Analytic Solution of Fractional-Order Heat- and Wave-Like Equations Using Generalized n-dimensional Differential Transform Method

Analytic Solution of Fractional-Order Heat- and Wave-Like Equations Using Generalized n-dimensional Differential Transform Method

Vipul K. Baranwal, Ram K. Pandey, Manoj P. Tripathi, and Om P. Singh

Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi-221005, India

Reprint requests to O. P. S.; E-mail:singhom@gmail.com

Z. Naturforsch.66a,581 – 590 (2011) / DOI: 10.5560/ZNA.2011-0020 Received February 18, 2011

In this paper, we have introduced a generalizedn-dimensional differential transform method to pro- pose a user friendly algorithm to obtain the closed form analytic solution forn-dimensional fractional heat- and wave-like equations. Three examples are given to establish the simplicity of the algorithm.

In Example5.3, we show that ten terms of the series representing the solution, even in fractional order, give a very accurate solution.

Key words:Generalizedn-Dimensional Differential Transform Method;n-Dimensional Fractional Heat- and Wave-Like Equation; Caputo Fractional Derivative.

Mathematics Subject Classification 2000:35L05, 35C05, 35Q80.

2. Introduction

The idea of fractional-order derivatives initially arose from a letter by Leibnitz to L’Hospital in 1695.

Fractional calculus has gained considerable popularity and importance during the past three decades, mainly due to its applications in numerous fields of science and engineering. One of the main advantages of using fractional-order differential equations in mathematical modelling is their non-local property. It is a well known fact that the integer-order differential operator is a local operator whereas the fractional-order differential oper- ator is non-local in the sense that the next state of the system depends not only upon its current state but also upon all of its proceeding states.

In the last decade, many authors have made notable contributions to both theory and application of frac- tional differential equations in areas as diverse as fi- nance [1–3], physics [4–7], control theory [8], and hydrology [9–13].

In this paper, we consider the following n- dimensional fractional heat- and wave-like equations which are the generalized form of the model in [14]:

∂^αu

∂t^α =f1(x₁,x2, . . . ,xn−1)∂²u

∂x²₁ +f₂(x₁,x₂, . . . ,xn−1)∂²u

∂x²₂+. . . (1)

+fn−1(x₁,x₂, . . . ,xn−1) ∂²u

∂x²_n−1+g(x₁,x₂, . . . ,xn−1,t), 0<x_i<a_i, i=1,2, . . . ,(n−1), 0<α≤2, t>0, subject to the initial conditions

u(x₁,x₂, . . . ,xn−1,0) =Ψ(x₁,x₂, . . . ,xn−1), u_t(x₁,x₂, . . . ,xn−1,0) =η(x₁,x₂, . . . ,xn−1), (2) where α is a parameter describing the fractional derivative. Fractional heat-like and wave-like equa- tions are obtained from (1) by restricting the parameter αin(0,1]and(1,2], respectively. The fractional wave- like equation can be used to describe different models in anomalous diffusive and sub diffusive systems, de- scription of fractional random walk, unification of dif- fusion and wave propagation phenomenon [15–18].

Several authors [14,19,20] applied the Adomian decomposition method (ADM), the variational it- eration method (VIM), and the homotopy analysis method (HAM) successfully to solve two- and three- dimensional fractional heat- and wave-like equations.

In 1986, a new powerful numerical technique named differential transform method (DTM), was developed by Zhao [21], to solve various scientific and engi- neering problems. Originally, he developed DTM to solve the electric circuit problems. DTM is based on

c

2011 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

582 V. K. Baranwal et al.·Analytic Solution of Fractional-Order Heat- and Wave-Like Equations the Taylor series expansion which constructs analyti-

cal solutions in the form of a polynomial. The tradi- tional higher-order Taylor series method requires sym- bolic computations, but the DTM does not require high symbolic computations. However, the solution is obtained by DTM in the form of polynomial series through an iterative procedure. Various applications of DTM are given in [22–26]. Recently Kurnaz et al. [27] have applied DTM for solving partial differen- tial equations. Arikoglu and Ozkol [28] developed the fractional differential transform method which is based on the classical differential transform method, on frac- tional power series, and on Caputo fractional deriva- tives. Odibat and Momani proposed the one- and two- dimensional generalized differential transform method (GDTM) to solve various ordinary/partial differential equations of integer and fractional order [29–31].

