Haar wavelet method for solving Fisher's equation - Semantic Scholar
Applied Mathematics and Computation 211 (2009) 284–292
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Haar wavelet method for solving Fisher’s equation G. Hariharan a,*, K. Kannan a, K.R. Sharma b a b
Department of Mathematics, SASTRA University, Thanjavur 613 402, Tamilnadu, India Department of Chemical Engineering, SASTRA University, Thanjavur 613 402, Tamilnadu, India
a r t i c l e
i n f o
Keywords: Reaction–diffusion equations Fisher’s equation Numerical simulation Haar wavelets Adomain decomposition method
a b s t r a c t In this paper, we develop an accurate and efficient Haar wavelet solution of Fisher’s equation, a prototypical reaction–diffusion equation. The solutions of Fisher’s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt Haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The problems of the propagation of nonlinear waves have fascinated scientists for over 200 years. The modern theory of nonlinear waves, like many areas of mathematics, had its beginnings in attempts to solve specific problems, the hardest among them being the propagation of waves in water. There was significant activity on this problem in the 19th century and the beginning of the 20th century, including the classic work of Stokes, Lord Rayleigh, Korteweg and de Vries, Boussinesque, Benard and Fisher to name some of the better remembered examples [15,30]. We would like to solve the well-known and classical Kolmogorov–Petrovski–Piscounov reaction–diffusion equation known as KPP equation. The solitons appear as a result of a balance between weak nonlinearity and dispersion. Soliton is defined as a nonlinear wave characterized by the following properties: (i) A localized wave propagates without change of its properties (shape, velocity, etc.). (ii) Localized wave are stable against mutual collisions and retain their identities. On the other hand, the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with compact support that are called campactons [22,25,26]. Unlike soliton that narrows as the amplitude increases, the compacton’s width is independent of the amplitude. However, when diffusion takes part instead of dispersion, energy release by nonlinearity balances energy consumption by diffusion, which results in traveling waves or fronts [14]. Traveling wave fronts are an important and much studied solution form for reaction–diffusion equations, with important applications to chemistry, biology and medicine [23]. Such solutions were first studied in the 1930s by Fisher for the scalar equation
oU o2 U ¼ 2 þ Uð1 UÞ: ot ox * Corresponding author. E-mail addresses: [email protected] (G. Hariharan), [email protected] (K. Kannan), [email protected] (K.R. Sharma). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.12.089
ð1Þ
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A traveling wave solution U(x, t) = U(n = x ct), propagating with a speed c, is restricted to be positive and bounded [5]. Therefore the boundary conditions for the traveling wave solution are usually
Uðn ! 1Þ ! 1;
Uðn ! 1Þ ! 0:
ð2Þ
In addition 0 6 U(x, t) 6 1 and c > 0 is the wave speed. Both Fisher and KPP found that Eq. (1) has an infinite number of traveling wave solutions of characteristic speeds c P 2. Fisher also carried out a very accurate and detailed numerical computation of the shock like profile of the traveling wave of minimum speed. Useful summaries of recent advances in this area have been provided by Wazwaz [25–29], Feng and Li [9], Boumenir [3], Whitham [30] and Zhou [31]. The well-known Fisher’s equation combines diffusion with logistic nonlinearity. Fisher proposed Eq. (1) as a model for the propagation of a mutantgene, with U denoting the density of an advantageous. This equation is encountered in chemical kinetics [19] and population dynamics, which includes problems such as nonlinear evolution of a population in a one-dimensional habitat, neutron population in a nuclear reaction. Moreover, the same equation occurs in logistic population growth models [5], flame propagation, neurophysiology, autocatalytic chemical reactions, and branching Brownian motion processes. The mathematical properties of Fisher’s Equation (FE) have been studied extensively and there have been numerous discussions in the literature. Excellent summaries have been provided in [5]. One of the first numerical solutions was presented in literature with a pseudo-spectral approach. Implicit and explicit finite differences algorithms have been reported by different authors such as Parekh and Puri and Twizell et al. A Galerkin finite element method was used by Tang and Weber whereas Carey and Shen [6] employed a least-squares finite element method. A collocation approach based on Whittaker’s sinc interpolation function [4] was also considered in [2]. The work in [10] considered a nonlocal form of FE. The exact solitary wave solution of Eq. (1) was not obtained until 1979. Noting the Painleve property of Eq. (1) and using the Laurent series [24], Ablowitz and Zepetella [1] looked for a solution of the form
