Local fractional Sumudu decomposition method for linear partial differential equations with local fractional derivative

D. Ziane; D. Baleanu; K. Belghaba; M. Hamdi Cherif

doi:10.1016/j.jksus.2017.05.002

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Review

31 (

); 83-88

doi:

10.1016/j.jksus.2017.05.002

Local fractional Sumudu decomposition method for linear partial differential equations with local fractional derivative

D. Ziane^{^a}, D. Baleanu^{^b}, K. Belghaba^{^a}, M. Hamdi Cherif^{^a,⁎}

a Laboratory of Mathematics and Its Applications (LAMAP), University of Oran1 Ahmed Ben Bella, Oran, 31000, Algeria

b Institute of Space Sciences, Magurele, 077125 Bucharest, Romania

⁎Corresponding author. mountassir27@yahoo.fr (M. Hamdi Cherif)

Received: 2017-4-10, Accepted: 2017-5-2, Published: 2017-01-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

In the paper, a combined form of the Sumudu transform method with the Adomian decomposition method in the sense of local fractional derivative, is proposed to solve fractional partial differential equations. This method is called the local fractional Sumudu decomposition method (LFSDM) and is used to describe the non-differentiable problems. It would be interesting to apply LFSDM to some well-known problems to see the benefits obtained.

Keywords

44A05

26A33

44A20

34K37

Local fractional derivative operator

Local fractional Sumudu decomposition method

Local fractional partial differential equations

1 Introduction

Historically, it has been established that the question of the numerical derivation of the fractional order of functions and its inverse integral operation has been discussed in various correspondences between Gottfried Leibniz (1646–1716), Guillaume de L’Hôpital (1661–1704). However, the question will remain confined to this fact and no major development was realized by these early precursors in this field of mathematics, it is only later, when studying certain phenomena in fluid mechanics, that it was observed the presence of an integral of order one-half in the equations of heat when it is desired, for example, to explain a lateral heat flux of a fluid flow as a function of the temporal evolution of the internal source. As a result, developments took place in different fields of study, in particular hydrodynamics, thermodynamics, diffusion theory, electrochemistry, to name but a few examples (Ziane et al., 2016).

In addition to the above, we find that the development of this branch has also led to the emergence of linear and nonlinear differential equations of fractional order, which became requires researchers to use conventional methods to solve them, for example, the Adomian decomposition method (Adomian et al., 1989; Adomian et al., 1990; Adomian et al., 1994 ), homotopy perturbation method (He et al., 1999; He et al., 2005; He et al., 2000 ) and variational iteration method (He et al., 1997; He et al., 1998; He et al., 1998 ). The reader wishing to learn more about the subject can consult (Singh et al., 2016; Kumar et al., 2017; Aslefallah and Shivanian, 2015; Hosseini et al., 2016; Abbasbandy et al., 2011; Soltani et al., 2016 ).

Recently, local fractional derivative and calculus theory has been introduced by the researcher in (Yang et al., 2011; Yang, 2012), which is set up on fractal geometry and which is the best method for describing the non-differentiable function defined on Cantor sets Yang and Hua, 2014. The fractional calculus is used in generalized Newtonian mechanics, the Maxwell’s equations and the Hamiltonian mechanics Golmankhaneh et al., 2015. This then led to the emergence of ordinary differential equations or partial differential equations relating to this new concept, which became known as local fractional differential equations or local fractional partial differential equations, prompting some researchers to use the above-mentioned methods to solve this new type of equations, among them we find, local fractional Adomian decomposition method (Yang et al., 2015; Baleanu et al., 5350; Yang et al., 2013 ), local fractional homotopy perturbation method (Yang et al., 2015; Zhang et al., 2015), local fractional variational iteration method (Yang et al., 2014; Yang et al., 2026), local fractional variational iteration transform method Yang et al., 3659 and local fractional Laplace decomposition method Jassim et al., 2015. Other authors have also been interested in this area of research (see Kumar et al., 2017; Singh et al., 2016).

