Homotopy Perturbation Method for Solving Partial Differential Equations with Variable Coefficients
Lin Jin
Modern Educational Technology Center Zhejiang Gongshang University Hangzhou, 310018, P. R. China
[email protected] Abstract
In this paper, we present the exact solutions of the partial differential equations in different dimensions with variable coefficients by using the homotopy perturbation method. The feature of this method is its flex- ibility and ability to solve parabolic-like equations and hyperbolic-like equations without the calculation of complicated Adomian polynomials or unrealistic nonlinear assumptions. The numerical results show that this method is a promising and powerful tool for solving the partial differential equations with variable coefficients.
Mathematics Subject Classification: 35K15, 35C05, 65D99, 65M99 Keywords: homotopy perturbation method, parabolic-like equation, hyperbolic-like equation, exact solution
1 Introduction
Many physical problems can be described by partial differential equations with variable coefficients in mathematical physics, and other areas of science and engineering. The investigation of exact or approximate solutions to these partial differential equations will help us to understand these physical phe- nomena better. There are some valuable efforts that focus on solving the partial differential equations arising in engineering and scientific applications [2, 6, 7, 8, 12, 13, 24, 28, 36]. Reviewing these improvements, the Adomian decomposition method [2, 12, 24, 36], the tanh method [6, 28], the extended tanh function method [7, 8] and other methods [13] are proposed to solve the partial differential equations. Among these solution methods, Adomian decomposition method is the most transparent method for solutions of the
partial differential equations, however, this method is involved in the calcu- lation of complicated Adomian polynomials which narrows down its applica- tions. To overcome this disadvantage of the Adomian decomposition method, we consider the homotopy perturbation method to solve the partial differential equations with variable coefficients.
The homotopy perturbation method proposed by J.H. He [14, 15] has been the subject of extensive studies, and applied to various linear and nonlinear problems [1, 3, 4, 5, 9, 10, 11, 16, 17, 18, 19, 20, 21, 26, 27, 29, 30, 31, 32, 33, 34, 35]. Unlike analytical perturbation methods, the significant feature of this method which doesn’t depend on a small parameter is that it provides the exact or approximate solutions to a wide range of nonlinear problems without unrealistic assumptions, linearization, discretization and the computation of the Adomian polynomials.
The goal of this work is to extend the homotopy perturbation method to solve the partial differential equations with variable coefficients. Based on the homotopy perturbation method, we present exact solutions of some parabolic-like equations and hyperbolic-like equations. It’s easy to see that this method gives the exact solutions without the calculation of Adomian polyno- mials which is inevitable to solve the partial differential equations with variable coefficients in [36], therefore, this method provides efficient approach to solve the partial differential equations.
The rest of this paper is organized as follows. In section 2, we give the analysis of the homotopy perturbation method. The parabolic-like equations and hyperbolic-like equations are introduced in sections 3 and 4. In section 5, we present numerical results to demonstrate the efficiency of the homo- topy perturbation method for some partial differential equations with variable coefficients. Finally, we give the conclusion in section 6.
2 Analysis of the homotopy perturbation method
To clarify the basic ideas of the homotopy perturbation method [14, 15], we consider the following nonlinear differential equation
A(u)− f(r) = 0, r ∈ Ω, (1)
with boundary conditions
B(u,du
dn) = 0, r ∈ Γ, (2)
where A is a general differential operator, B is a boundary operator, u is a known analytical function, and Γ is the boundary of the domain Ω.
The operator A can be divided into two parts L and N , where L is linear, while N is nonlinear. Therefore (1) can be rewritten as follows
L(u) + N (u)− f(r) = 0. (3)
By the homotopy technique proposed by Liao [23], we can construct a homotopy v(r, p) : Ω× [0, 1] → R which satisfies
H(v, p) = (1− p)[L(v) − L(u0)] + p[A(v)− f(r)] = 0, (4) or
H(v, p) = L(v)− L(u0) + pL(u0) + p[N (v)− f(r)] = 0, (5) where r ∈ Γ and p ∈ [0, 1] is an embedding parameter, u0 is an initial approxi- mation of (1), which satisfies the boundary conditions. By (4), it easily follows that
H(v, 0) = L(v)− L(u0) = 0, (6)
H(v, 1) = A(v)− f(r) = 0, (7)
and the changing process of p from zero to unity is just that of H(v, p) from L(v)−L(u0) to A(v)−f(r). In topology, this is called deformation, L(v)−L(u0) and A(v)− f(r) are called homotopic.
