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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:

Lαu(x, t ) + Rαu(x, t ) = g (x, t), (1)

where Lα=

∂tnα, 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

eLα{Lαu(x, t )} + eLα{Rαu(x, t )} = eLα©g (x, t)ª. (2)

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

snαLeα{u(x, t )} − s(n−1)αu(x, 0) − s(n−2)αu(α)(x, 0) − ··· − u((n−1)α)(x, 0) + eLα{Rαu(x, t )} = eLα{g (x, t )}, (3)

E-mail address:hassan.kamil28@yahoo.com

(2)

or

eLα{u(x, t )} = 1

sαu(x, 0) + 1

s2αu(α)(x, 0) + ··· + 1

snαu((n−1)α)(x, 0) + 1

snαLeα{g (x, t )} − 1

snαLeα{Rαu(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−1α µ 1

snαLeα©g (x, t) − Rαu(x, t )ª

. (5)

Derivative by α

∂tα both side Eq. (5), we have

αu(x, t )

∂tα = u(α)(x, 0) + ··· + t(n−2)α

Γ(1 + (n − 2)α)u((n−1)α)(x, 0) + α

∂tαeL−1α µ 1

snαeLα©g (x, t) − Rαu(x, t )ª

(6)

We now structure the correctional local fractional function in the form

um+1(x, t ) = um(x, t ) + 1 Γ(1 + α)

Z t 0

λ(ξ)α Γ(1 + α)









αum(x,ξ)

∂ξαα

∂ξαL−1α µ 1

snαLα©g (x,ξ) − Rαum(x,ξ)ª

u(mα)(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

δαum+1(x, t ) = δαum(x, t )+δα 1 Γ(1 + α)

Z t 0

λ(ξ)α Γ(1 + α)









αum(x,ξ)

∂ξαα

∂ξαL−1α µ 1

sLα©g (x,ξ) − Rαum(x,ξ)ª

−u(mα)(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 um+1(x, t ) is given by

δαum+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

um+1(x, t ) = um(x, t ) − 1 Γ(1 + α)

Z t 0









αum(x,ξ)

∂ξαα

∂ξαL−1α µ 1

snαLα©g (x,ξ) − Rαum(x,ξ)ª

−u(mα)(x, 0) + ··· + ξ(n−2)α

Γ(1 + (n − 2)α)um((n−1)α)(x, 0)









(dξ)α. (12)

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

m→∞um(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 )

∂tαu(x, t )

∂x2α = 0, (14)

(3)

subject to the initial value

u(x, 0) = xα

Γ(1 + α)+ cosα(xα). (15)

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

um+1(x, t ) = um(x, t ) − 1 Γ(1 + α)

Z t 0

·αum(x,ξ)

∂ξαα

∂ξαeL−1α µ 1 sαeLα

½um(x,ξ)

∂x2α

¾¶¸

(dξ)α, m ≥ 0. (16)

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

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

u0(x, t ) = xα

Γ(1 + α)+ cosα(xα), (17)

u1(x, t ) = u0(x, t ) − 1 Γ(1 + α)

Z t 0

·αu0(x,ξ)

∂ξαα

∂ξαLe−1α µ 1 sαLeα

½2αu0(x,ξ)

∂x2α

¾¶¸

(dξ)α

= xα

Γ(1 + α)+ cosα(xα)

·

1 − tα Γ(1 + α)

¸

, (18)

u2(x, t ) = u1(x, t ) − 1 Γ(1 + α)

Z t 0

·αu1(x,ξ)

∂ξαα

∂ξαLe−1α µ 1 sαLeα

½2αu1(x,ξ)

∂x2α

¾¶¸

(dξ)α

= xα

Γ(1 + α)+ cosα(xα)

·

1 − tα

Γ(1 + α)+ t2α Γ(1 + 2α)

¸

, (19)

u3(x, t ) = u2(x, t ) − 1 Γ(1 + α)

Z t 0

·αu2(x,ξ)

∂ξαα

∂ξαLe−1α µ 1 sαLeα

½2αu2(x,ξ)

∂x2α

¾¶¸

(dξ)α

= xα

Γ(1 + α)+ cosα(xα)

·

1 − tα

Γ(1 + α)+ t

Γ(1 + 2α)t Γ(1 + 3α)

¸

, (20)

...

um(x, t ) = xα

Γ(1 + α)+ cosα(xα)

"

m

X

k=0

(−1)k t Γ(1 + kα)

#

(21)

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

= xα

Γ(1 + α)+ cosα(xα)

" X

k=0

(−1)k tkα Γ(1 + kα)

#

= xα

Γ(1 + α)+ Eα(−tα)cosα(xα). (22)

Example 3.2.