UðzÞ ¼
6 a1 þ þ a0 a1 z þ . . . ; z2 z
ð3Þ
where z = x ct. The solution can then be found by solving nontrivial recursion relations, where the exact solution
Uðx; tÞ ¼
2 1 1 5 b 1 þ tanh pffiffiffi x pffiffiffi t þ 2 2 2 2 6 6
ð4Þ
is readily obtained, where b is a constant. Wang [24] recovered the same solution by introducing the transformation U = w2/a and used the equation
dw ¼ awð1 wÞ dz
ð5Þ
to obtain an exact solution for the generalized Fisher’s equation
oU o2 U ¼ 2 þ Uð1 U a Þ: ot ox
ð6Þ
The exact solution of Eq. (6) obtained by Wang [24] was given by
Uðx; tÞ ¼
2=a 1 a aþ4 b 1 þ ; tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t þ 2 2 2 2a þ 4 2 2a þ 4
ð7Þ
where b is a constant. Malfliet [19] showed that traveling wave solutions of complicated nonlinear wave equations were found with the aid of tanh functions. With the tanh method, Wang [24] formally derived
Uðx; tÞ ¼
2 1 1 5 pffiffiffi x pffiffiffi t 1 tanh 4 2 6 6
ð8Þ
as a solution to Fisher’s equation (1) which represents a shock waves structures. Brazhnik and Tyson [5] have shown that for quadratic Fisher equation in two spatial dimensions that, along with a plane wave, there exist several other traveling waves with nontrivial front geometry. Explicit solutions and approximations have been obtained in [5]. Mansour [20] showed that traveling wave solutions of a nonlinear reaction–diffusion-chemotaxis model for bacterial pattern formation. Olmos and Shizgal [21] have shown that a pseudo-spectral method of solution of Fisher’s equation. Abdul-Majid Wazwaz showed that analytical study on Burgers, Fisher, and Huxley equations and combined forms of these equations. As stated before, several studies in the literature, employing a large variety of methods, have been conducted to derive explicit solutions for Fisher’s equation (1) and for the generalized Fisher’s equation (6). For more details about these investigations, the reader is advised to see Refs. [5,14,23,24] and the references therein.
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This paper is devoted to study the Fisher’s equation, generalized Fisher’s equation and nonlinear diffusion equation of the Fisher’s type. We introduce a Haar wavelet method for solving Fisher’s equation (1) with the initial and boundary conditions (2), which will exhibit several advantageous features: (i) Very high accuracy fast transformation and possibility of implementation of fast algorithms compared with other known methods. (ii) The simplicity and small computation costs, resulting from the sparsity of the transform matrices and the small number of significant wavelet coefficients. (iii) The method is also very convenient for solving the boundary value problems, since the boundary conditions are taken care of automatically. Beginning from 1980s, wavelets have been used for solution of partial differential equations (PDE). The good features of this approach are possibility to detect singularities, irregular structure and transient phenomena exhibited by the analyzed equations. Most of the wavelet algorithms can handle exactly periodic boundary conditions. The wavelet algorithms for solving PDE are based on the Galerkin techniques or on the collocation method. Evidently all attempts to simplify the wavelet solutions for PDE are welcome one possibility for this is to make use of the Haar wavelet family. Haar wavelets (which are Daubechies of order 1) consists of piecewise constant functions and are therefore the simplest orthonormal wavelets with a compact support. A drawback of the Haar wavelets is their discontinuity. Since the derivatives do not exist in the breaking points it is not possible to apply the Haar wavelets for solving PDE directly. There are two possibilities for getting out of this situation. One way is to regularize the Haar wavelets with interpolating splines (e.g. B-splines or Deslaurier-Dabuc interpolating wavelets). This approach