The objective of this study is coupling the Adomian decomposition method (ADM) with Sumudu transform in the sense of local fractional derivative. Then we apply this modified method to solve some examples related to linear local fractional partial differential equations.

2 Preliminaries

In this section, we present the basic theory of local fractional calculus and we focus specifically on the following concepts: local fractional derivative, local fractional integral, and local fractional Sumudu transform. Some important results are cited.

2.1

2.1 Local fractional derivative

Definition 2.1

The local fractional derivative of $Φ (r)$ of order $ν$ at $r = r_{0}$ is defined by (Yang et al., 2011; Yang, 2012)

(2.1)

Φ^{(ν)} (r) = {\frac{d^{ν} Φ}{{dr}^{ν}}|}_{r = r_{0}} = \frac{Δ^{ν} (Φ (r) - Φ (r_{0}))}{{(r - r_{0})}^{ν}},

where

(2.2)

Δ^{ν} (Φ (r) - Φ (r_{0})) ≅ Γ (1 + ν) [(Φ (r) - Φ (r_{0}))] .

For any $r \in (α, β)$ , there exists $Φ^{(ν)} (r) = D_{r}^{ν} Φ (r),$ denoted by $Φ (r) \in D_{r}^{ν} (α, β) .$

Local fractional derivative of high order is written in the form

(2.3)

Φ^{(m ν)} (r) = \overset{m times}{\overset{︷}{D_{r}^{(ν)} \dots D_{r}^{(ν)} Φ (r)}},

and local fractional partial derivative of high order is

(2.4)

\frac{\partial^{m ν} Φ (r)}{\partial r^{m ν}} = \overset{m times}{\overset{︷}{\frac{\partial^{ν}}{\partial r^{ν}} \dots \frac{\partial^{ν}}{\partial r^{ν}} Φ (r)}} .

2.2

2.2 Local fractional integral

Definition 2.2

The local fractional integral of $Φ (r)$ of order $ν$ in the interval $[α, β]$ is defined as (Yang et al., 2011; Yang, 2012)

(2.5)

\begin{matrix} _{α} I_{β}^{(ν)} Φ (r) = & \frac{1}{Γ (1 + ν)} \int_{α}^{β} Φ (τ) {(d τ)}^{ν} \\ = & \frac{1}{Γ (1 + ν)} \lim_{Δ τ ⟶ 0} \sum_{j = 0}^{N - 1} f (τ_{j}) {(Δ τ_{j})}^{ν}, \end{matrix}

where

Δ τ_{j} = τ_{j + 1} - τ_{j}, Δ τ = \max \{Δ τ_{0}, Δ τ_{1}, Δ τ_{2}, \dots\}

and

[τ_{j}, τ_{j + 1}], τ_{0} = α, τ_{N} = β

, is a partition of the interval

[α, β]

. For any

r \in (α, β)

, there exists

_{α} I_{r}^{(ν)} Φ (r),

denoted by

Φ (r) \in I_{r}^{(ν)} (α, β) .

2.3

2.3 Some important results

Definition 2.3

The local fractional Laplace transform of $Φ (r)$ of order $ν$ is

(2.6)

L_{ν} \{Φ (r)\} = \frac{1}{Γ (1 + ν)} \int_{0}^{\infty} E_{ν} (- s^{ν} r^{ν}) Φ (r) {(dr)}^{ν} .

Definition 2.4

In fractal space, the Mittag–Leffler function, sine function and cosine function are defined as (Yang et al., 2011; Yang, 2012)

(2.7)

E_{ν} (r^{ν}) = \sum_{m = 0}^{+ \infty} \frac{r^{m ν}}{Γ (1 + m ν)}, 0 < ν ⩽ 1,

(2.8)

\sin_{ν} (r^{ν}) = \sum_{m = 0}^{+ \infty} {(- 1)}^{m} \frac{r^{(2 m + 1) ν}}{Γ (1 + (2 m + 1) ν)}, 0 < ν ⩽ 1,

(2.9)

\cos_{ν} (r^{ν}) = \sum_{m = 0}^{+ \infty} {(- 1)}^{m} \frac{r^{2 m ν}}{Γ (1 + 2 m ν)}, 0 < ν ⩽ 1 .