In this paper, we extend the two-dimensional GDTM [29–31] ton-dimensions and apply it to solve n-dimensional fractional heat- and wave-like equa- tions. The accuracy and applicability of the above method is established by means of several examples.

3. Fractional Calculus

We give some basic definitions and properties of fractional calculus [32–35] that are prerequisite for further development.

Definition 2.1. A real function f(x),x>0, is said to be in a spaceC_µ,µ∈Rif there exists a real number p(<µ)such thatf(x) =x^pf1(x)wheref1(x)∈C[0,∞), and is said to be in the spaceC_m^µif f^(m)∈C_µ,m∈N. Definition 2.2.The Riemann–Liouville fractional inte- gral operator of orderα≥0 of a function f∈C_µ,µ≥

−1 is defined as J_a^αf(x) = 1

Γ(α)

x Z

a

(x−t)^α−1f(t)dt, α>0, x>0.

(3)

Forα,β >0,a≥0, andγ≥ −1, the operatorJ_a^α has the following properties:

1. J_a^α(x−a)^γ= Γ(1+γ)

Γ(1+γ+α)(x−a)^γ+α, 2. J_a^αJ_a^βf(x) =J_a^α+βf(x),

3. J_a^αJ_a^βf(x) =J_a^βJ_a^αf(x).

(4)

Definition 2.3.The fractional derivative of orderα of a function f(x)in the Caputo sense is defined as D^α_af(x) =J_a^m−αD^α_af(x)

= 1

Γ(m−α)

x Z

a

(x−t)^m−α−1f^m(t)dt (5) form−1<α≤m, m∈N, x>a, f ∈C₋₁^m .

The following properties of the operatorD^α_a are well known:

D^α_a(x−a)^γ=







Γ(1+γ)

Γ(1+γ−α)(x−a)^γ−α, forα≤γ

0, forα>γ,

(6) D^α_aJ_a^αf(x) = f(x), (7) J_a^αD^α_af(x) = f(x)−

m−1 k=0

∑

f^(k)(a)(x−a)^k

k! , x>0. (8) The following theorem involving generalized Taylor’s formula is needed for the further development of the theory.

Theorem 3.1. Ifu(x,y) = f(x)g(y), f(x) =x^λg(x), λ>−1, andg(x)has the generalized power series ex- pansiong(x) =∑^∞_n=0a_n·(x−x₀)^nα with the radius of convergenceR>0,then for 0<α≤1, x∈(0,R), D^γ_x

0D^β_x

0f(x) =D^γ+β_x

0 f(x), (9)

when either of the two conditions hold:

(a) β<λ+1 andγ is arbitrary or

(b) β ≥λ+1, γ is arbitrary, and a_n =0 for n= 0,1, . . . ,m−1,wherem−1<β ≤m.

Proof is given in [31].

4. Generalizedn-Dimensional Differential Transform Method

We have used the following symbolic notations for convenience:

(i) (x₁,x₂, . . . ,x_n)≡(x)_n, (ii) α1,α2, . . . ,αn≡(α)_n, (iii) (∗,x₂, . . . ,x_n)≡(∗,x¯_n).

The generalized n-dimensional differential transform of a functionu(x)_nis defined as

U_(α)n(k)_n= 1

n

∏

i=1

Γ(α_ik_i+1)

"

n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! u(x)_n

#

(˜x)n

.

(10) The inversion of (10) is given by

u(x)_n=

∞

∑

k1,k2,...,kn=0

"

U_(α)n(k)_n

n

∏

i=1

(x_i−x˜_i)^kiαi

#

. (11)

Forα=1,∀ithe generalizedn-dimensional differen- tial transform reduces to the classical n-dimensional differential transform. For the special case whenu(x)_n can be split as

u(x)_n=

n

∏

i=1

f_i(x_i),

thenu(x)_n=

n

∏

i=1

"

∞

∑

ki=0

Fα_i(k_i)·(x_i−x˜_i)^kiαi

# ,

(12)

whereF_αi(k_i)are the generalized one-dimensional dif- ferential transforms of f_i(x_i),1≤i≤n.From (11) and (12) we deduce thatU_(α)n(k)_n=∏ⁿi=1F_αi(k_i).

Now we give some theorems outlining the different properties ofu(x)_nandU_(α)n(k)_n.These theorems are then-dimensional generalisations of the corresponding theorems of [29–31].