The embedding parameter p is introduced much more naturally, unaffected by artificial factors. Furthermore, it can be considered as a small parameter for 0 < p ≤ 1. By applying the perturbation technique used in [25, 22], we assume that the solution of (4) can be expressed as
v = v0 + pv1+ p2v2+· · · . (8) Therefore, the approximate solution of (1) can be readily obtained as fol- lows:
u = lim
p→1v = v0+ v1+ v2+· · · . (9) The convergence of this method has been proved by J.H. He in [14, 15].
3 Parabolic-like equation
Consider the parabolic-like equation in three dimensions of the form
ut+ f1(x, y, z)uxx+ f2(x, y, z)uyy+ f3(x, y, z)uzz = 0, (10)
with the initial condition
u(x, y, z, 0) = f4(x, y, z). (11) According to the homotopy perturbation method, we construct the homotopy Ω× [0, 1] → R which satisfies
ut− y0t(x, y, z) + py0t(x, y, z) + p(f1(x, y, z)uxx (12) +f2(x, y, z)uyy+ f3(x, y, z)uzz) = 0,
with the initial approximation y0(x, y, z) = u(x, y, z, 0) = f4(x, y, z).
Assume that the form of the solution to (10) is as follows
u = u0+ pu1+ p2u2+· · · . (13) Substituting the solution (13) into Eq. (12), and equating the terms of the same power of p, we have
p0 : u0t− y0t(x, y, z) = 0, y0(x, y, z) = f4(x, y, z),
p1 : u1t+ f1(x, y, z)u0xx+ f2(x, y, z)u0yy + f3(x, y, z)u0zz +y0t(x, y, z) = 0, u1(x, y, z, 0) = 0,
p2 : u2t+ f1(x, y, z)u1xx+ f2(x, y, z)u1yy + f3(x, y, z)u1zz = 0, u2(x, y, z, 0) = 0,
· · ·
pn : unt+ f1(x, y, z)un−1xx+ f2(x, y, z)un−1yy+ f3(x, y, z)un−1zz = 0, un(x, y, z, 0) = 0.
Letting u0 = y0(x, y, z), and solving the above equations results in the approx- imate solution
u = lim
n→∞(u0+ u1+ u2+· · · + un). (14) which converges to the exact solution if such a solution exists.
4 Hyperbolic-like equation
Consider the three dimensional hyperbolic-like equation of the form
utt+ g1(x, y, z)uxx+ g2(x, y, z)uyy+ g3(x, y, z)uzz = 0, (15)
subject to the initial condition
u(x, y, z, 0) = g4(x, y, z), ut(x, y, z, 0) = g5(x, y, z). (16) Similarly, we can construct the homotopy Ω× [0, 1] → R which satisfies
utt− y0tt(x, y, z, t) + py0tt(x, y, z, t) + p(g1(x, y, z)uxx (17) +g2(x, y, z)uyy+ g3(x, y, z)uzz) = 0,
with the initial approximation y0(x, y, z, t) = g4(x, y, z) + tg5(x, y, z).
Suppose that the solution of (15) can be expressed as
u = u0+ pu1+ p2u2+· · · . (18) Substituting the solution (18) into Eq. (17), and equating the terms of the identical power of p gives the following results
p0 : u0tt− y0tt(x, y, z) = 0, y0(x, y, z, t) = g4(x, y, z) + tg5(x, y, z), p1 : u1tt+ g1(x, y, z)u0xx+ g2(x, y, z)u0yy+ g3(x, y, z)u0zz
+y0tt(x, y, z, t) = 0, u1(x, y, z, 0) = 0,
p2 : u2tt+ g1(x, y, z)u1xx+ g2(x, y, z)u1yy+ g3(x, y, z)u1zz= 0, u2(x, y, z, 0) = 0,
· · ·
pn : untt+ g1(x, y, z)un−1xx+ g2(x, y, z)un−1yy+ g3(x, y, z)un−1zz = 0, un(x, y, z, 0) = 0.
Solving the above equations results in the approximate solution u = lim
n→∞(u0+ u1+ u2+· · · + un). (19) Note that the approximate solution converges to the exact solution if such an exact solution exists.
5 Numerical examples
In this section, we will present the exact solutions of the parabolic-like equa- tions and hyperbolic-like equations investigated by A.M. Wazwaz [36] to assess the efficiency of the homotopy perturbation method.
Example 1. Consider the following one-dimensional parabolic-like equation with variable coefficients in the form [36]
ut(x, t)− x2
2 uxx(x, t) = 0, (20)
subject to the initial condition u(x, 0) = x2.
According to the homotopy perturbation method, we can construct the homotopy Ω× [0, 1] → R which satisfies
ut(x, t)− y0t+ py0t− p(x2
2uxx(x, t)) = 0 (21) with the initial approximation y0 = u(x, 0) = x2.
