Classification and Approximate Functional Separable Solutions to the Generalized Diffusion Equations with Perturbation

Classification and Approximate Functional Separable Solutions to the Generalized Diffusion Equations with Perturbation

Fei-Yu Ji^aand Shun-Li Zhang^b

aCollege of Science, Xi’an University of Architecture and Technology, Xi’an 710055, China

bDepartment of Mathematics, Northwest University, Xi’an 710069, China Reprint requests to F.-Y. J.; E-mail:feiyuji@xauat.edu.cn

Z. Naturforsch.68a,621 – 628 (2013) / DOI: 10.5560/ZNA.2013-0058 Received May 17, 2013 / published online October 2, 2013

In this paper, the generalized diffusion equation with perturbationut=A(u,ux)uxx+εB(u,ux)is studied in terms of the approximate functional variable separation approach. A complete classification of these perturbed equations which admit approximate functional separable solutions is presented.

Some approximate solutions to the resulting perturbed equations are obtained by examples.

Key words:Diffusion Equation; Approximate Functional Separable Solutions; Approximate Functional Variable Separation Approach.

PACS numbers:02.30.Jr; 02.20.Sv; 02.30.Ik

1. Introduction

The symmetry group method plays an important role in reducing and finding exact solutions of par- tial differential equations (PDEs). Some of these meth- ods have been employed to seek exact solutions of PDEs for decades, for example, the Lie symmetry method [1–5], the generalized conditional symme- try (GCS) method [6,7], and the potential symme- try method [8], etc. Moreover, some variable sepa- ration approaches were presented, and have been ef- fectively used to construct exact solutions of PDEs, such as the classical method [9], the differential geom- etry method [10], the ansatz-based method [11–13], the formal variable separation approach [14], and the multi-linear variable separation approach [15,16]. As far as the symmetry group and the ansatz of the so- lution form of PDEs are concerned, we point out two types of the approaches, namely, the functional variable separation approach (FVSA) [17] and the derivative-dependent functional variable separation ap- proach (DDFVSA) [18]. Both methods are used to in- vestigate variable separation of the generalized nonlin- ear evolution equations.

On the other hand, in recent years, more and more researchers have been engaging in the study of the nonlinear evolution equations with a small parameter that were arising from science, technology, and en-

gineering. In order to solve such perturbed systems, there are some approximate methods that are com- monly used, for instance, the numerical and the pertur- bation methods [19], the approximate conditional sym- metry method [20], the approximate potential symme- try method [21], the approximate generalized condi- tional symmetry method (AGCS) [22], the approxi- mate homotopy direct reduction method [23], and the approximate direct reduction method [24].

Recently, we introduced the concept of the ap- proximate functional separable solutions (AFSSs), and proposed the approximate functional variable separa- tion approach (AFVSA) that based on AGCS, and it was applied to discuss the perturbed evolution equa- tions [25,26]. In this paper, we consider the approx- imate functional variable separation of the following generalized diffusion equations with perturbation:

ut=A(u,u_x)u_xx+εB(u,ux), (1) whereA(u,u_x)andB(u,u_x)are smooth functions of the indicated variables,εis a small parameter.

The outline of the paper is as follows. In Section2, a complete classification to the generalized diffusion equations with perturbation which admit AFSSs is ob- tained. In Section3, some AFSSs to the resulting per- turbed equations are constructed by way of examples.

The last section is reserved for conclusion and discus- sion.

© 2013 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

622 F.-Y. Ji and S.-L. Zhang·Approximate Functional Separable Solutions to Generalized Diffusion Equations 2. Classification of (1) Which Admits the AFSSs

Consider akth-order differential system[E], which is perturbed up to the first order in the small parameter ε, viz.

E^β x,u,u₍₁₎, . . . ,u_(k);ε

≡E₀^β x,u,u₍₁₎, . . . ,u_(k) +εE₁^β x,u,u₍₁₎, . . . ,u_(k)

=0, β=1, . . . ,q,

(2)

wherex= x¹,x², . . . ,xⁿ

,u= u¹,u², . . . ,u^m ,E_i^βare smooth functions in their arguments, ε is a small pa- rameter,u_(i)(i=1, . . . ,k)is the collection ofith-order partial derivatives, and

D_i= ∂

∂xⁱ+u^α_i ∂

∂u^α+u^α_{i j} ∂

∂u^α_j +· · ·, i=1, . . . ,n, denotes the operator of total derivative with respect to xⁱ.