Theorem 4.1. If u(x)_n = v(x)_n ± w(x)_n, then U_(α)n(k)_n=V_(α)n(k)_n±W_(α)n(k)_n.

Theorem 4.2. If u(x)_n =cv(x)_n, then U_(α)n(k)_n = cV_(α)n(k)_n,wherecis a scalar.

Theorem 4.3. Foru(x)_n=v(x)_n·w(x)_n,

U_(α)n(k)_n = ∑^k_a¹₁₌₀∑^k_a²₂₌₀. . .∑^k_aⁿ_n=0V_(α)n(a₁,k_n−a_n)

·W_(α)n(k₁−a₁,a_n).

Proof.The theorem is proved by using induction onn.

The assertion follows trivially forn=1,as U_α1(k₁) =

k1

∑

a1=0

V_α1(k₁−a₁)W_α1(a₁). (13) Assuming the theorem holds forn=m,

U_(α)m(k)_m=

k₁

∑

a1=0 k₂

∑

a2=0

. . .

km

∑

am=0

V_(α)m(a₁,km−am)W_(α)m(k₁−a1,am).

(14)

The inverse of above follows from (11) and is given as u(x)_m=

∞

∑

k₁=k₂=...km=0 k1

∑

a1=0 k2

∑

a2=0

. . .

km

∑

am=0

V_(α)m(a₁,k_m−a_m)

·W_(α)m(k₁−a₁,a_m)

m

∏

i=1

(x_i−x˜_i)^kiαi

u(x)_m=

"

∞

∑

k1=k₂=...k_m=0

V_(α)m(k)_m

m

∏

i=1

(x_i−x˜_i)^kiαi

#

·

"

∞

∑

k1=k₂=...k_n=0

W_(α)m(k)_m

m

∏

i=1

(x_i−x˜_i)^kiαi

#

, (15)

sinceu(x)_n=v(x)_n·w(x)_n.Replacingmbym+1,we obtain

u(x)_m+1=

"

∞

∑

k1=k₂=...k_m+1=0

V_(α)

m+1(k)_m+1

m+1

∏

i=1

(x_i−x˜_i)^kiαi

#

·

"

∞

∑

k₁=k₂=...k_m+1=0

W_(α)m+1(k)_m+1

m+1

∏

i=1

(x_i−x˜_i)^kiαi

# .

(16) LetV_(α)¹

m(k)_m=

"

∞

∑

km+1=0

V_(α)m+1(k)_m+1(x_m+1−x˜m+1)^km+1αm+1

# (17)

andW_(α)¹

m(k)_m=

"

∞

∑

km+1=0

W_(α)m+1(k)_m+1(x_m+1−x˜_m+1)^km+1αm+1

#

. (18)

Using (14) – (18), we have u(x)_m+1=

∞

∑

k₁=k₂=...km=0 k1

∑

a₁=0 k2

∑

a₂=0

. . .

km

∑

am=0

V_(α)¹

m(a₁,k_m−a_m)

·W_(α)¹

m(k₁−a₁,a_m)

m

∏

i=1

(x_i−x˜_i)^kiαi

=

∞

∑

k1=k₂=...k_m=0 k₁ a

∑

₁=0

k₂ a

∑

₂=0

. . .

km

a

∑

m=0

∞

∑

km+1=0

V_(α)m+1

·(a₁,k_m−am,km+1)(x_m+1−x˜m+1)^km+1αm+1

584 V. K. Baranwal et al.·Analytic Solution of Fractional-Order Heat- and Wave-Like Equations

·

∞

∑

k_m+1=0

W_(α)m+1(k₁−a₁,a_m,k_m+1)

·(x_m+1−x˜m+1)^km+1αm+1 m

∏

i=1

(x_i−x˜_i)^kiαi. (19) Using (14) in (19), we have

u(x)_m+1=

∞

∑

k₁=k₂=...km+1=0 k1

∑

a1=0 k2

∑

a2=0

. . .

km+1

∑

am+1=0

V_(α)m+1

·(a₁,km+1−a_m+1)W_(α)

m+1(k₁−a1,am+1)

·

m+1

∏

i=1

(x_i−x˜_i)^kiαi

. (20)

Substituting (13) into (20), the validity of the theorem holds forn=m+1,thus proving the theorem by in- duction.

From now onwards 0<α_i≤1,andi=1,2, . . . ,n.

