2. Local fractional variational iteration Transform Method

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International Journal of Advances in Applied Mathematics and Mechanics

Local fractional variational iteration transform method to solve partial differential equations arising in mathematical physics

Research Article

Hassan Kamil Jassim∗

Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq

Received 06 June 2015; accepted (in revised version) 16 August 2015

Abstract: In this paper, we investigate solutions of one-dimensional and two-dimensional diffusion and wave equations on Cantor sets within the local fractional derivatives by using local fractional variational iteration transform method.

This method is coupled by the Yang-Laplace transform and variational iteration method. Illustrative examples are included to demonstrate the validity and applicability of the presented method.

MSC: 26A33 • 34A12 • 35R11

Keywords:Diffusion equation • Wave equation • Local fractional variational iteration method • Yang-Laplace transform • Local fractional derivative

© 2015 The Author. This is an open access article under the CC BY-NC-ND license(https://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

The local fractional calculus was successfully applied to describe the non-differentiable problems arising in mathematical physics [1,2], such as the diffusion and wave equations on Cantor sets. These problems were studied by several authors by using local fractional decomposition method [3–5], local fractional variational iteration method [5–

7], local fractional series expansion method [8], local fractional functional decomposition method [9], local fractional Laplace variational iteration method [10], and local fractional Laplace decomposition method [11]. In this paper, our aims are to present the coupling method of local fractional Laplace transform and variational iteration method, which is called as the local fractional variational iteration transform method, and to use it to solve diffusion and wave equations with local fractional derivative.

2. Local fractional variational iteration Transform Method

To illustrate the basic idea of the LFVITM for the local fractional partial differential equation as:

L_αu(x, t ) + R_αu(x, t ) = g (x, t), (1)

where L_α= ∂^nα

∂t^nα, n ∈ N is the linear local fractional derivative operator, Rαdenotes a lower order local fractional derivative operator and g (x, t ) is the nondifferentiable source term.

Applying the Yang-Laplace transform on both sides of (1), we get

eL_α{L_αu(x, t )} + eL_α{R_αu(x, t )} = eL_α©g (x, t)ª. (2)

Using the property of the Yang-Laplace transform, we have

s^nαLe_α{u(x, t )} − s^(n−1)αu(x, 0) − s^(n−2)αu^(α)(x, 0) − ··· − u^((n−1)α)(x, 0) + eL_α{R_αu(x, t )} = eL_α{g (x, t )}, (3)

∗ E-mail address:[email protected]

or

eL_α{u(x, t )} = 1

s^αu(x, 0) + 1

s^2αu^(α)(x, 0) + ··· + 1

s^nαu^((n−1)α)(x, 0) + 1

s^nαLe_α{g (x, t )} − 1

s^nαLe_α{R_αu(x, t )} . (4) Operating with the Yang-Laplace inverse on both sides on Eq. (4) gives

u(x, t ) = u(x,0) + ··· + t^(n−1)α

Γ(1 + (n − 1)α)u^((n−1)α)(x, 0) + eL⁻¹_α µ 1

s^nαLe_α©g (x, t) − Rαu(x, t )ª

¶

. (5)

Derivative by ∂^α

∂t^α both side Eq. (5), we have

∂^αu(x, t )

∂t^α = u^(α)(x, 0) + ··· + t^(n−2)α

Γ(1 + (n − 2)α)u^((n−1)α)(x, 0) + ∂^α

∂t^αeL⁻¹_α µ 1

s^nαeL_α©g (x, t) − R_αu(x, t )ª

¶

(6)

We now structure the correctional local fractional function in the form

u_m+1(x, t ) = um(x, t ) + 1 Γ(1 + α)

Z t 0

λ(ξ)^α Γ(1 + α)











∂^αu_m(x,ξ)

∂ξ^α − ∂^α

∂ξ^αL⁻¹_α µ 1

s^nαL_α©g (x,ξ) − R_αu_m(x,ξ)ª

¶

u⁽_m^α)(x, 0) + ··· + ξ^(n−2)α

Γ(1 + (n − 2)α)u^((n−1)α)_m (x, 0)











(dξ)^α. (7)

Making the local fractional variation, we get

δ^αu_m+1(x, t ) = δ^αum(x, t )+δ^α 1 Γ(1 + α)