Let

V=η ∂

∂u≡η(x,t,u;ε) ∂

∂u (3)

be an evolutionary vector field andηits characteristic.

Definition 1. The approximate solutionu=u(x,t;ε) of (2) is said to be an approximate functional separable solution (AFSS) if it satisfies (2) and the ansatz

f(u) +εg(u) =ψ(x) +φ(t) +ε ω(x) +θ(t)

+O ε²

, (4)

where f(u,u_x) and g(u,u_x) are smooth functions of u and u_x, and ψ(x), φ(t), ω(x), and θ(t) are some smooth functions ofxandt.

In particular, with respect to any perturbed (1+1)- dimensional nonlinear evolution equations, we have definition as follows:

Definition 2. The evolutionary vector field (3) is said to be an AGCS of the perturbed nonlinear evolution equation

u_t=K(x,t,u;ε) (5)

if and only if

V^(k) u_t−K(x,t,u;ε) _[W]T

[E]=O ε²

, (6)

wheneveru_t=K(x,t,u;ε), whereV^(k)denotes thekth- order prolongation to (3),K andη are differentiable

functions oft,xandu,ux,uxx, . . . ,[W]indicates the set of all differential consequences ofη=O ε²

with re- spect tox, that is,D_x^jη=O ε²

,j=0,1,2, . . . ,and[E]

denotes the solution manifold of (5).

Considering to the results in [26], we have the fol- lowing theorem.

Theorem 1. Equation (5) possesses AFSS (4) if and only if it admits the AGCS

V=η ∂

∂u≡

uxt+

p(u) +εq(u) uxut

∂

∂u, (7) where

p(u) = ln(f⁰)0

, q(u) = g⁰

f⁰ 0

, (8)

where f ≡ f(u), g≡g(u), and the prime denotes the corresponding order derivative with respect to u.

Next we classify to (1) that admits AGCS (3) by means of AFVSA. By Definition2and Theorem1, we know that classification of (1) which possesses the AF- SSs is equivalent to obtaining the AGCS of (3) satisfy- ing

V⁽²⁾

ut−A(u,ux)u_xx−εB(u,ux) _[W]T

[E]

=

D_tη−(A_uη+A_uxD_xη)u_xx

−ε(B_uη+B_uxD_xη)−AD²_xη

[W]^T[E].

(9)

Using expressions Dⁱ_xη=0, i=0,1,2, . . . ,and (1), and excluding the higher-order derivative of u in (9), we obtain an expression of independent derivative of u, which is

D_tη|_[W]^T_[E]=ε

Ξ1u³_xx+Ξ2u²_xx+Ξ3u_xx+Ξ4

+Ξ₅u³_xx+Ξ₆u²_xx+Ξ₇u_xx=O ε² ,

(10) where

Ξi≡Ξi(u,u_x) =O(ε), i=1,2,3,4, Ξj≡Ξj(u,u_x) =O ε²

, j=5,6,7. (11)

Since the expressions for Ξ_j (j=1,2, . . . ,7) are lengthy, we omit them here.

By solving over-determined system of differential equations (11), we obtain the classification theorem as follows:

F.-Y. Ji and S.-L. Zhang·Approximate Functional Separable Solutions to Generalized Diffusion Equations 623 Theorem 2. Suppose A(u,ux)6=0 and B(u,u_x)6=

constant, the perturbed equation u_t=A(u,u_x)u_xx+εB(u,u_x)

admits AFSS of the (4) if and only if it is equivalent to one of the following equations, up to first order inε:

(1) u_t=c²₁u^αu^α_xu_xx +ε

F₁

ux

u

+c₃

u^2α+1+c₂u

, (12)

η=uxt−1

uuxut; (13)