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

αu(x, , y, t )

∂tα −1 2

µ2αu(x, y, t )

∂x2α +2αu(x, y, t )

∂y2α

= 0, (23)

and the initial condition

u(x, y, 0) = Eα(xα+ yα). (24)

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

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

− 1

Γ(1 + α) Z t

0

·αum(x, y,ξ)

∂ξαα

∂ξαLe−1α µ 1 sαLeα½ 1

2

µ2αum(x, y,ξ)

∂x +2αum(x, y,ξ)

∂y

¶¾¶¸

(dξ)α. (25)

(4)

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

u0(x, y, t ) = Eα(xα+ yα), u1(x, y, t ) = u0(x, y, t ) − 1

Γ(1 + α) Z t

0

·αu0(x, y,ξ)

∂ξαα

∂ξαLe−1α µ 1 sαLeα½ 1

2

µ2αu0(x, y,ξ)

∂x2α +2αu0(x, y,ξ)

∂y2α

¶¾¶¸

(dξ)α

= Eα(xα+ yα)

·

1 + tα Γ(1 + α)

¸ ,

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

Z t 0

·αu1(x, y,ξ)

∂ξαα

∂ξαLe−1α µ 1 sαLeα½ 1

2

µ2αu1(x, y,ξ)

∂x2α +2αu1(x, y,ξ)

∂y2α

¶¾¶¸

(dξ)α

= Eα(xα+ yα)

·

1 + tα

Γ(1 + α)+ t Γ(1 + 2α)

¸ ,

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

Z t 0

·αu2(x, y,ξ)

∂ξαα

∂ξαLe−1α µ 1 sαLeα½ 1

2

µu2(x, y,ξ)

∂x2α +u2(x, y,ξ)

∂y2α

¶¾¶¸

(dξ)α

= Eα(xα+ yα)

·

1 + tα

Γ(1 + α)+ t2α

Γ(1 + 2α)+ t3α Γ(1 + 3α)

¸ , ...

um(x, y, t ) = Eα(xα+ yα)

"

m

X

k=0

t Γ(1 + kα)

# .

Finally, the solution is

u(x, y, t ) = limm→∞um(x, y, t )

=Eα(xα+ yα)

"

X k=0

t Γ(1 + kα)

#

=Eα(xα+ yα) + tα). (26)

Example 3.3.

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

αu(x, t )

∂t2α2αu(x, t )

∂x2α = 0, (27)

subject to the initial value

u(x, 0) = 0,∂αu(x, 0)

∂tα = Eα(xα). (28)

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

um+1(x, t ) = um(x, t ) − 1 Γ(1 + α)

Z t 0

·αum(x,ξ)

∂ξαα

∂ξαeL−1α µ 1

s2αLeα

½2αum(x,ξ)

∂x2α

¾¶

− Eα(xα)

¸

(dξ)α. (29)

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

Γ(1 + α)Eα(xα) . Using this selection into the correction functional

(5)

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

Γ(1 + α)Eα(xα), u1(x, t ) = u0(x, t ) − 1

Γ(1 + α) Z t

0

·αu0(x,ξ)

∂ξαα

∂ξαLe−1α µ 1

s2αeLα

½2αu0(x,ξ)

∂x2α

¾¶

− Eα(xα)

¸ (dξ)α

= Eα(xα)

· tα

Γ(1 + α)+ t Γ(1 + 3α)

¸ ,

u2(x, t ) = u1(x, t ) − 1 Γ(1 + α)

Z t

0

·αu1(x,ξ)

∂ξαα

∂ξαLe−1α µ 1

s2αeLα

½2αu1(x,ξ)

∂x2α

¾¶

− Eα(xα)

¸ (dξ)α

= Eα(xα)

· tα

Γ(1 + α)+ t3α

Γ(1 + 3α)+ t5α Γ(1 + 5α)

¸ ,

u3(x, t ) = u2(x, t ) − 1 Γ(1 + α)

Z t 0

·αu1(x,ξ)

∂ξαα

∂ξαLe−1α µ 1

seLα

½2αu2(x,ξ)

∂x

¾¶

− Eα(xα)

¸ (dξ)α

= Eα(xα)

· tα

Γ(1 + α)+ t

Γ(1 + 3α)+ t

Γ(1 + 5α)+ t Γ(1 + 7α)

¸ , ...

um(x, t ) = Eα(xα)

"

m

X

k=0

t(2k+1)α Γ(1 + (2k + 1)α)

# .

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

=Eα(xα)

"

X k=0

t(2k+1)α Γ(1 + (2k + 1)α)

#

=Eα(xα) sinhα(tα). (30)

Example 3.4.