has been applied by Cattani [7], but the regularization process considerably complicates the solution and the main advantage of the Haar wavelets – the simplicity gets to some extent lost. The other way is to make use of the integral method, which was proposed by Chen and Hsiao [8]. Lepik [16–18] had solved higher order as well as nonlinear ODEs by using Haar wavelet method. There are discussions by other researchers [12,13]. The paper is organized the following way. For completeness sake the Haar wavelet method is presented in Section 2. Function approximation is presented in Section 3. The method of solution the PDE is proposed in Section 4. Some numerical examples are presented in Section 5. Concluding remarks are given in Section 6. 2. Haar wavelets Haar functions have been used from 1910 when they were introduced by the Hungarian mathematician Haar [11]. The Haar transform is one of the earliest examples of what is known now as a compact, dyadic, orthonormal wavelet transform. The Haar function, being an odd rectangular pulse pair, is the simplest and oldest orthonormal wavelet with compact support. In the mean time, several definitions of the Haar functions and various generalizations have been published and used. They were intended to adopt this concept to some practical applications as well as to extend its in applications to different classes of signals. Haar functions appear very attractive in many applications as for example, image coding, edge extraction, and binary logic design. Recently, Haar wavelets have been applied extensively for signal processing in communications and physics research, and have proved to be a wonderful mathematical tool. After discretizing the differential equations in a conventional way like the finite difference approximation, wavelets can be used for algebraic manipulations in the system of equations obtained which lead to better condition number of the resulting system. The previous work in system analysis via Haar wavelets was led by Chen and Hsiao [8], who first derived a Haar operational matrix for the integrals of the Haar function vector and put the application for the Haar analysis into the dynamical systems. Then, the pioneer work in state analysis of linear time delayed systems via Haar wavelets was laid down by Hsiao [12], who first proposed a Haar product matrix and a coefficient matrix. Hsiao and Wang proposed a key idea to transform the time-varying function and its product with states into a Haar product matrix. Kalpana and Raja Balachandar [13] presented Haar wavelet based method of analysis for observer design in the generalized state space or singular system of transistor circuits. The orthogonal set of Haar function hi(t) is shown in Fig. 1. This is a group of square waves with magnitudes of ±1 in certain intervals and zeros elsewhere. For applications of the Haar transform in logic design, efficient ways of calculating the Haar spectrum from reduced forms of Boolean functions are needed. The Haar wavelet family for t 2 [0, 1] is defined as follows:
8
for t 2 mk ; kþ0:5 ; > < 1; kþ0:5m kþ1
hi ðtÞ ¼ 1; for t 2 m ; m ; > : 0; elsewhere:
ð9Þ
Integer m = 2j (j = 0, 1, 2, . . . , J) indicates the level of the wavelet; k = 0, 1, 2, . . . , m 1 is the translation parameter. Maximal level of resolution is J. The index i is calculated according the formula i = m + k + 1; in the case of minimal values. m = 1, k = 0
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G. Hariharan et al. / Applied Mathematics and Computation 211 (2009) 284–292
S.No.
Integrals of Haar functions
Haar functions
1.
h0(t)
h1(t)
2.
1
1 0 1
t
1 1
0
h2(t)
0
1
1
0
t
-1
h3(t)
4.
1
0
t
-1
h4(t)
5.
1
0
t
-1
h5(t)
6.
1
1
0
t
-1
h6(t)
7.
1
1
0
t
-1
h7(t)
8.
1
1
0 -1
t
1
0.25 0
1
1
t
t
0.25 0
1
1
t
-1
3.
0 0.5
0.125 0
0.125 0
0.125 0
0.125 0
1
1
1
1
1
t
t
t
t
t
t
Fig. 1. First eight Haar functions and their integrals.
we have i = 2, the maximal value of i is i = 2M = 2J+1. It is assumed that the value i = 1 corresponds to the scaling function for which h1 1 in [0, 1]. Let us define the collocation points tl = (l 0.5)/2M, (l = 1, 2, . . . , 2M) and discretise the Haar function hi(t); in this way we get the coefficient matrix H(i, l) = (hi(tl)), which has the dimension 2M 2M. The operational matrix of integration P, which is a 2M square matrix, is defined by the equation
ðPHÞil ¼
Z
tl
0
Z
ðQHÞil ¼
tl
hi ðtÞdt; Z t dt hi ðtÞdt:
0
ð10Þ ð11Þ
0
The elements of the matrices H, P and Q can be evaluated according to 8, 10 and 11.