(2) u_t=c₁e^γuu^α_xu_xx+ε

c₁c₂(2α+1) α+2 +c₁c₃e^{γ(α−1)u/α}

u^α+2_x +F₂(u_x)e^γu γ +c₄

, η=u_xt+ε

c₂e^−γu+c₃e^−γu/α u_xu_t, α |c₂|+|c₃|

6=0 ; (3) ut=c1e^αu

u²_x uxx+ε

F2(u_x)e^αu

α −3c1c2ln(u_x) +c₁c₃e^3αu/2+c₄

, η=u_xt+ε

h

c₂e^αu/2+c₃e^−αui u_xu_t,

|c₂|+|c₃| 6=0 ; (4) u_t=c₁u^α_xu_xx+ε

2c₁c₂(α+1)u^α+2_x α+2 +c₄

u+F₂(u_x)

,

(14)

η=u_xt+ε(c₂u+c₃)u_xu_t, α |c₂|+|c₃|

6=0 ; (15)

(5) u_t=c₁

u²_xu_xx+εh

c₄−2c₁c₂ln(u_x)

u+F₂(u_x)i , η=u_xt+ε(c₂u+c₃)uu_xu_t, |c₂|+|c₃|

6=0 ; (6) u_t=c₁e^αuu_xx+ε

1

2c₁c₂u²_x+F₂(u_x)e^αu+c₃

, η=u_xt+εc₂e^−αuu_xu_t, c₂α6=0 ;

(7) ut=c1uxx+ε

uF3(u)− Z _u

ξF3(ξ)dξ

u²_x +F₂(u_x) +c₂u

,

η=uxt+ε

uF₃(u)−^RuξF₃(ξ)dξ c₁

uxut;

(8) u_t=c1u^αu_xx+ε

F1

ux

u

+c4

u^α+1 +

c₁c₂(α−6)u²_x 2(α−2) +c₃

u

,

(16)

η=u_xt+

−1

u+εc2u^1−α

u_xu_t; (17) (9) u_t=c₁u²u_xx+ε

F₁

u_x u

+c₃

u³ +

2c₁c₂ln(u)u²_x+c₄ u

,

η=u_xt+

−1 u+εc2

u

u_xu_t; (10) ut=u^α

u²_xuxx+ε

"

F1

u_x u

+c5

u^α−1

−3c₁c₃uln(u_x) +c1c2αu^3α2−2−c4u α−2

# ,

η=u_xt+

−1 u+ε

c2u^12α−2+c3u^1−α

u_xu_t,

|c₂|+|c₃| 6=0 ;

(11) u_t=c₁(u−c)^αu^β_xu_xx+εB(u,u_x), c₁β(β+2)6=0,

η=u_xt+

− 1 u−c+ε

c2(u−c)^1−α +c₃(u−c)⁻

α+2β β

u_xu_t, where B=B(u,u_x)satisfies Bu+c1





c3α(u−c)

α β−α−3β β

β

+ c₂h

α(2β+1)−6(β+1)i β+2



u^β+2_x

+B_uxu_x−(1+α+β)B−(α+β)H+K

u−c +L=0,

where H=H(u), K=K(u), and L=L(u)satisfy (u−c)K⁰−K−(α+β)

(u−c)H⁰−H

−(u−c)²L⁰=0.

(12) ut=A(u,ux)u_xx+εB(u,ux), η=uxt+

p(u) +εq(u) uxut,

where A=A(u,u_x), B=B(u,u_x), p=p(u), and q= q(u)satisfy

624 F.-Y. Ji and S.-L. Zhang·Approximate Functional Separable Solutions to Generalized Diffusion Equations A(p⁰−p²) +c1u^c_x²=0, c16=0,

h

2Ap²+pAu−2Ap⁰

Au−ApA_uuxi ux

−2A² p⁰−p²

−A(pA_u−A_uu)−A²_u=0, Ah

qA_u+4Apq−2Aq⁰

A_ux−qAA_uuxi u²_u

x

+h

(4pq−2q⁰)−A²(pB_uxux+qA_u) +A pA_uxB_ux

−pBAu_xu_x

+pBA²_uxi

u_x−BA_ux(pA+A_uA_ux) +A AB_uux−A_uB_ux+BA_uux

=0, A²h

(2pq−q⁰)A_u−A q⁰⁰−2pq⁰−2p⁰qi u³_x +

B

pA_u−2p⁰A+3p²A

+pAB_u A_ux

−A

A(B_uxp⁰+B_uuxp) +pBA_uux u²_x +h

A²B_uu+A(pA−A_u)B_u+

A² 2p²−p⁰ +A(A_uu−2pA_u)−A²_u

Bi u_x=0 ;

whereα,β,γ, c, and c_i(i∈Z)are arbitrary constants in their sets of definition when not specified.