Suppose that the solution of (20) can be represented as
u = u0+ pu1+ p2u2+· · · . (22) Substituting (22) into (21), and equating the terms of the same power of p, it follows that
p0 : u0t(x, t)− y0t = 0, y0 = x2, p1 : u1t(x, t)−x2
2 u0xx(x, t) + y0t= 0, u1(x, 0) = 0, p2 : u2t(x, t)−x2
2 u1xx(x, t) = 0, u2(x, 0) = 0,
· · ·
pn: unt(x, t)− x2
2 un−1xx(x, t) = 0, un(x, 0) = 0.
By choosing u0(x, t) = y0, and solving the above equations, we obtain the following approximations
u1(x, t) = tx2, u2(x, t) = t2
2!x2, · · · , un(x, t) = tn
n!x2. (23) Then the exact solution of (20) is given by
u(x, t) = lim
n→∞(1 + t + t2
2! +· · · + tn
n!)x2 = etx2, (24) which is exactly the same solution obtained in [36] by using the Adomian de- composition method, however, the homotopy perturbation method provides the exact solution without the computation of the complicated Adomian poly- nomials, therefore, this method can be seen as a more readily implemented and promising method for solving the partial differential equations with variable coefficients.
Example 2. Consider the two-dimensional parabolic-like equation with vari- able coefficients [36]
ut(x, y, t)− y2
2uxx(x, y, t)−x2
2uyy(x, y, t) = 0, (25) subject to the initial condition u(x, y, 0) = y2.
Similarly, by using the homotopy perturbation method, we have p0 : u0t(x, y, t)− y0t = 0, y0 = y2,
p1 : u1t(x, y, t)− y2
2u0xx(x, y, t)− x2
2 u0yy(x, y, t) + y0t= 0, u1(x, y, 0) = 0,
p2 : u2t(x, y, t)− y2
2u1xx(x, y, t)− x2
2 u1yy(x, y, t) = 0, u2(x, y, 0) = 0,
· · ·
pn: unt(x, y, t)−y2
2un−1xx(x, y, t)− x2
2 un−1yy(x, y, t) = 0, un(x, y, 0) = 0.
With the initial approximation u0(x, y, t) = y0, by solving the above equations, we have
u1(x, y, t) = tx2, u2(x, y, t) = t2
2!y2, u3(x, y, t) = t3
3!x2, · · · , (26) u2n(x, y, t) = t2n
(2n)!y2, u2n+1(x, y, t) = t2n+1 (2n + 1)!x2. Then the exact solution of (25) is given by
u(x, y, t) = lim
n→∞(t +t3
3! +· · · + t2n+1
(2n + 1)!)x2 + (1 +t2
2! +· · · + t2n (2n)!)y2
= x2sinh t + y2cosh t.
Example 3. Consider the three-dimensional parabolic-like equation with vari- able coefficients [36]
ut− (xyz)4− 1
36(x2uxx+ y2uyy+ z2uzz) = 0, (27) subject to the initial condition u(x, y, z, 0) = 0.
Similar to the manipulation of the previous examples, we obtain p0 : u0t(x, y, z, t)− y0t= 0, y0 = 0,
p1 : u1t(x, y, z, t)− (xyz)4− 1
36(x2u0xx(x, y, z, t) + y2u0yy(x, y, z, t) +z2u0zz(x, y, z, t)) + y0t = 0, u1(x, y, z, 0) = 0,
p2 : u2t(x, y, z, t)− 1
36(x2u1xx(x, y, z, t) + y2u1yy(x, y, z, t) +z2u1zz(x, y, z, t)) = 0, u2(x, y, z, 0) = 0,
· · ·
pn: unt(x, y, z, t)− 1
36(x2un−1xx(x, y, z, t) + y2un−1yy(x, y, z, t) +z2un−1zz(x, y, z, t)) = 0, un(x, y, z, 0) = 0.
Letting u0(x, y, z, t) = y0, and solving the above equations, it leads to that the exact solution of (27) can be expressed as
u(x, y, z, t) = lim
n→∞(t +t2
2! +· · · +tn
n!)x4y4z4 = x4y4z4(et− 1).
Example 4. Consider the one-dimensional hyperbolic-like equation with vari- able coefficients as [36]
utt(x, t)−x2
2uxx(x, t) = 0, (28)
with the initial condition u(x, 0) = x, ut(x, 0) = x2. Similarly, we have
p0 : u0tt(x, t)− y0tt= 0, y0 = x + tx2, p1 : u1tt(x, t)−x2
2 u0xx(x, t) + y0tt= 0, u1(x, 0) = 0, p2 : u2tt(x, t)−x2
2 u1xx(x, t) = 0, u2(x, 0) = 0,
· · ·
pn : untt(x, t)− x2
2 un−1xx(x, t) = 0, un(x, 0) = 0.