The properties of local fractional derivatives and integral of non-differentiable functions are given by (Yang et al., 2011; Yang, 2012)

(2.10)

\frac{d^{ν}}{{dr}^{ν}} \frac{r^{m ν}}{Γ (1 + m ν)} = \frac{r^{(m - 1) ν}}{Γ (1 + (m - 1) ν)},

(2.11)

\frac{d^{ν}}{{dr}^{ν}} E_{ν} (r^{ν}) = E_{ν} (r^{ν}),

(2.12)

\frac{d^{ν}}{{dr}^{ν}} \sin_{ν} (r^{ν}) = \cos_{ν} (r^{ν}),

(2.13)

\frac{d^{ρ}}{{dr}^{ρ}} \cos_{ρ} (r^{ρ}) = - \sin_{ρ} (r^{ρ}),

(2.14)

_{0} I_{r}^{(ν)} \frac{r^{m ν}}{Γ (1 + m ν)} = \frac{r^{(m + 1) ν}}{Γ (1 + (m + 1) ν)} .

2.4

2.4 Local fractional Sumudu transform

We present here the definition of local fractional Sumudu transform (denoted in this paper by $^{LF} S_{ν}$ ) and some properties concerning this transformation Srivastava et al., 1763. If there is a new transform operator $^{LF} S_{ν} : Φ (r) ⟶ Ϝ (u)$ , namely

(2.15)

^{LF} S_{ν} \{\sum_{m = 0}^{\infty} a_{m} r^{m ν}\} = \sum_{m = 0}^{\infty} Γ (1 + m ν) a_{m} u^{m ν} .

As typical examples, we have

(2.16)

^{LF} S_{ν} \{E_{ν} (i^{ν} r^{ν})\} = \sum_{m = 0}^{\infty} i^{ν m} u^{ν m} .

(2.17)

^{LF} S_{ν} \{\frac{r^{ν}}{Γ (1 + ν)}\} = u^{ν} .

Definition 2.5

The local fractional Sumudu transform of $Φ (r)$ of order $ν$ is

(2.18)

^{LF} S_{ν} \{Φ (r)\} = Ϝ_{ν} (u) = \frac{1}{Γ (1 + ν)} \int_{0}^{\infty} E_{ν} (- u^{- ν} r^{ν}) \frac{Φ (r)}{u^{ν}} {(dr)}^{ν}, 0 < ν ⩽ 1

Following (2.18), its inverse formula is defined by

(2.19)

^{LF} S_{ν}^{- 1} \{Ϝ_{ν} (u)\} = Φ (r), 0 < ν ⩽ 1 .

Theorem 2.1

(linearity). If $^{LF} S_{ν} \{Φ (r)\} = Ϝ_{ν} (u)$ and $^{LF} S_{ν} \{φ (r)\} = Ψ_{ν} (u)$ , then one has

(2.20)

^{LF} S_{ν} \{Φ (r) + φ (r)\} = Ϝ_{ν} (u) + Ψ_{ν} (u) .

Proof

From the definition (2.18), we obtain the result. □

Theorem 2.2

(local fractional Laplace-Sumudu duality). If $L_{ν} \{Φ (r)\} = Φ_{s}^{L, ν} (s)$ and $^{LF} S_{ν} \{Φ (r)\} = F_{ν} (u)$ , then one has

(2.21)

^{LF} S_{ν} \{Φ (r)\} = \frac{1}{u^{ν}} L_{ν} \{Φ (\frac{1}{r})\},

(2.22)

L_{ν} \{Φ (r)\} = \frac{S_{ν} \{Φ (\frac{1}{s})\}}{s^{ν}} .