3. Construction of AFSSs for the Resulting Equations

Using the AFVSA, we present some AFSSs of the resulting equations by the following some examples.

Example 1. Equation (12) enjoys the following AF- SSs,

u=

1+ε ω(x) +θ(t) a₂+a₃

·exp

ψ(x) +φ(t)−a₁−a₄ a₂+a₃

, a26=0, whereψ(x),φ(t),ω(x), andθ(t)satisfy

(1) α=0. ψ(x) =−

√ λ c₁ x+ a₂

2c₁

·ln

"

c²₁

4λ b₁exp 2√ λx c1a2

!

−b₂

!2# ,

φ(t) =λt a₂+b₃, a²₂c²₁

a²₂ω⁰⁰(x) +2a₂ψ⁰(x)ω⁰(x)−a₃(ψ(x))²

+a⁵₂

F1

ψ⁰(x) a₂

+c3

+λ1=0, θ(t) = c2a⁵₂−λ1

t

a⁴₂ +b4, |b₁|+|b₂| 6=0

; (2) α=0, λ<0.

ψ(x) =a₂ 2 ln

"

−c²₁ λ

b₁sin

√

−λx c₁a₂

−b₂cos √

−λx c₁a₂

2# ,

φ(t) =λt a₂+b3, a²₂c²₁

a²₂ω⁰⁰(x) +2a₂ψ⁰(x)ω⁰(x)−a₃(ψ(x))² +a⁵₂

F1

ψ⁰(x) a₂

+c3

+λ1=0, θ(t) = c₂a⁵₂−λ1

t

a⁴₂ +b₄, |b₁|+|b₂| 6=0

; (3) α=1, λ6=0.

Z ψ(x) c₁a₂e^ξ/a2 r

a2

h

2λe^2a1/a2ln(ξ)+h₁a2c²₁i

dξ=±x+h₂,

φ(t) =1 2a₂ln

− a³₂ 2λ(t+h₃)

, a²₂

c²₁ ψ(x)

a²₂ω⁰⁰(x) +2a₂ψ⁰(x)ω⁰(x)

−a3(ψ⁰(x))² +a⁴₂

F1

ψ⁰(x) a₂

+c3

·e^{2(ψ(x)−a1)/a2}+a³₂λ(ψ⁰(x))⁻¹ω⁰(x) +a₂λ

h

2a₂ ω(x)−a₄

−2a₃ ψ(x)−a₁

−a2a3

i

+λ1=0, θ(t) =1

2a₃

ln

− a³₂ λ(t+h₃)

+1

+ a₂c₂t² 2(t+h₃) +

2a²₂c₂h₃λ−a²₂a₃λln(2) +λ₁

t+2h₄a²₂λ 2a²₂λ(t+h₃) . (4) α6=0,1, λ 6=0.

Z ψ(x)(

c1(α−1)e^ξ/a2dξ

a5c²₁(α−1)²

− λ ξ^α

2(α−1)ξ a₂

α/2

exp2a₁α−(α−1)ξ a₂

F.-Y. Ji and S.-L. Zhang·Approximate Functional Separable Solutions to Generalized Diffusion Equations 625

·H(ξ) −¹₂)

=±x+a₆,

φ(t) = 1 2αln

"

− a^α+2₂ 2λ α(t+a₇)

# ,

a²₂

c²₁ ψ(x)α

a²₂ω⁰⁰(x) +2a₂ψ⁰(x)ω⁰(x)

−a₃ ψ⁰(x)2 +a^α+3₂

F₁

ψ⁰(x) a2

+c₃

·e^{2α(ψ(x)−a1)/a2}+a³₂λ α ψ⁰(x)−1

ω⁰(x) +a₂λ α

h

2a2(ω(x)−a4)−2a3(ψ(x)−a1)

−a₂a₃i

+λ1=0, a^α+4₂ θ⁰(t)−a₂c₂

e^{2α φ(t)/a2}

−2a₂α λ a₂θ(t)−a₃φ(t)

+λ1=0, where

H(ξ) =WhittakerM

−α 2,1−α

2 ,2(α−1)ξ a2

, whereα,λ,λ1,a_i, andb_j (i=1, . . . ,7,j=1, . . . ,4) are arbitrary constants in their sets of definition when not specified. where and hereafter the prime denotes the corresponding order derivative with respect to x ort.