Theorem 4.4. For u(x)_n = D^α_x˜ⁱ

iv(x)_n,U_(α)n(k)_n =

Γ(α_i(k_i+1)+1)

Γ(αiki+1) V_(α)n(k₁,k₂, . . . ,ki−1,k_i+1,k_i+1, . . .k_n).

Proof.From (10) we have U_(α)n(k)_n= 1

n

∏

i=1

Γ(α_iki+1)

" _n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! u(x)_n

#

(˜x)_n

= Γ(α_i(k_i+1) +1) Γ(α_i(k_i+1) +1)∏ⁿ

i=1

Γ(α_ik_i+1)

·

"

n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! D^α_x˜ⁱ

iv(x)_n

#

(x)˜n

=Γ(α_i(k_i+1) +1) Γ(α_ik_i+1)

·V_(α)n(k₁,k2, . . . ,ki−1,ki+1,ki+1, . . .k_n).

Theorem 4.5. Ifu(x)_n=D^α_x˜¹

1D^α_x˜²

2 . . .D^α_x˜ⁿ

nv(x)_n,then U_(α)n(k)_n=

n

∏

j=1

Γ(α_j(k_j+1) +1)

n

∏

j=1

Γ(α_jk_j+1)

·V_(α)n(k₁+1,k₂+1, . . . ,k_n+1).

Proof.From (10) we have U_(α)n(k)_n= 1

n

∏

i=1

Γ(α_ik_i+1)

"

n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! u(x)_n

#

(˜x)n

=

n

∏

j=1

Γ(α_j(k_j+1) +1)

n

∏

j=1

Γ(α_j(k_j+1) +1)∏ⁿ

i=1

Γ(α_ik_i+1)

·

"

n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! D^α_x˜¹

1D^α_x˜²

2. . .D^α_x˜ⁿ

nv(x)_n

#

(˜x)n

=

n

∏

j=1

Γ(α_j(k_j+1) +1)

n

∏

j=1

Γ(α_jk_j+1)

·V_(α)n(k₁+1,k₂+1, . . . ,k_n+1).

Theorem 4.6. If u(x)_n = ∏ⁿ_i=1(x_i−x˜_i)^miαi, then U_(α)n(k)_n=∏ⁿ_i=1δ(k_i−m_i).

Proof.From (11) we have u(x)_n=

n

∏

i=1

(x_i−x˜_i)^miαi,

=

∞

∑

k1=0

∞

∑

k2=0

. . .

∞

∑

kn=0 n

∏

i=1

(δ(k_i−m_i)(x_i−x˜i)^kiαi)

! .

So, applying the inverse differential transform (10), we getU_(α)n(k)_n=∏ⁿi=1δ(k_i−m_i).

Theorem 4.7. Let u(x)_n = ∏ⁿi=1f_i(x_i),f_i(x_i) = x^λ_i h_i(x_i),λ>−1,h_i(x_i)has the generalized Taylor se- ries expansionh_i(x_i) =∑^∞_n=0a_n(x_i−x˜_i)^nαi,and either of the two conditions hold:

(a) β<λ+1 andγ is arbitrary or

(b) β ≥λ+1, γ is arbitrary, and an =0 for n= 0,1, . . . ,m−1, wherem−1<β≤m.

Then the generalized n-dimensional differential transform (10) becomes

U_(α)n(k)_n= 1

n

∏

j=1

Γ(α_jk_j+1)

·













n

∏

j=1 j6=i

(D^α_x˜^j

j)^kj





(D^α_x˜^iki

i )u(x)_n







(x)˜n

.

Proof. The proof follows immediately from the fact thatD^γ_x˜¹

iD^γ_x˜²

i f_i(x_i) =D^γ_x˜^1+γ2

i f_i(x_i)under the conditions given in Theorem3.1.

In Theorems 4.8–4.10, the functions fi(x_i)satisfy the conditions given in Theorem3.1.

Theorem 4.8. Let u(x)_n =D^γ_x˜

iv(x)_n,m−1 < γ ≤ m,v(x)_n=∏ⁿ_i=1f_i(x_i),then

U_(α)n(k)_n=Γ(α_ik_i+γ+1) Γ(α_ik_i+1)

·V_(α)n(k₁,k₂, . . . ,ki−1,k_i+γ/α_i,k_i+1, . . .k_n).