Proof

By using the formulas (2.6) and (2.18), we get the results of this theorem. □

Theorem 2.3

(1) (local fractional Sumudu transform of local fractional derivative). If $^{LF} S_{ν} \{Φ (r)\} = Ϝ_{ν} (u)$ , then one has

(2.23)

^{LF} S_{ν} \{\frac{d^{ν} Φ (r)}{{dr}^{ν}}\} = \frac{Ϝ_{ν} (u) - Ϝ (0)}{u^{ν}} .

As the direct result of (2.23), we have the following results. If

^{LF} S_{ν} \{Φ (r)\} = Ϝ_{ν} (u)

, we obtain

(2.24)

^{LF} S_{ν} \{\frac{d^{n ν} Φ (r)}{{dr}^{n ν}}\} = \frac{1}{u^{n ν}} [Ϝ_{ν} (u) - \sum_{k = 0}^{n - 1} u^{k ν} Φ^{(k ν)} (0)] .

When n

= 2

, from (2.24), we obtain

(2.25)

^{LF} S_{ν} \{\frac{d^{2 ν} Φ (r)}{{dr}^{2 ν}}\} = \frac{1}{u^{2 ν}} [Ϝ_{ν} (u) - Φ (0) - u^{ν} Φ^{(ν)} (0)]

(2) (local fractional Sumudu transform of local fractional integral). If

S_{ν} \{Φ (r)\} = Ϝ_{ν} (u)

, then we have

(2.26)

^{LF} S_{ν} \{_{0} I_{r}^{(ν)} Φ (r)\} = u^{ν} Ϝ_{ν} (u) .

Proof

(see Srivastava et al. (1763)). □

Theorem 2.4

(local fractional convolution). If $^{LF} S_{ν} \{Φ (r)\} = Ϝ_{ν} (u)$ and $^{LF} S_{ν} \{φ (r)\} = Ψ_{ν} (u)$ , then one has

(2.27)

^{LF} S_{ν} \{Φ (r) * φ (r)\} = u^{ν} Ϝ_{ν} (u) Ψ_{ν} (u),

with

Φ (r) * φ (r) = \frac{1}{Γ (1 + ν)} \int_{0}^{\infty} Φ (τ) φ (r - τ) {(dr)}^{ν} .

Proof

(see Srivastava et al. (1763)). □

3 Local fractional Sumudu decomposition method

Let us consider the following linear operator with local fractional derivative

(3.1)

L_{ν} U (r, τ) + R_{ν} U (r, τ) = g (r, τ),

where

L_{α} = \frac{\partial^{m α}}{\partial t^{m α}}

(m \in N^{*})

denotes linear local fractional derivative operator of order

m ν, R_{ν}

denotes linear local fractional derivative operator of order less than

L_{ν}

, and

g (r, τ)

is the non-differentiable source term.

Taking the local fractional Sumudu transform (denoted in this paper by $^{LF} S_{ν}$ ) on both sides of (3.1), we obtain

(3.2)

^{LF} S_{ν} [L_{ν} U (r, τ)] +^{LF} S_{ν} [R_{ν} U (r, τ)] =^{LF} S_{ν} [g (r, τ)] .

Using the property of the local fractional Sumudu transform, it follows

(3.3)

^{LF} S_{ν} [U (r, τ)] = \sum_{k = 0}^{m - 1} u^{k ν} \frac{\partial^{k ν} U (r, 0)}{\partial t^{k ν}} + u^{m ν} (^{LF} S_{ν} [g (r, τ)]) - u^{m ν} (^{LF} S_{ν} [R_{ν} U (r, τ)]) .