Example 2. Some AFSSs to (14) is given by (4), with f(u) =d₂u+d₁, d₂6=0,

g(u) =1

6c₂d₂u³+1

2c₃d₂u²+d₃u+d₄, whereψ(x),φ(t),ω(x), andθ(t)are expressed by (1) α6=−1,−2, c₁c₂α ρ6=0.

ψ(x) =

d₂^α/(α+1)ρ^1/(α+1)

h(α+1)

·(x−d₅)i(α+2)/(α+1)

c^1/(α+1)₁ (α+2) −¹

2

+ ρ₂

2c₂d₂α ρ(α+2)+ c4d₂² c₂α ρ−d₂c₃

c₂ +d₁, φ(t) =ρt+d₆,

2c₁d₂^1−α(ψ⁰(x))^αh

d2(α+2)

d2ω⁰⁰(x)−c3(ψ⁰(x))²

−c₂α(d₁−ψ(x))(ψ⁰(x))²i

+2d₂³(α+2)

"

d₂F₂ ψ⁰(x)

d₂

+ρ α(ψ⁰(x))⁻¹ω⁰⁰(x)

#

−c2d2α ρ(α+2)ψ²(x) +2d₂(α+2)h

α ρ(c₂d1

−c₃d₂) +c₄d₂²i

ψ(x)−d₂(α+2)h

d₁(c₂d₁α ρ +2c₄d²₂)−2d₂ρ α(d₁c₃−d₃)i

+ρ1=0, θ(t) =−c₂α ρ³t³

6d²₂ −ρ

2c₂d₂d₆ρ α(α+2) +ρ2 t² 4d³₂(α+2)

−

c₂d₂d²₆ρ α(α+2) +d₆ρ2+ρ1 t 2d³₂(α+2) +d₇. (2) α=−1, c₁c₂ρ6=0.

ψ(x) =k1e^ρx/(c1d2)−2d₂²(d₂c4+c3ρ) +ρ2

c₂d₂ρ +d1, (k₁6=0)

φ(t) =ρt+k₂, 2d₂²

"

ψ⁰(x)⁻¹

c₁d₂ω⁰⁰(x)−ρ ω⁰(x) +d₂F₂

· ψ⁰(x)

d₂ #

+c2ρ ψ(x)2

+2c₁d2 c2d1−d2c3

−c₂ψ(x)

ψ⁰(x) +2

c₃d₂ρ−c₂d₁ρ+c₄d₂² ψ(x)

−2d2(c₄d1d2−d3ρ+c3d1ρ)+c₁d²₁+d₂⁻¹ρ1=0, θ(t) =c₂ρ³t³

6d²₂ +

ρ 2c₂d₂k₂ρ−ρ2 t²−2h

ρ1

+k₂ ρ₂−c₂d₂k₂ρi t

4d₂³

−1

+k₃. (3) α6=−1,−2, c₂=0, c₁ρ6=0.

ψ(x) =

d

α (α+1)

2 ρ

1 (α+1)h

(α+1)(x+r₁)i^(α+2)_(α+1)

·

c^1/(α+1)₁ (α+2) −1

+r₂, φ(t) =ρt+r₃,

2c₁d₂^2−α(α+2)(ψ⁰(x))^α d₂ω⁰⁰(x)−c₃(ψ⁰(x))² +2d³₂(α+2)

d₂F₂

ψ⁰(x) d₂

+ρ α(ψ⁰(x))⁻¹

·ω⁰⁰(x)

+2d₂²(α+2)(c₄d2−α ρc3)ψ(x)

−2d²₂(α+2)

c₄d₁d₂−ρ α(d₁c₃−d₃)

+ρ₁=0, θ(t) =ρ(d₂c₄−c₃α ρ)t²

2d₂

626 F.-Y. Ji and S.-L. Zhang·Approximate Functional Separable Solutions to Generalized Diffusion Equations

−

2r₃d₂²(α+2)(c₃ρ α−c₄d₂) +ρ₁ t 2d₂³ α+2 +r₄. (4) α=−1, c₂=0, c₁ρ6=0.