1 H2 ¼
1
! ;
1 1 2
1
1
2 1
1 P2 ¼ 4 1
1
; 1
0
1
0
3
2
7 6 6 1 1 1 1 7 7 6 7; H4 ¼ 6 7 6 6 1 1 0 0 7 5 4 0
!
8 4 2 2
6 64 1 6 6 P4 ¼ 16 6 61 4
0
2
1
0
1 1
1
0
32 16 8 8 4 4 4 4 6 16 0 8 8 4 4 4 4 7 7 6 7 6 6 4 4 0 0 4 4 0 0 7 7 6 4 0 0 4 4 0 0 7 1 6 7 6 4 P8 ¼ 7: 6 7 64 6 1 1 2 0 0 0 0 0 7 6 6 1 1 2 0 0 0 0 0 7 7 6 7 6 4 1 1 0 2 0 0 0 0 5 1
1
0
2
0
0
0
0
7 2 7 7 7; 7 0 7 5 0
3
2
3
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Chen and Hsiao [8] showed that the following matrix equation for calculating the matrix P of order m holds
PðmÞ ¼
1 2m
2mPðm=2Þ
Hðm=2Þ
H1 ðm=2Þ
O
where O is a null matrix of order
m 2
! ;
m2 ,
HmXm D½hm ðt 0 Þhm ðt1 Þ hm ðtm1 Þ
ð12Þ
T 1 and mi 6 t < i þ m1 and H1 mxm ¼ m H mxm diagðrÞ It should be noted that calculations for P(m) and H(m) must be carried out only once; after that they will be applicable for solving whatever differential equations. Since H and H1 contain many zeros, this phenomenon makes the Haar transform must faster than the Fourier transform, and it is even faster than the Walsh transform. This is one of the reason for rapid convergence of the Haar wavelet series.
3. Function approximation Any function y(x) 2 L2[0, 1) can be decomposed as
yðxÞ ¼
1 X
cn hn ðxÞ;
ð13Þ
n¼0
where the coefficients cn are determined by
c n ¼ 2j
Z
1
yðxÞhn ðxÞdx;
ð14Þ
0
R1
where n = 2j + k, j P 0, 0 6 k < 2j. Specially c0 ¼ 0 yðxÞdx. The series expansion of y(x) contains an infinite terms. If y(x) is piecewise constant by itself, or may be approximated as piecewise constant during each subinterval, then y(x) will be terminated at finite terms, that is
yðxÞ ¼
m1 X
cn hn ðxÞ ¼ cTðmÞ hðmÞ ðxÞ;
ð15Þ
n¼0
where the coefficients cTðmÞ and the Haar function vector h(m)(x) are defined as
cTðmÞ ¼ ½c0 ; c1 ; . . . ; cm1 and h(m)(x) = [h0(x), h1(x), . . . , hm1(x)]T where ‘T’ means transpose and m = 2j. 4. Method of solution of Fisher’s equation Consider the Fisher’s equation
ou o2 u þ uð1 uÞ ¼ ot ox2
ð16Þ
with the initial condition u(x, 0) = f(x), 0 6 x 6 1 and the boundary conditions u(0, t) = g0(t), u(1, t) = g1(t), 0 < t 6 T. Let us divide the interval (0, 1] into N equal parts of length Dt = (0, 1]/N and denote ts = (s 1)Dt, s = 1, 2, . . . , N. We assume that u_ 00 ðx; tÞ can be expanded in terms of Haar wavelets as formula