In taking the inverse local fractional Sumudu transform of both sides of (3.3), it follows

(3.4)

U (r, τ) = \sum_{k = 0}^{m - 1} u^{k ν} \frac{\partial^{k ν} U (r, 0)}{\partial t^{k ν}} \frac{t^{k ν}}{Γ (1 + k ν)} +^{LF} S_{ν}^{- 1} (u^{m ν} (^{LF} S_{ν} [g (r, τ)])) -^{LF} S_{ν}^{- 1} (u^{m ν} (^{LF} S_{ν} [R_{ν} U (r, τ)])) .

According to the Adomian decomposition method Adomian et al., 1989, we decompose the unknown function U as an infinite series given by

(3.5)

U (r, τ) = \sum_{n = 0}^{\infty} U_{n} (r, τ) .

Substituting (3.5) in (3.4), we get

(3.6)

\begin{matrix} \sum_{n = 0}^{\infty} U_{n} (r, τ) = & \sum_{k = 0}^{m - 1} [\frac{\partial^{k ν} U (r, 0)}{\partial t^{k ν}} \frac{t^{k ν}}{Γ (1 + k ν)}] +^{LF} S_{ν}^{- 1} (u^{m ν} (^{LF} S_{ν} [g (r, τ)])) \\ -^{LF} S_{ν}^{- 1} (u^{m ν} (^{LF} S_{ν} [R_{ν} \sum_{n = 0}^{\infty} U_{n} (r, τ)])) . \end{matrix}

On comparing both sides of (3.6), it then comes

(3.7)

\begin{matrix} U_{0} (r, τ) = & \sum_{k = 0}^{m - 1} [\frac{\partial^{k ν} U (r, 0)}{\partial t^{k ν}} \frac{t^{k ν}}{Γ (1 + k ν)}] +^{LF} S_{ν}^{- 1} (u^{m ν LF} S_{ν} [g (r, τ)]), \\ U_{1} (r, τ) = & -^{LF} S_{ν}^{- 1} (u^{m ν} (^{LF} S_{ν} [R_{ν} (U_{0} (r, τ))])), \end{matrix}

\begin{matrix} U_{2} (r, τ) = & -^{LF} S_{ν}^{- 1} (u^{m ν} (^{LF} S_{ν} [R_{ν} (U_{1} (r, τ))])), \\ U_{3} (r, τ) = & -^{LF} S_{ν}^{- 1} (u^{m ν} (^{LF} S_{ν} [R_{ν} (U_{2} (r, τ))])), \\ ⋮ \end{matrix}

The local fractional recursive relation in its general form is

(3.8)

\begin{matrix} U_{0} (r, τ) = \sum_{k = 0}^{m - 1} \frac{\partial^{k ν} U (r, 0)}{\partial t^{k ν}} \frac{t^{k ν}}{Γ (1 + k ν)} +^{LF} S_{ν}^{- 1} (u^{m ν LF} S_{ν} [g (r, τ)]), \\ U_{n} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{m ν LF} S_{ν} [R_{ν} (U_{n - 1} (r, τ))]), \end{matrix}

where

0 < ν ⩽ 1, n \in N^{*}

and

m = 1, 2, 3, \dots

4 Applications

In this section, we will implement the proposed method local fractional Sumudu decomposition method (LFSDM) for solving some examples.

Example 4.1

First, we consider the following local fractional partial differential equation

(4.1)

\begin{matrix} \frac{\partial^{ν} U (r, τ)}{\partial τ^{ν}} + \frac{\partial^{2 ν} U (r, τ)}{\partial r^{2 ν}} - U (r, τ) = 0, \\ U (r, 0) = \sin_{ν} (r^{ν}) . \end{matrix}

From (3.8) and (4.1), the successive approximations are

(4.2)

\begin{matrix} U_{0} (r, τ) = \sin_{ν} (r^{ν}), \\ U_{n} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{n - 1} (r, τ)}{\partial r^{2 ν}} - U_{n - 1} (r, τ)])), n ⩾ 1 . \end{matrix}

According to the successive formula (4.2), we have

(4.3)

\begin{matrix} U_{0} (r, τ) = \sin_{ν} (r^{ν}), \\ U_{1} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{0} (r, τ)}{\partial r^{2 ν}} - U_{0} (r, τ)])), \\ U_{2} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{1} (r, τ)}{\partial r^{2 ν}} - U_{1} (r, τ)])), \\ U_{3} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{1} (r, τ)}{\partial r^{2 ν}} - U_{1} (r, τ)])), \\ ⋮ \end{matrix}

and so on.