ψ(x) =h₂e^ρx/(c1d2)+h₁, (h₂6=0) φ(t) =ρt+h₃,

d₂ ψ⁰(x)−1

c₁d₂ω⁰⁰(x)−ρ ω⁰(x)

−d₂

c₁c₃ψ⁰(x)−d₂F₂ ψ⁰(x)

d2

+ (c₄d2+c₃ρ)(ψ(x)−d1)+ρd₃+2⁻¹d⁻²₂ ρ1=0, θ(t) =ρ(c₃ρ+d₂c₄)t²

2d₂ +

2d₂²h₃(c₄d₂+c₃ρ)−ρ1

t 2d³₂ +h₄,

whereα, ρ, ρ1, d_i, k_j, r_n, and h_n (i=1, . . . ,7,j= 1, . . . ,3,n=1, . . . ,4) are arbitrary constants in their sets of definition.

Example 3. Some AFSSs to (16) is determined by (4), with

f(u) =s₂ln(u) +s₁, s₂6=0, g(u) =c₂s₂u^2−α

(α−2)² +s₃ln(u) +s₄,

whereψ(x),φ(t),ω(x), andθ(t)are determined by (1) α6=0,1,2, c₁c₂λ6=0.

Z ψ(x)(

c₁c₂s₂α(α−2)e^(ξ−s1)/s2

c₁c₂α(2−α)

2c₂s²₂α λexp

(s₁−ξ)(α−2) s₂

+λ₂ −¹₂)

dξ =±x+s₅, φ(t) =s2

α ln

− s²₂ α λ(t+s₆)

, 2s₂

s²₂

c₁ω⁰⁰(x) +s₂F₁ ψ⁰(x)

s₂

+c₁

2s₂ω⁰(x)

−s₃ψ⁰(x)

ψ⁰(x) +c₄s³₂

e^{α(ψ(x)−s1)/s2}+2α λh s₂

·(ω(x)−s₄)−s₃(ψ(x)−s₁)i

−λ1(α−2)⁻²=0, θ(t) =s₃

α ln

− s²₂ α λ(t+s₆)

+ λ2s

4−3α α

2

α(t+s6)^α−2_α 4(−λ)α²(α−1)(α−2)²

+ c₃s₂t² 2(t+s₆)+s₃

α

+

2c₃s²₂λ α(α−2)²s₆−λ₁

t+2s₂s₇λ α(α−2)² 2s₂λ α(α−2)²(t+s₆) . (2) α=0, c26=0.

(i) c₁λ>0. ψ(x) =−

s λ c₁x+1

2s2

·ln



 c₁

4λ k₁exp 2 s2

s λ c1

x

! +k₂

!2

,

φ(t) =λt s2

+k₃, s²₂

c₁ω⁰⁰(x) +s₂F₁ ψ⁰(x)

s₂

+c₁ 2s₂ω⁰(x)

−s₃ψ⁰(x)

ψ⁰(x) +c₄s³₂−8⁻¹s⁻¹₂ λ1=0, θ(t) = 8c₃s⁴₂+λ₁

t 8s³₂ +k₄; (ii) c₁λ <0.

ψ(x) =1 2s₂ln

(

−c₁ λ

"

h₁sin q

−c⁻¹₁ λx s₂

!

−h₂cos q

−c⁻¹₁ λx s2

!#2) ,

φ(t) =λt s₂+h₃, s²₂

c1ω⁰⁰(x) +s2F1

ψ⁰(x) s₂

+c1

2s₂ω⁰(x)

−s₃ψ⁰(x)

ψ⁰(x) +c₄s³₂−8⁻¹s⁻¹₂ λ1=0, θ(t) = 8c₃s⁴₂+λ1

t 8s³₂ +h₄. (3) α6=0,2, c₂=0, c₁λ6=0.