Z t 0

λ(ξ)^α Γ(1 + α)











∂^αu_m(x,ξ)

∂ξ^α − ∂^α

∂ξ^αL⁻¹_α µ 1

s^nαL_α©g (x,ξ) − R_αu_m(x,ξ)ª

¶

−u⁽m^α)(x, 0) + ··· + ξ^(n−2)α

Γ(1 + (n − 2)α)u^((n−1)α)_m (x, 0)











(dξ)^α. (8)

The extremum condition of u_m+1(x, t ) is given by

δ^αu_m+1(x, t ) = 0. (9)

In view of (9), we have the following stationary conditions:

1 + λ(ξ)^α

Γ(1 + α|ξ=t= 0,

· λ(ξ)^α Γ(1 + α

¸_(α)

|ξ=t= 0. (10)

This is turn gives λ(ξ)^α

Γ(1 + α= −1 (11)

Substituting (11) into (7), we obtained

u_m+1(x, t ) = um(x, t ) − 1 Γ(1 + α)

Z t 0











∂^αum(x,ξ)

∂ξ^α − ∂^α

∂ξ^αL⁻¹_α µ 1

s^nαL_α©g (x,ξ) − Rαu_m(x,ξ)ª

¶

−u⁽m^α)(x, 0) + ··· + ξ^(n−2)α

Γ(1 + (n − 2)α)u_m^((n−1)α)(x, 0)











(dξ)^α. (12)

Finally, the solution u(x, t ) is given by u(x, t ) = lim

m→∞u_m(x, t ) (13)

3. Illustrative examples

In this section several examples for diffusion and wave equations on Cantor sets is presented in order to demon- strate the simplicity and the efficiency of the above method.

Example 3.1.

Let us consider the following one-dimensional diffusion equation with local fractional derivative in the form

∂^αu(x, t )

∂t^α −∂^2αu(x, t )

∂x^2α = 0, (14)

subject to the initial value

u(x, 0) = x^α

Γ(1 + α)+ cosα(x^α). (15)

In view of (12) and (14) the local fractional iteration algorithm can be written as follows:

u_m+1(x, t ) = um(x, t ) − 1 Γ(1 + α)

Z t 0

·∂^αum(x,ξ)

∂ξ^α − ∂^α

∂ξ^αeL⁻¹_α µ 1 s^αeL_α

½∂^2αum(x,ξ)

∂x^2α

¾¶¸

(dξ)^α, m ≥ 0. (16)

We can use the initial condition to select u0(x, t ) = x^α

Γ(1 + α)+ cosα(x^α) . Using this selection into the correction func- tional (16) gives the following successive approximations

u₀(x, t ) = x^α

Γ(1 + α)+ cosα(x^α), (17)

u₁(x, t ) = u0(x, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu₀(x,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1 s^αLe_α

½∂^2αu₀(x,ξ)

∂x^2α

¾¶¸

(dξ)^α

= x^α

Γ(1 + α)+ cosα(x^α)

·

1 − t^α Γ(1 + α)

¸

, (18)

u₂(x, t ) = u1(x, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu₁(x,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1 s^αLe_α

½∂^2αu₁(x,ξ)

∂x^2α

¾¶¸

(dξ)^α

= x^α

Γ(1 + α)+ cosα(x^α)

·

1 − t^α

Γ(1 + α)+ t^2α Γ(1 + 2α)

¸

, (19)

u₃(x, t ) = u2(x, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu₂(x,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1 s^αLe_α

½∂^2αu₂(x,ξ)

∂x^2α

¾¶¸

(dξ)^α

= x^α

Γ(1 + α)+ cosα(x^α)

·

1 − t^α

Γ(1 + α)+ t^2α

Γ(1 + 2α)− t^3α Γ(1 + 3α)

¸

, (20)

...

um(x, t ) = x^α

Γ(1 + α)+ cosα(x^α)

"

m

X

k=0

(−1)^k t^kα Γ(1 + kα)

#

(21)

Finally, the solution is u(x, t ) = lim_m→∞um(x, t )

= x^α

Γ(1 + α)+ cosα(x^α)

"_∞ X

k=0

(−1)^k t^kα Γ(1 + kα)

#

= x^α

Γ(1 + α)+ Eα(−t^α)cos_α(x^α). (22)

Example 3.2.