From the formulas (4.3), the first terms of local fractional Sumudu decomposition method are given by

(4.4)

\begin{matrix} U_{0} (r, τ) = \sin_{ν} (r^{ν}), \\ U_{1} (r, τ) = \sin_{ν} (r^{ν}) \frac{2 τ^{ν}}{Γ (1 + ν)}, \\ U_{2} (r, τ) = \sin_{ν} (r^{ν}) \frac{4 τ^{2 ν}}{Γ (1 + 2 ν)}, \\ U_{3} (r, τ) = \sin_{ν} (r^{ν}) \frac{8 τ^{3 ν}}{Γ (1 + 3 ν)}, \\ ⋮ \\ U_{n} (r, τ) = \sin_{ν} (r^{ν}) \frac{{(2 τ^{ν})}^{n}}{Γ (1 + n ν) .} \end{matrix}

Then the local fractional series form is

(4.5)

U (r, τ) = \sin_{ν} (r^{ν}) (1 + \frac{2 τ^{ν}}{Γ (1 + ν)} + \frac{{(2 τ^{ν})}^{2}}{Γ (1 + 2 ν)} + \frac{{(2 τ^{ν})}^{3}}{Γ (1 + 3 ν)} + \dots + \frac{{(2 τ^{ν})}^{n}}{Γ (1 + n ν)} + \dots)

Hence the exact solution of (4.1) by local fractional Sumudu decomposition method has the form

(4.6)

\begin{matrix} U (r, τ) = & \sin_{ν} (r^{ν}) \sum_{n = 0}^{\infty} \frac{{(2 τ^{ν})}^{n}}{Γ (1 + n ν)} \\ = & \sin_{ν} (r^{ν}) E_{ν} (2 τ^{ν}) . \end{matrix}

Example 4.2

Second, we consider the local fractional Laplace equation as Yang et al., 2026

(4.7)

\begin{matrix} \frac{\partial^{2 ν} U (r, τ)}{\partial τ^{2 ν}} + \frac{\partial^{2 ν} U (r, τ)}{\partial r^{2 ν}} = 0, \\ U (r, 0) = 0, \frac{\partial^{ν} U (r, 0)}{\partial τ^{ν}} = - E_{ν} (r^{ν}) . \end{matrix}

From (3.8) and (4.7), the formula of successive approximations is

(4.8)

\begin{matrix} U_{0} (r, τ) = - E_{ν} (r^{ν}) \frac{τ^{ν}}{Γ (1 + ν)}, \\ U_{n} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{2 ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{n - 1} (r, τ)}{\partial r^{2 ν}}])), n ⩾ 1 . \end{matrix}

According to the successive formula (4.8), we obtain

(4.9)

\begin{matrix} U_{0} (r, τ) = - E_{ν} (r^{ν}), \\ U_{1} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{2 ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{0} (r, τ)}{\partial r^{2 ν}}])), \\ U_{2} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{2 ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{1} (r, τ)}{\partial r^{2 ν}}])), \\ U_{3} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{2 ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{2} (r, τ)}{\partial r^{2 ν}}])), \\ ⋮ \\ U_{n} (r, τ) = -^{LF} S_{ν}^{- 1} (u^{2 ν} (^{LF} S_{ν} [\frac{\partial^{2 ν} U_{n - 1} (r, τ)}{\partial r^{2 ν}}])) . \end{matrix}

From the formulas (4.9), the first terms of local fractional Sumudu decomposition method are given by