Z ψ(x)(

c₁(α−2)

c₁(α−2)h

b^1/s₂ ²e^−2ξ/s2

−2λe^{α(s1−ξ)/s2}i−¹₂)

dξ =±x+b₃, φ(t) =s₂

α ln

− s²₂ α λ(t+b₁)

, 2s₂

s²₂

c₁ω⁰⁰(x) +s₂F₁ ψ⁰(x)

s2

+c₁ 2s₂ω⁰(x)

F.-Y. Ji and S.-L. Zhang·Approximate Functional Separable Solutions to Generalized Diffusion Equations 627

−s₃ψ⁰(x)

ψ⁰(x) +c4s³₂

e^{α(ψ(x)−s1)/s2}+2α λh s2

·(ω(x)−s4)−s3(ψ(x)−s1)i

−λ1(α−2)⁻²=0, θ(t) =s3

α ln

− s²₂ α λ(t+b₁)

+ s2c3t² 2(t+b₁)+s3

α

+

2b₁c₃s²₂α λ(α−2)²−λ₁

t+2b₄s₂λ α(α−2)² 2s₂λ α(α−2)²(t+b1) . (4) α=1, c₁c₂λ 6=0.

Z ψ(x) c1c2s2e^(ξ−s1)/s2 q

c1c2

2c2s²₂λe^(ξ−s1)/s2+λ₂

dξ=±x+a₁,

φ(t) =s₂ln

− s²₂ λ(t+a₂)

, 2s₂

s²₂

c1ω⁰⁰(x) +s2F1

ψ⁰(x) s₂

+c1(2s₂ω⁰(x)

−s₃ψ⁰(x))ψ⁰(x) +c₄s³₂

e^{(ψ(x)−s1)/s2} +2λh

s2(ω(x)−s4)−s3(ψ(x)−s1)i

−λ1=0, θ(t) =1

2

2s₃− s₂λ₂ λ²(t+a₂)

ln

− s²₂ λ(t+a₂)

+ a3

t+a₂+1

2s₂c₃(t+a₂)− λ1

2s₂λ +s₃. (5) α=1, c₂=0, c₁λ 6=0.

ψ(x) =s2ln

λe^s1/s2x²−2c₁s2v2x+2c₁s2v3

2c₁s²₂

,

φ(t) =s2ln

− s²₂ λ(t+v₁)

, 2s₂

s²₂

c₁ω⁰⁰(x) +s₂F₁ ψ⁰(x)

s₂

+c₁

2s₂ω⁰(x)

−s₃ψ⁰(x)

ψ⁰(x) +c₄s³₂

e^{(ψ(x)−s1)/s2}

+2λh

s2(ω(x)−s4)−s₃(ψ(x)−s1)i

−λ₁=0, θ(t) =s₃ln

− s²₂ λ(t+v₁)

+s₃ +s²₂c₃λt²+ 2s²₂c₃v₁λ−λ1

t+2s₂v₄λ 2s2λ(t+v1) , where α, λ, λ₁, λ₂, s_i, k_j, b_j, v_j, h_j, and a_n (i= 1, . . . ,7,j =1, . . . ,4,n =1, . . . ,3) are arbitrary con- stants in their sets of definition.

4. Conclusion and Discussion

In summary, by utilizing the AFVSA, we have clas- sified the generalized diffusion equations with per- turbation which admit AFSSs. AFSSs of some re- sulting equations are constructed. In general, these AFSSs cannot be obtained by other approximate methods. Other types of perturbed nonlinear evolu- tion equations may be studied by the AFVSA, and some interesting results will be presented sooner or later.

There are still two interesting topics to be investi- gated later: (i) How to apply the AFVSA to other types of nonlinear evolution equations, such as the higher di- mensional equations, and the system of equations with perturbation? (ii) How to extend the AFVSA so as to obtain more accurate approximate solutions?

Acknowledgement

The work is partly supported by the National NSF of China (No. 11371293), the Youth Science and Tech- nology Fund of Xi’an University of Architecture and Technology (No. QN1328), and the Talent of Science and Technology Fund of Xi’an University of Architec- ture and Technology (No. DB12077).

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