Consider the following two-dimensional diffusion equation with local fractional derivative in the form

∂^αu(x, , y, t )

∂t^α −1 2

µ∂^2αu(x, y, t )

∂x^2α +∂^2αu(x, y, t )

∂y^2α

¶

= 0, (23)

and the initial condition

u(x, y, 0) = E_α(x^α+ y^α). (24)

Applying Eqs. (12) and (23), we obtain the correction function can be written as follows:

u_m+1(x, y, t ) =um(x, y, t )

− 1

Γ(1 + α) Z t

0

·∂^αu_m(x, y,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1 s^αLe_α½ 1

2

µ∂^2αu_m(x, y,ξ)

∂x^2α +∂^2αu_m(x, y,ξ)

∂y^2α

¶¾¶¸

(dξ)^α. (25)

We can use the initial condition to select u₀(x, y, t ) = E_α(x^α+ y^α) . Using this selection into the correction functional (25) gives the following successive approximations

u₀(x, y, t ) = E_α(x^α+ y^α), u₁(x, y, t ) = u0(x, y, t ) − 1

Γ(1 + α) Z _t

0

·∂^αu₀(x, y,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1 s^αLe_α½ 1

2

µ∂^2αu₀(x, y,ξ)

∂x^2α +∂^2αu₀(x, y,ξ)

∂y^2α

¶¾¶¸

(dξ)^α

= Eα(x^α+ y^α)

·

1 + t^α Γ(1 + α)

¸ ,

u₂(x, y, t ) = u1(x, y, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu₁(x, y,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1 s^αLe_α½ 1

2

µ∂^2αu₁(x, y,ξ)

∂x^2α +∂^2αu₁(x, y,ξ)

∂y^2α

¶¾¶¸

(dξ)^α

= Eα(x^α+ y^α)

·

1 + t^α

Γ(1 + α)+ t^2α Γ(1 + 2α)

¸ ,

u3(x, y, t ) = u2(x, y, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu2(x, y,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1 s^αLe_α½ 1

2

µ∂^2αu2(x, y,ξ)

∂x^2α +∂^2αu2(x, y,ξ)

∂y^2α

¶¾¶¸

(dξ)^α

= Eα(x^α+ y^α)

·

1 + t^α

Γ(1 + α)+ t^2α

Γ(1 + 2α)+ t^3α Γ(1 + 3α)

¸ , ...

um(x, y, t ) = Eα(x^α+ y^α)

"

m

X

k=0

t^kα Γ(1 + kα)

# .

Finally, the solution is

u(x, y, t ) = lim_m→∞um(x, y, t )

=Eα(x^α+ y^α)

"

X∞ k=0

t^kα Γ(1 + kα)

#

=Eα(x^α+ y^α) + t^α). (26)

Example 3.3.

Let us consider the one-dimensional wave equation involving local fractional operator

∂^αu(x, t )

∂t^2α −∂^2αu(x, t )

∂x^2α = 0, (27)

subject to the initial value

u(x, 0) = 0,∂^αu(x, 0)

∂t^α = Eα(x^α). (28)

In view of (12) and (27) the local fractional iteration algorithm can be written as follows:

u_m+1(x, t ) = um(x, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu_m(x,ξ)

∂ξ^α − ∂^α

∂ξ^αeL⁻¹_α µ 1

s^2αLe_α

½∂^2αu_m(x,ξ)

∂x^2α

¾¶

− Eα(x^α)

¸

(dξ)^α. (29)

We can use the initial condition to select u₀(x, t ) = t^α

Γ(1 + α)E_α(x^α) . Using this selection into the correction functional

(29) gives the following successive approximations u₀(x, t ) = t^α

Γ(1 + α)E_α(x^α), u₁(x, t ) = u0(x, t ) − 1

Γ(1 + α) Z t

0

·∂^αu₀(x,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1

s^2αeL_α

½∂^2αu₀(x,ξ)

∂x^2α

¾¶

− Eα(x^α)

¸ (dξ)^α

= Eα(x^α)

· t^α

Γ(1 + α)+ t^3α Γ(1 + 3α)

¸ ,

u₂(x, t ) = u1(x, t ) − 1 Γ(1 + α)