Proof.From (10) we have U_(α)n(k)_n= 1

n

∏

i=1

Γ(α_ik_i+1)

"

n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! u(x)_n

#

(˜x)n

= Γ(α_ik_i+γ+1) Γ(α_ik_i+γ+1)∏ⁿ

i=1

Γ(α_ik_i+1)

·

"

n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! D^γ_x˜

iv(x)_n

#

(˜x)_n

=Γ(α_ik_i+γ+1) Γ(α_ik_i+1)

·V_(α)n(k₁,k₂, . . . ,ki−1,k_i+γ/α_i,k_i+1, . . .k_n).

Theorem 4.9. Ifu(x)_n=∏ⁿi=1f_i(x_i),thenU_(α)n(k)_n=

1

∏ⁿ_i=1Γ(α_ik_i+1)

h

∏ⁿ_i=1D^α_x˜^iki

i

u(x)_ni

(x)˜n

. Theorem 4.10. Letu(x)_n=D^γ_x˜¹

1D^γ_x˜²

2. . .D^γ_x˜ⁿ

nv(x)_n,m_i− 1<γi≤m_i,v(x)_n=∏ⁿ_i=1f_i(x_i),then

U_(α)n(k)_n=

n

∏

i=1

Γ(α_ik_i+γi+1)

n

∏

i=1

Γ(α_ik_i+1)

·

V_(α)n(k₁+γ₁/α₁,k₂+γ2/α₂, . . . ,k_n+γn/α_n) .

Proof.From (10) we have U_(α)n(k)_n= 1

n

∏

i=1

Γ(α_ik_i+1)

"

n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! u(x)_n

#

(˜x)n

=

n

∏

i=1

Γ(α_ik_i+γi+1)

n

∏

i=1

Γ(α_ik_i+γ_i+1)

n

∏

i=1

Γ(α_ik_i+1)

·

" _n

∏

i=1

(D^α_x˜ⁱ

i)^ki

! D^γ_x˜¹

1D^γ_x˜²

2. . .D^γ_x˜ⁿ

nv(x)_n

#

(x)˜_n

=

n

∏

i=1

Γ(α_ik_i+γi+1)

n

∏

i=1

Γ(α_ik_i+1)

[V_(α)n(k₁+γ₁/α₁,k₂

+γ₂/α₂, . . . ,k_n+γn/α_n)].

5. Numerical Examples

Let x= (x₁,x₂, . . . ,x_n)∈ Rⁿ⁻¹, α˜ = (α₁,α₂, . . . , αn−1) ∈ (0,1]ⁿ⁻¹ and k = (k₁,k₂, . . . ,k_n) ∈ Nⁿ⁻¹₀ , whereNⁿ⁻¹₀ =N∪ {0}.We use the following standard notations:x^α₁^1k1,x^α₂^2k2, . . . ,x^α_n−1^n−1kn−1 =x^αk˜ . In the fol- lowing examplesu(x,t)denotes the exact solution of the problem under consideration and is given as u(x,t) =

∞

∑

k₁=0

∞

∑

k₂=0

. . .

∞

∑

kn−1=0

∞

∑

h=0

U_(α)n−1,β(k,h)x^α˜kt^hβ. (21) Further, we define the error by

E_m^α=|u(x,t)−u˜_m(x,t)|, (22) where ˜u_m(x,t)is the approximate solution containing mterms obtained by truncating the solution series (21).

Example 5.1. Consider the following n-dimensional heat-like equation:

∂^αu

∂t^α =γ

n−1

∑

i=1

∂²u

∂x²_i , 0<x_i<c_i, i=1,2, . . . ,(n−1), 0<α≤1, t>0,

(23)

subject to the initial condition u(x,0) =

n−1

∏

i=1

sinxi, (24)

havingu(x,t) =e^{−(n−1)γt}∏ⁿ⁻¹_i=1sinx_i as the exact solu- tion forα=1.