u_ 00 ðx; tÞ ¼
m1 X
cs ðnÞhn ðxÞ ¼ cTðmÞ hðmÞ ðxÞ;
ð17Þ
n¼0
where and 0 means differentiation with respect to t and x, respectively, the row vector cTðmÞ is constant in the subinterval t 2 (ts, ts+1]. Integrating formula (17) with respect to t from ts to t and twice with respect to x from 0 to x, we obtain
u00 ðx; tÞ ¼ ðt ts ÞcTðmÞ hðmÞ ðxÞ þ u00 ðx; ts Þ; uðx; tÞ ¼ ðt
t s ÞcTðmÞ Q ðmÞ hðmÞ ðxÞ
ð18Þ 0
0
þ uðx; t s Þ uð0; t s Þ þ x½u ð0; tÞ u ð0; t s Þ þ uð0; tÞ;
_ tÞ ¼ cTðmÞ Q ðmÞ hðmÞ ðxÞ þ xu_ 0 ð0; tÞ þ uð0; _ uðx; tÞ: By the boundary conditions, we obtain
uð0; ts Þ ¼ g 0 ðts Þ; uð1; t s Þ ¼ g 1 ðt s Þ; _ _ uð0; tÞ ¼ g 00 ðtÞ; uð1; tÞ ¼ g 01 ðtÞ:
ð19Þ ð20Þ
G. Hariharan et al. / Applied Mathematics and Computation 211 (2009) 284–292
289
Putting x = 1 in formulae (19) and (20), we have
u0 ð0; tÞ u0 ð0; ts Þ ¼ ðt t s ÞcTðmÞ Q ðmÞ hðmÞ ðxÞ þ g 1 ðtÞ g 0 ðtÞ g 1 ðts Þ þ g 0 ðt s Þ;
ð21Þ
u_ 0 ð0; tÞ ¼ g 01 ðtÞ cTðmÞ Q ðmÞ hðmÞ ðxÞ g 00 ðtÞ:
ð22Þ
Substituting formulae (21) and (22) into formulae (18)–(20), and discretizising the results by assuming x ? xl, t ? ts+1 we obtain
u00 ðxl ; t sþ1 Þ ¼ ðt sþ1 t s ÞcTðmÞ hðmÞ ðxl Þ þ u00 ðxl ; ts Þ;
ð23Þ
uðxl ; tsþ1 Þ ¼ ðt sþ1 ts ÞcTðmÞ Q ðmÞ hðmÞ ðxl Þ þ uðxl ; t s Þ g 0 ðt s Þ þ g 0 ðtsþ1 Þ þ xl ½ðt sþ1 t s ÞcTðmÞ PðmÞ f þ g l ðt sþ1 Þ g 0 ðt sþ1 Þ g 1 ðt s Þ þ g 0 ðt s Þ; _ l ; tsþ1 Þ ¼ cTðmÞ Q ðmÞ hðmÞ ðxÞ þ g 00 ðt sþ1 Þ þ xl ½cTðmÞ PðmÞ f þ g 01 ðtsþ1 Þ g 00 ðtsþ1 Þ; uðx
ð24Þ ð25Þ
where the vector f is defined as
f ¼ ½1; 0; . . . ; 0 T : |fflfflfflffl{zfflfflfflffl} ðm1Þelements
In the following the scheme
_ l ; tsþ1 Þ ¼ u00 ðxl ; t sþ1 Þ þ uðxl ; tsþ1 Þð1 uðxl ; t sþ1 ÞÞ uðx
ð26Þ
which leads us from the time layer ts to ts+1 is used. Substituting Eqs. (23)–(25) into Eq. (26), we gain
cTðmÞ Q ðmÞ hðmÞ ðxl Þ þ xl ½cTðmÞ PðmÞ f þ g 01 ðt sþ1 Þ g 00 ðt sþ1 Þ þ g 00 ðt sþ1 Þ ¼ u00 ðxl ; tsþ1 Þ þ uðxl ; tsþ1 Þ½1 uðxl ; tsþ1 Þ: From formula (27) the wavelet coefficients
cTðmÞ
ð27Þ
can be successively calculated.