Z _t

0

·∂^αu1(x,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1

s^2αeL_α

½∂^2αu1(x,ξ)

∂x^2α

¾¶

− Eα(x^α)

¸ (dξ)^α

= Eα(x^α)

· t^α

Γ(1 + α)+ t^3α

Γ(1 + 3α)+ t^5α Γ(1 + 5α)

¸ ,

u₃(x, t ) = u2(x, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu₁(x,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1

s^2αeL_α

½∂^2αu₂(x,ξ)

∂x^2α

¾¶

− Eα(x^α)

¸ (dξ)^α

= Eα(x^α)

· t^α

Γ(1 + α)+ t^3α

Γ(1 + 3α)+ t^5α

Γ(1 + 5α)+ t^7α Γ(1 + 7α)

¸ , ...

u_m(x, t ) = E_α(x^α)

"

m

X

k=0

t^(2k+1)α Γ(1 + (2k + 1)α)

# .

Finally, the solution is u(x, t ) = lim_m→∞um(x, t )

=Eα(x^α)

"

X∞ k=0

t^(2k+1)α Γ(1 + (2k + 1)α)

#

=Eα(x^α) sinh_α(t^α). (30)

Example 3.4.

Consider the following two-dimensional wave equation with local fractional derivative in the form

∂^αu(x, , y, t )

∂t^2α − 2

µ∂^2αu(x, y, t )

∂x^2α +∂^2αu(x, y, t )

∂y^2α

¶

= 0, (31)

and the initial condition

u(x, y, 0) = sin_α(x^α) sin_α(y^α),∂^αu(x, y, 0)

∂t^α = 0. (32)

Applying Eqs. (12) and (31), we obtain the correction function can be written as follows:

u_m+1(x, y, t ) =um(x, y, t )

− 1

Γ(1 + α) Z t

0

·∂^αu_m(x, y,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1

s^2αeL_α

½ 2

µ∂^2αu_m(x, y,ξ)

∂x^2α +∂^2αu_m(x, y,ξ)

∂y^2α

¶¾¶¸

(dξ)^α. (33) We can use the initial condition to select u₀(x, y, t ) = sinα(x^α) sin_α(y^α) . Using this selection into the correction func- tional (33) gives the following successive approximations

u₀(x, y, t ) = sin_α(x^α) sin_α(y^α), u₁(x, y, t ) = u0(x, y, t ) − 1

Γ(1 + α) Z t

0

·∂^αu₀(x, y,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1

s^2αLe_α

½ 2

µ∂^2αu₀(x, y,ξ)

∂x^2α +∂^2αu₀(x, y,ξ)

∂y^2α

¶¾¶¸

(dξ)^α

= sinα(x^α) sin_α(y^α)

·

1 − 4t^2α Γ(1 + 2α)

¸ ,

u₂(x, y, t ) = u1(x, y, t ) − 1 Γ(1 + α)

Z t 0

·∂^αu₁(x, y,ξ)

∂ξ^α − ∂^α

∂ξ^αLe⁻¹_α µ 1

s^2αLe_α

½ 2

µ∂^2αu₁(x, y,ξ)

∂x^2α +∂^2αu₁(x, y,ξ)

∂y^2α

¶¾¶¸

(dξ)^α

= sinα(x^α) sin_α(y^α)

·

1 − 4t^2α

Γ(1 + 2α)+ 8t^4α Γ(1 + 4α)

¸ , ...

u_m(x, y, t ) = sin_α(x^α) sin_α(y^α)

"

m

X

k=0

(−1)^k 2^2kt^2kα Γ(1 + 2kα)

# .

Finally, the solution is

u(x, y, t ) = lim_m→∞um(x, y, t )

= sinα(x^α) sin_α(y^α)

"

X∞ k=0

(−1)^k 2^2kt^2kα Γ(1 + 2kα)

#

= sinα(x^α) sin_α(y^α) cos_α(2t^α). (34)

4. Conclusions

In this work the diffusion and wave equations on Cantor sets within the local fractional differential operators had been analyzed using the local fractional variational iteration transform method. The non-differentiable solutions are obtained. The present method is a powerful tool for solving many differential equations on Cantor sets within the local fractional derivative.

Acknowledgments

The author acknowledges Ministry of Higher Education and Scientific Research in Iraq for its support this work.

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