Takingα1=1,α2=1, . . . ,αn−1=1,β=α,and ap- plying the generalizedn-dimensional transform to both sides of (23) and (24), we get

Γ(α(h+1) +1)

Γ(αh+1) U1,1,...,1,α(k,h+1) =

γ[(k₁+1)(k₁+2)U1,1,...,1,α(k₁+2,k₂, . . . ,kn−1,h) + (k₂+1)(k₂+2)U1,1,...,1,α(k₁,k₂+2,k₃, . . . ,kn−1,h)

586 V. K. Baranwal et al.·Analytic Solution of Fractional-Order Heat- and Wave-Like Equations +. . .+ (k_n−1+1)(k_n−1+2)U1,1,...,1,α

·(k₁,k2, . . . ,kn−2,kn−1+2,h)] (25)

and

U1,1,...,1,α(k,0) =(−1)^{(k1+k2+...+kn−1−(n−1))/2}

k₁!k₂!. . .k_n! , (26)

respectively.

Substitutingh=0,1,2,3, . . .in the recurrence rela- tion (25) and using (26), we obtain the different com- ponents ofU1,1,...,1,α(k,h)as follows:

U1,1,...,1,α(k,h) =(−(n−1)γ)^h Γ(αh+1)

·(−1)^{(k1+k2+...+kn−1−(n−1))/2}

k1!k₂!. . .k_n! .

The solutionu(x,t)of (23) is given as u(x,t) =

∞

∑

k1=0

∞

∑

k2=0

. . .

∞

∑

kn−1=0

∞

∑

h=0

U1,1,...,1,α(k,h)x^kt^hα

=E_α(−(n−1)γt^α)

n−1

∏

i=1

sinx_i, (27)

whereEα(z)is the Mittag–Leffler function defined by E_α(z) =∑^∞n=0 zⁿ

Γ(αn+1),α>0,z∈C.

Forα =1,the solution (27) of the fractional-order partial differential equation (PDE) reduces to the exact solution of the integer-order PDE

u(x,t) =

n−1

∏

i=1

sinx_i

∞

∑

h=0

(−(n−1)γt)^h h!

=e^{−(n−1)γt}

n−1

∏

i=1

sinxi.

Taking n=3,c₁=c₂=2π, and γ =1 in (23) and (27), we obtain the analytical solution of ^∂_∂t^α_α^u=^∂2u

∂x²₁+

∂²u

∂x²₂, 0<x₁,x₂<2π, 0<α ≤1, t>0, as u(x,t) = sinx₁sinx₂E_α(−2t^α),[14,19].

Similarly, Example 1 in [27] follows as a special case of our general solution (27) by substitutingn=4 andα=1.

Example 5.2. Next, we apply our algorithm to the fol- lowingn-dimensional heat-like equation:

∂^αu

∂t^α =

n−1

∏

i=1

x⁴_i+ 1 12(n−1)

n−1

∑

i=1

x²_i∂²u

∂x²_i

, (28)

0<xi<1, i=1,2, . . . ,(n−1), 0<α≤1, t>0,

subject to the initial condition

u(x,0) =0, (29)

havingu(x,t) =∏ⁿ⁻¹_i=1x⁴_i(e^t−1)as the exact solution forα=1.

Takingα₁=1,α₂=1, . . . ,αn−1=1,β=α, and ap- plying the generalizedn-dimensional transform to both sides of (28) and (29), we obtain

Γ(α(h+1) +1)

Γ(αh+1) U1,1,...,1,α(k,h+1) =

n−1

∏

i=1

δ(k_i−4)δ(h)

+ 1

12(n−1)

k₁

∑

a1=0 k₂

∑

a2=0

. . .

kn−1

∑

an−1=0 h

∑

b=0

(k₁−a₁+1)

·(k₁−a1+2)δ(a₁−2)

n−1

∏

j=2

δ(k_j−a_j)δ(h−b)

·U1,1,...,1,α(k₁−a₁+2,a₂,a₃, . . . ,an−1,b)

+

(a₂+1)(a₂+2)δ(a₁)δ(k₂−a₂−2)

·

n−1

∏

j=3

δ(k_j−a_j)δ(h−b)U1,1,...,1,α

·(k₁−a₁,a₂+2,a₃, . . . ,an−1,b)

+

(a₃+1)(a₃+2)δ(a₁)δ(k₃−a₃−2)

·

n−1

∏

j=2 j6=3

δ(k_j−a_j)δ(h−b)U1,1,...,1,α

·(k₁−a₁,a₂,a₃+2,a₄, . . . ,an−1,b)

+. . . +

(a_n−1+1)(a_n−1+2)δ(a₁)δ(k_n−1−an−1−2)

·

n−2

∏

j=2

δ(k_j−a_j)δ(h−b)U1,1,...,1,α

·(k₁−a₁,a₂,a₃, . . . ,an−2,an−1+2,b)

(30) andU1,1,...,1,α(k,0) =0,respectively. (31) Substitutingh=0,1,2,3, . . .in the recurrence rela- tion (30) and using (31), the different components of

U1,1,...,1,α(k,h)are obtained as

U1,1,...,1,α(k,h) =









 1

Γ(αh+1), k₁=k₂=. . .