If we choose u0(x, t) = y0, and solve the above equations, it’s easy to obtain that
u1(x, t) = t3
3!x2, u2(x, t) = t5
5!x2, · · · , un(x, t) = t2n+1
(2n + 1)!x2. (29)
Then the exact solution of (28) can be formulated as u(x, t) = lim
n→∞x + (t + t3
3!+· · · + t2n+1
(2n + 1)!)x2 = x + x2sinh t.
Example 5. Consider the two-dimensional hyperbolic-like equation with vari- able coefficients [36]
utt(x, y, t)− x2
12uxx(x, y, t)− y2
12uyy(x, y, t) = 0, (30) with the initial condition u(x, y, 0) = x4, ut(x, y, 0) = y4.
By applying the homotopy perturbation method, we have p0 : u0tt(x, y, t)− y0tt= 0, y0 = x4+ ty4,
p1 : u1tt(x, y, t)− x2
12u0xx(x, y, t)− y2
12u0yy(x, y, t) + y0tt= 0, u1(x, y, 0) = 0,
p2 : u2tt(x, y, t)− x2
12u1xx(x, y, t)− y2
12u1yy(x, y, t) = 0, u2(x, y, 0) = 0,
· · ·
pn : untt(x, y, t)− x2
12un−1xx(x, y, t)− y2
12un−1yy(x, y, t) = 0, un(x, y, 0) = 0.
Beginning with the initial approximation y0 = x4+ ty4, and solving the above equations, it follows that
u1(x, y, t) = t2
2!x4+t3
3!y4, u2(x, y, t) = t4
4!x4+t5
5!y4,· · · , (31) un(x, y, t) = t2n
(2n)!x4+ t2n+1 (2n + 1)!y4. Therefore, the approximate solution of (30) reads
u(x, y, t) = lim
n→∞x4(1 + t2
2! +· · · + t2n
(2n)!) + y4(t + t3
3! +· · · + t2n+1 (2n + 1)!)
= x4cosh t + y4sinh t which gives the exact solution.
Example 6. Consider the three-dimensional hyperbolic-like equation with variable coefficients [36]
utt− (x2+ y2+ z2)− 1
2(x2uxx+ y2uyy+ z2uzz) = 0, (32)
with the initial condition u(x, y, z, 0) = 0, ut(x, y, z, 0) = x2+ y2− z2. Similarly, by using the homotopy perturbation method, we have p0 : u0tt(x, y, z, t)− y0tt= 0, y0 = t(x2+ y2− z2),
p1 : u1tt(x, y, z, t)− (x2+ y2+ z2)−1
2(x2u0xx(x, y, z, t) + y2u0yy(x, y, z, t) +z2u0zz(x, y, z, t)) + y0tt= 0, u1(x, y, z, 0) = 0,
p2 : u2tt(x, y, z, t)− 1
2(x2u1xx(x, y, z, t) + y2u1yy(x, y, z, t) +z2u1zz(x, y, z, t)) = 0, u1(x, y, z, 0) = 0,
· · ·
pn : untt(x, y, z, t)−1
2(x2un−1xx(x, y, z, t) + y2un−1yy(x, y, z, t) +z2un−1zz(x, y, z, t)) = 0, un(x, y, z, 0) = 0.
Starting with the initial approximation y0 = t(x2 + y2− z2), and solving the above equations, we can obtain the following approximations
u1(x, y, z, t) = t2
2!(x2+ y2+ z2) + t3
3!(x2+ y2− z2), (33) u2(x, y, z, t) = t4
4!(x2+ y2+ z2) + t5
5!(x2+ y2− z2),· · · , un(x, y, t) = t2n
(2n)!(x2+ y2+ z2) + t2n+1
(2n + 1)!(x2+ y2− z2).
Therefore, we have
u(x, y, z, t) = lim
n→∞(t +t2
2! +· · · + t2n
2n!)(x2+ y2) +(−t + t2
2!+· · · − t2n+1 (2n + 1)!)z2
= (x2 + y2)et+ z2e−t− (x2 + y2+ z2), which is the exact solution of (32).
6 Conclusions
The homotopy perturbation method has been applied for solving the partial differential equations with variable coefficients. Compared with the Adomian decomposition method [36], this method provides the exact solutions of the
parabolic-like equations and hyperbolic-like equations without the tedious cal- culation of the Adomian polynomials. Therefore, this method can be seen as a promising and powerful tool for solving the partial differential equations in different dimensions with variable coefficients.
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Received: February 21, 2008
