frac{{\partial }^{2\text{γ}}\text{v}\left(\text{s},\text{x}\right)}{\partial {\text{x}}^{2\text{γ}}}\}$$

(36)

$$=\frac{{\text{x}}^{2\text{γ}}}{{\text{s}}^{\text{γ}}\text{Г}\left(2\text{γ}+1\right)}-\frac{{\text{x}}^{\text{γ}}}{{\text{s}}^{3\text{γ}}\text{Г}\left(\text{γ}+1\right)}={\text{v}}_1\left(\text{s},\text{x}\right).$$

For the third iteration, (34)–(36) are utilized and we get

(37)

Continuing this process up to nth approximation, we get

$${\mathrm{Ɫ}}_{\text{γ}}\{{\text{v}}_{\text{n}+1}\left(\text{t},\text{x}\right)\}=\frac{{\text{x}}^{2\text{γ}}}{{\text{s}}^{\text{γ}}\text{Г}\left(2\text{γ}+1\right)}-\frac{{\text{x}}^{\text{γ}}}{{\text{s}}^{3\text{γ}}\text{Г}\left(\text{γ}+1\right)}.$$

(38)

Applying ${{Ɫ}_{\text{γ}}}^{-1}$ on (38), we deduce the following result as the solution of (32), (33), by the help of proposed method LFLVIM and our scheme in (22):

$$\text{v}\left(\text{t},\text{x}\right)=\frac{{\text{x}}^{2\text{γ}}}{\text{Г}\left(2\text{γ}+1\right)}-\frac{{\text{t}}^{2\text{γ}}{\text{x}}^{\text{γ}}}{\text{Г}\left(2\text{γ}+1\right)\text{Г}\left(\text{γ}+1\right)}.$$

(39)

5. Conclusions

This paper describes an iteration scheme based on the LFLVIM for the solutions of LFWEs which is a powerful technique and the efficiency of the iterative scheme is examined by two illustrative examples. The solutions obtained are graphically presented by the figures, Figure 1, Figure 2 respectively for $\text{γ}=\frac{\ln 2}{\ln 3}$. The prescribed technique is a better approach for the approximation of LFWEs in particular and LFEs in general.

Figure 1. Exact solution of Equation (23) for $\text{γ}=\frac{\ln 2}{\ln 3}$.

Figure 1. Exact solution of Equation (23) for $\text{γ}=\frac{\ln 2}{\ln 3}$.

Figure 2. Exact solution of Equation (32) for $\text{γ}=\frac{\ln 2}{\ln 3}.$

Figure 2. Exact solution of Equation (32) for $\text{γ}=\frac{\ln 2}{\ln 3}.$