=kn−1=4

0, otherwise.

Thus the solutionu(x,t)of (28) is given by u(x,t) =

∞

∑

k1=0

∞

∑

k2=0

. . .

∞

∑

kn−1=0

∞

∑

h=0

U1,1,...,1,α(k,h)x^kt^hα

= (E_α(t^α)−1)

n−1

∏

i=1

x⁴_i. (32)

Forα=1,the solution (32) reduces to u(x,t) =x⁴

∞ h=0

∑

1

Γ(h+1)t^h−1

!

= (e^t−1)

n−1

∏

i=1

x⁴_i,

which is the solution of the integer-order PDE.

The differential equation (28) and its solution (32) become

∂^αu

∂t^α =x⁴₁x⁴₂x⁴₃+ 1 36

x²₁∂²u

∂x²₁+x²₂∂²u

∂x²₂+x²₃∂²u

∂x²₃

,

0<x₁,x₂,x₃<1, 0<α≤1, t>0,

andu(x₁,x₂,x₃,t) =x⁴₁x⁴₂x⁴₃(E_α(t^α)−1), respectively, for n=4,which is the same as the solution obtained by other methods [14,19].

Example 5.3. Now, we consider the following n- dimensional wave-like equation with initial conditions:

∂^αu

∂t^α =

n−1 i=1

∑

x²_i+1 2

n−1 i=1

∑

x²_i∂²u

∂x²_i

, (33)

0<x_i<1, i=1,2, . . . ,(n−1), 1<α≤2, t>0, u(x,0) =0, u_t(x,0) =

n−2 i=1

∑

x²_i −x²_n−1, (34) havingu(x,t) = ∑ⁿ⁻²_i=1x²_i

(e^t−1) +x²_n−1(e^−t−1)as the exact solution forα=2.

We solve (33) for various values ofα.

(a)α=2

Taking αi=1,1≤i≤n−1, β =1, applying the generalized n-dimensional transform to both sides of

(33) – (34), and using theorem (17), we get Γ(h+3)

Γ(h+1)U1,1,...,1,1(k,h+2) =

n−1

∑

i=1







δ(k_i−2)

n−1

∏

j=1 j6=i

δ(k_j)δ(h)







+1 2

k1

∑

a1=0 k2

∑

a2=0

. . .

kn−1

∑

an−1=0 h

∑

b=0

(k₁−a₁+1)

·(k₁−a₁+2)δ(a₁−2)

n−1

∏

j=2

δ(k_j−a_j)δ(h−b)

·U1,1,...,1,1(k₁−a₁+2,a₂,a₃, . . . ,an−1,b)

+

(a₂+1)(a₂+2)δ(a₁)δ(k₂−a₂−2)

·

n−1

∏

j=3

δ(k_j−a_j)δ(h−b)U1,1,...,1,1

·(k₁−a₁,a₂+2,a₃, . . . ,an−1,b)

+

(a₃+1)(a₃+2)δ(a₁)δ(k₃−a₃−2)

·

n−1

∏

j=2 j6=3

δ(k_j−a_j)δ(h−b)U1,1,...,1,1

·(k₁−a1,a2,a3+2,a₄, . . . ,an−1,b)

+. . . +

(an−1+1)(an−1+2)δ(a₁)δ(kn−1−an−1−2)

·

n−2

∏

j=2

δ(k_j−a_j)δ(h−b)U1,1,...,1,1

·(k₁−a1,a2,a3, . . . ,an−2,an−1+2,b)

, (35)

U1,1,...,1,1(k,0) =0, (36)

U1,1,...,1,1(k,1) =V₁(k,1) +V₂(k,1) +. . .+Vn−1(k,1), where

Vj(k,1) =

(1, k_j=2,k_i=0, i6=j 0, otherwise,

1≤i≤n−1, 1≤ j≤n−2, Vn−1(k,1) =

−1, k₁=k₂=. . .=kn−2=0,kn−1=2,

0, otherwise.