5. Test problems Problem 1. Consider the following problem:
ou o2 u þ uð1 uÞ; ¼ ot ox2
0<x 0
with the data u(1, t) = u(1, t) = 0 and the initial condition u(x, 0) = 0. The exact solution of the model problem is given by
uðx; tÞ ¼ 1
1 2 cosh x 16 X ð1Þn cos½ð2n 1Þðpx=2Þ 2p exp 1 þ ð2n 1Þ t : cosh 1 p n¼1 ð2n 1Þ½ð2n 1Þ2 p2 þ 4 4
Problem 5. Consider the generalized Fisher’s equation
ou o2 u þ uð1 u6 Þ ¼ ot ox2 1 subject to the initial condition uðx; 0Þ ¼ ð1þeð3=2Þx . Þ1=3
The solution in a closed form is given by
uðx; tÞ ¼
1=3 1 3 5 1 tanh x t þ : 2 4 2 2
In a similar manner, we can show that for the generalized Fisher’s equation
ut ¼ uxx þ uð1 ua Þ;
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the solution by Wang [7] is given by
uðx; tÞ ¼
2=a 1 a aþ4 b 1 : tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t þ þ 2 2 2 2a þ 4 2 2a þ 4
6. Conclusion The theoretical elegance of the Haar wavelet approach can be appreciated from the simple mathematical relations and their compact derivations and proofs. It has been well demonstrated that in applying the nice properties of Haar wavelets, the partial differential equations can be solved conveniently and accurately by using Haar wavelet method systematically. Haar wavelets approach for the Fisher’s equation and the generalized Fisher’s equation is proposed. In comparison with existing numerical schemes used to solve Fisher’s equation, the scheme in this paper is an improvement over other methods in terms of accuracy. Another benefit of our method is that the scheme presented here, with some modifications, seems to be easily extended to solve model equations including more mechanical, physical or biophysical effects, such as nonlinear convection, reaction, linear diffusion and dispersion. Acknowledgement The authors of this paper wish to thank the reviewers for their useful comments towards the improvements of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Haar wavelet method for solving Fisher’s equation G. Hariharan a,*, K. Kannan a, K.R. Sharma b a b
Department of Mathematics, SASTRA University, Thanjavur 613 402, Tamilnadu, India Department of Chemical Engineering, SASTRA University, Thanjavur 613 402, Tamilnadu, India
a r t i c l e
i n f o
Keywords: Reaction–diffusion equations Fisher’s equation Numerical simulation Haar wavelets Adomain decomposition method
a b s t r a c t In this paper, we develop an accurate and efficient Haar wavelet solution of Fisher’s equation, a prototypical reaction–diffusion equation. The solutions of Fisher’s equation are characterized by propagating fronts that can be very steep for large values of the reaction rate coefficient. There is an ongoing effort to better adapt Haar wavelet methods to the solution of differential equations with solutions that resemble shock waves or fronts typical of hyperbolic partial differential equations. Moreover the use of Haar wavelets is found to be accurate, simple, fast, flexible, convenient, small computation costs and computationally attractive. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction The problems of the propagation of nonlinear waves have fascinated scientists for over 200 years. The modern theory of nonlinear waves, like many areas of mathematics, had its beginnings in attempts to solve specific problems, the hardest among them being the propagation of waves in water. There was significant activity on this problem in the 19th century and the beginning of the 20th century, including the classic work of Stokes, Lord Rayleigh, Korteweg and de Vries, Boussinesque, Benard and Fisher to name some of the better remembered examples [15,30]. We would like to solve the well-known and classical Kolmogorov–Petrovski–Piscounov reaction–diffusion equation known as KPP equation. The solitons appear as a result of a balance between weak nonlinearity and dispersion. Soliton is defined as a nonlinear wave characterized by the following properties: (i) A localized wave propagates without change of its properties (shape, velocity, etc.). (ii) Localized wave are stable against mutual collisions and retain their identities. On the other hand, the delicate interaction between nonlinear convection with genuine nonlinear dispersion generates solitary waves with compact support that are called campactons [22,25,26]. Unlike soliton that narrows as the amplitude increases, the compacton’s width is independent of the amplitude. However, when diffusion takes part instead of dispersion, energy release by nonlinearity balances energy consumption by diffusion, which results in traveling waves or fronts [14]. Traveling wave fronts are an important and much studied solution form for reaction–diffusion equations, with important applications to chemistry, biology and medicine [23]. Such solutions were first studied in the 1930s by Fisher for the scalar equation
oU o2 U ¼ 2 þ Uð1 UÞ: ot ox * Corresponding author. E-mail addresses: [email protected] (G. Hariharan), [email protected] (K. Kannan), [email protected] (K.R. Sharma). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.12.089