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

αu(x, , y, t )

∂t2α − 2

µ2αu(x, y, t )

∂x2α +2αu(x, y, t )

∂y2α

= 0, (31)

and the initial condition

u(x, y, 0) = sinα(xα) sinα(yα),αu(x, y, 0)

∂tα = 0. (32)

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

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

− 1

Γ(1 + α) Z t

0

·αum(x, y,ξ)

∂ξαα

∂ξαLe−1α µ 1

s2αeLα

½ 2

µ2αum(x, y,ξ)

∂x2α +2αum(x, y,ξ)

∂y2α

¶¾¶¸

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

u0(x, y, t ) = sinα(xα) sinα(yα), u1(x, y, t ) = u0(x, y, t ) − 1

Γ(1 + α) Z t

0

·αu0(x, y,ξ)

∂ξαα

∂ξαLe−1α µ 1

s2αLeα

½ 2

µ2αu0(x, y,ξ)

∂x2α +2αu0(x, y,ξ)

∂y2α

¶¾¶¸

(dξ)α

= sinα(xα) sinα(yα)

·

1 − 4t2α Γ(1 + 2α)

¸ ,

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

Z t 0

·αu1(x, y,ξ)

∂ξαα

∂ξαLe−1α µ 1

s2αLeα

½ 2

µ2αu1(x, y,ξ)

∂x2α +2αu1(x, y,ξ)

∂y2α

¶¾¶¸

(dξ)α

= sinα(xα) sinα(yα)

·

1 − 4t2α

Γ(1 + 2α)+ 8t4α Γ(1 + 4α)

¸ , ...

um(x, y, t ) = sinα(xα) sinα(yα)

"

m

X

k=0

(−1)k 22kt2kα Γ(1 + 2kα)

# .

(6)

Finally, the solution is

u(x, y, t ) = limm→∞um(x, y, t )

= sinα(xα) sinα(yα)

"

X k=0

(−1)k 22kt2kα Γ(1 + 2kα)

#

= sinα(xα) sinα(yα) cosα(2tα). (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.

References

[1] X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher Limited, Hong Kong, 2011.

[2] X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, 2012.

[3] X. J. Yang, D. Baleanu, W. P. Zhong, Approximation solutions for diffusion equation on Cantor time-space, Pro- ceeding of the Romanian Academy A, vol.14, no.2 (2013) 127–133.

[4] Z. P. Fan, H. K. Jassim, R. K. Rainna, and X. J. Yang, Adomian Decomposition Method for Three-Dimensional Diffusion Model in Fractal Heat Transfer Involving Local Fractional Derivatives, Thermal Science 19(1) (2015) 137–141.

[5] D. Baleanu, J. A. Machado, C. Cattani, M. C. Baleanu, X. J. Yang, Local fractional variational iteration and de- composition methods for wave equation on Cantor sets within local fractional operators, Abstract and Applied Analysis, vol. 2014, Article ID 535048 (2014) 1–6

[6] X. J. Yang, et al., Local Fractional Variational Iteration Method for Diffusion and Wave Equations on Cantor Sets, Romanian Journal of Physics 59(1-2) (2014)36–48.

[7] S. Xu, X. Ling, Y. Zhao, H. K. Jassim, A Novel Schedule for Solving the Two-Dimensional Diffusion in Fractal Heat Transfer, Thermal Science 19(1) (2015) 99–103.

[8] A. M. Yang, X. J. Yang, Z. B. Li, Local fractional series expansion method for solving wave and diffusion equations on Cantor sets. Abstract and Applied Analysis, vol. 2013, Article ID 351057 (2013) 1–5.

[9] Y. Cao, W. G. Ma, L. C. Ma, Local fractional functional method for solving diffusion equations on Cantor sets, Abstract and Applied Analysis, vol. 2014, Article ID 803693 (2014) 1–6.

[10] H. K. Jassim, C. Unlu, S. P. Moshokoa, C. M. Khalique, Local Fractional Laplace Variational Iteration Method for Solving Diffusion and Wave Equations on Cantor Sets within Local Fractional Operators, Mathematical Problems in Engineering, vol. 2015, Article ID 309870 (2015) 1–9.

[11] H. K. Jassim, Local Fractional Laplace Decomposition Method for Nonhomogeneous Heat Equations Arising in Fractal Heat Flow with Local Fractional Derivative, International Journal of Advances in Applied Mathematics and Mechanics 2(4) (2015) 1–7.

References

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