(4.10)

\begin{matrix} U_{0} (r, τ) = - E_{ν} (r^{ν}), \\ U_{1} (r, τ) = E_{ν} (r^{ν}) \frac{τ^{3 ν}}{Γ (1 + 3 ν)}, \\ U_{2} (r, τ) = - E_{ν} (r^{ν}) \frac{τ^{5 ν}}{Γ (1 + 5 ν)}, \\ U_{3} (r, τ) = E_{ν} (r^{ν}) \frac{τ^{7 ν}}{Γ (1 + 7 ν)}, \\ ⋮ \\ U_{n} (r, τ) = E_{ν} (r^{ν}) {(- 1)}^{n + 1} \frac{τ^{(2 n + 1) ν}}{Γ (1 + (2 n + 1) ν)} . \end{matrix}

Then the local fractional series form is

(4.11)

U (r, τ) = - E_{ν} (r^{ν}) (1 - \frac{τ^{3 ν}}{Γ (1 + 3 ν)} + \frac{τ^{5 ν}}{Γ (1 + 5 ν)} - \dots + {(- 1)}^{n} \frac{τ^{(2 n + 1) ν}}{Γ (1 + (2 n + 1) ν)} + \dots)

Hence the exact solution of (4.7) by local fractional Sumudu decomposition method is given by

(4.12)

\begin{matrix} U (r, τ) = & - E_{ν} (r^{ν}) \sum_{n = 0}^{\infty} {(- 1)}^{n} \frac{τ^{(2 n + 1) ν}}{Γ (1 + (2 n + 1) ν)} \\ = & - E_{ν} (r^{ν}) \sin_{ν} (τ^{ν}) . \end{matrix}

Example 4.3

Finally, we consider the following local fractional partial differential equation with the initial conditions

(4.13)

\frac{\partial^{4 ν} U (r, τ)}{\partial τ^{4 α}} - 2^{ν} \frac{\partial^{3 ν} U (r, τ)}{\partial r^{3 ν}} = 1,

(4.14)

\begin{matrix} U (r, 0) = E_{ν} ({(2 r)}^{ν}), \frac{\partial^{ν} U (r, 0)}{\partial τ^{ν}} = 0, \\ \frac{\partial^{2 ν} U (r, 0)}{\partial τ^{2 ν}} = - 2^{2 ν} E_{ν} ({(2 r)}^{ν}), \frac{\partial^{3 ν} U (r, 0)}{\partial τ^{3 ν}} = 0 . \end{matrix}

From (3.8) and (4.13), we get the following formula

(4.15)

\begin{matrix} U_{0} (r, τ) = E_{ν} ({(2 r)}^{ν}) - 2^{2 ν} E_{ν} ({(2 r)}^{ν}) \frac{τ^{2 ν}}{Γ (1 + 2 ν)} + \frac{τ^{4 ν}}{Γ (1 + 4 ν)}, \\ U_{n} (r, τ) =^{LF} S_{ν}^{- 1} (u^{4 ν} (^{LF} S_{ν} [2^{ν} \frac{\partial^{3 ν} U_{n - 1} (r, τ)}{\partial r^{3 ν}}])), n ⩾ 1 . \end{matrix}

Using the formula (4.15), we obtain the following successive approximations

(4.16)

\begin{matrix} U_{0} (r, τ) = E_{ν} ({(2 r)}^{ν}) (1 - 2^{2 ν} \frac{τ^{2 ν}}{Γ (1 + 2 ν)}) + \frac{τ^{4 ν}}{Γ (1 + 4 ν)}, \\ U_{1} (r, τ) =^{LF} S_{ν}^{- 1} (u^{4 ν} (^{LF} S_{ν} [2^{ν} \frac{\partial^{3 ν} U_{0} (r, τ)}{\partial r^{3 ν}}])), \\ U_{2} (r, τ) =^{LF} S_{ν}^{- 1} (u^{4 ν} (^{LF} S_{ν} [2^{ν} \frac{\partial^{3 ν} U_{1} (r, τ)}{\partial r^{3 ν}}])), \\ U_{3} (r, τ) =^{LF} S_{ν}^{- 1} (u^{4 ν} (^{LF} S_{ν} [2^{ν} \frac{\partial^{3 ν} U_{2 + *} (r, τ)}{\partial r^{3 ν}}])), \\ ⋮ \\ U_{n} (r, τ) =^{LF} S_{ν}^{- 1} (u^{4 ν} (^{LF} S_{ν} [2^{ν} \frac{\partial^{3 ν} U_{n - 1} (r, τ)}{\partial r^{3 ν}}])) . \end{matrix}

According to the formulas (4.16), the first terms of local fractional Sumudu decomposition method have the form

(4.17)

\begin{matrix} U_{0} (r, τ) = E_{ν} ({(2 r)}^{ν}) (1 - 2^{2 ν} \frac{τ^{2 ν}}{Γ (1 + 2 ν)}) + \frac{τ^{4 ν}}{Γ (1 + 4 ν)}, \\ U_{1} (r, τ) = E_{ν} ({(2 r)}^{ν}) (2^{4 ν} \frac{τ^{4 ν}}{Γ (1 + 4 ν)} - 2^{6 ν} \frac{τ^{6 α}}{Γ (1 + 6 ν)}), \\ U_{2} (r, τ) = E_{ν} ({(2 r)}^{ν}) (2^{8 ν} \frac{τ^{8 ν}}{Γ (1 + 8 ν)} - 2^{10 ν} \frac{τ^{10 ν}}{Γ (1 + 10 ν)}), \\ U_{3} (r, τ) = E_{ν} ({(2 r)}^{ν}) (2^{12 ν} \frac{τ^{12 ν}}{Γ (1 + 12 ν)} - 2^{14 ν} \frac{τ^{14 ν}}{Γ (1 + 14 ν)}) \\ ⋮ \end{matrix}

Then the local fractional series form is

(4.18)

\begin{matrix} U (r, τ) = & \frac{τ^{4 ν}}{Γ (1 + 4 ν)} + E_{ν} ({(2 r)}^{ν}) (1 - \frac{{(2 τ)}^{2 ν}}{Γ (1 + 2 ν)} + \frac{{(2 τ)}^{4 ν}}{Γ (1 + 4 ν)} - \frac{{(2 τ)}^{6 ν}}{Γ (1 + 6 ν)} \\ + \frac{{(2 τ)}^{8 ν}}{Γ (1 + 8 ν)} - \frac{{(2 τ)}^{10 ν}}{Γ (1 + 10 ν)} + \dots + {(- 1)}^{n} \frac{{(2 τ)}^{2 n ν}}{Γ (1 + 2 n ν)} + \dots) \end{matrix}

Hence the exact solution of (4.13) by local fractional Sumudu decomposition method take the form

(4.19)

\begin{matrix} U (r, τ) = & \frac{τ^{4 ν}}{Γ (1 + 4 ν)} + E_{ν} ({(2 r)}^{ν}) \sum_{n = 0}^{\infty} {(- 1)}^{n} \frac{{(2 τ)}^{2 n ν}}{Γ (1 + 2 n ν)} \\ = & \frac{τ^{4 ν}}{Γ (1 + 4 ν)} + E_{ν} ({(2 r)}^{ν}) \cos ({(2 τ)}^{ν}) . \end{matrix}

5 Conclusion

The coupling of Adomian decomposition method (ADM) and the Sumudu transform method in the sense of local fractional derivative, proved very effective to solve linear local fractional partial differential equations. The proposed algorithm provides the solution in a series form that converges rapidly to the exact solution if it exists. From the obtained results, it is clear that the LFSDM yields very accurate solutions using only a few iterates. As a result, the conclusion that comes through this work is that LFSDM can be applied to other linear local fractional partial differential equations of higher order, due to the efficiency and flexibility in the application as can be seen in the proposed examples.

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