ð1Þ
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A traveling wave solution U(x, t) = U(n = x ct), propagating with a speed c, is restricted to be positive and bounded [5]. Therefore the boundary conditions for the traveling wave solution are usually
Uðn ! 1Þ ! 1;
Uðn ! 1Þ ! 0:
ð2Þ
In addition 0 6 U(x, t) 6 1 and c > 0 is the wave speed. Both Fisher and KPP found that Eq. (1) has an infinite number of traveling wave solutions of characteristic speeds c P 2. Fisher also carried out a very accurate and detailed numerical computation of the shock like profile of the traveling wave of minimum speed. Useful summaries of recent advances in this area have been provided by Wazwaz [25–29], Feng and Li [9], Boumenir [3], Whitham [30] and Zhou [31]. The well-known Fisher’s equation combines diffusion with logistic nonlinearity. Fisher proposed Eq. (1) as a model for the propagation of a mutantgene, with U denoting the density of an advantageous. This equation is encountered in chemical kinetics [19] and population dynamics, which includes problems such as nonlinear evolution of a population in a one-dimensional habitat, neutron population in a nuclear reaction. Moreover, the same equation occurs in logistic population growth models [5], flame propagation, neurophysiology, autocatalytic chemical reactions, and branching Brownian motion processes. The mathematical properties of Fisher’s Equation (FE) have been studied extensively and there have been numerous discussions in the literature. Excellent summaries have been provided in [5]. One of the first numerical solutions was presented in literature with a pseudo-spectral approach. Implicit and explicit finite differences algorithms have been reported by different authors such as Parekh and Puri and Twizell et al. A Galerkin finite element method was used by Tang and Weber whereas Carey and Shen [6] employed a least-squares finite element method. A collocation approach based on Whittaker’s sinc interpolation function [4] was also considered in [2]. The work in [10] considered a nonlocal form of FE. The exact solitary wave solution of Eq. (1) was not obtained until 1979. Noting the Painleve property of Eq. (1) and using the Laurent series [24], Ablowitz and Zepetella [1] looked for a solution of the form
UðzÞ ¼
6 a1 þ þ a0 a1 z þ . . . ; z2 z
ð3Þ
where z = x ct. The solution can then be found by solving nontrivial recursion relations, where the exact solution
Uðx; tÞ ¼
2 1 1 5 b 1 þ tanh pffiffiffi x pffiffiffi t þ 2 2 2 2 6 6
ð4Þ
is readily obtained, where b is a constant. Wang [24] recovered the same solution by introducing the transformation U = w2/a and used the equation
dw ¼ awð1 wÞ dz
ð5Þ
to obtain an exact solution for the generalized Fisher’s equation
oU o2 U ¼ 2 þ Uð1 U a Þ: ot ox
ð6Þ
The exact solution of Eq. (6) obtained by Wang [24] was given by
Uðx; tÞ ¼
2=a 1 a aþ4 b 1 þ ; tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t þ 2 2 2 2a þ 4 2 2a þ 4
ð7Þ
where b is a constant. Malfliet [19] showed that traveling wave solutions of complicated nonlinear wave equations were found with the aid of tanh functions. With the tanh method, Wang [24] formally derived
Uðx; tÞ ¼
2 1 1 5 pffiffiffi x pffiffiffi t 1 tanh 4 2 6 6
ð8Þ
as a solution to Fisher’s equation (1) which represents a shock waves structures. Brazhnik and Tyson [5] have shown that for quadratic Fisher equation in two spatial dimensions that, along with a plane wave, there exist several other traveling waves with nontrivial front geometry. Explicit solutions and approximations have been obtained in [5]. Mansour [20] showed that traveling wave solutions of a nonlinear reaction–diffusion-chemotaxis model for bacterial pattern formation. Olmos and Shizgal [21] have shown that a pseudo-spectral method of solution of Fisher’s equation. Abdul-Majid Wazwaz showed that analytical study on Burgers, Fisher, and Huxley equations and combined forms of these equations. As stated before, several studies in the literature, employing a large variety of methods, have been conducted to derive explicit solutions for Fisher’s equation (1) and for the generalized Fisher’s equation (6). For more details about these investigations, the reader is advised to see Refs. [5,14,23,24] and the references therein.
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This paper is devoted to study the Fisher’s equation, generalized Fisher’s equation and nonlinear diffusion equation of the Fisher’s type. We introduce a Haar wavelet method for solving Fisher’s equation (1) with the initial and boundary conditions (2), which will exhibit several advantageous features: (i) Very high accuracy fast transformation and possibility of implementation of fast algorithms compared with other known methods. (ii) The simplicity and small computation costs, resulting from the sparsity of the transform matrices and the small number of significant wavelet coefficients. (iii) The method is also very convenient for solving the boundary value problems, since the boundary conditions are taken care of automatically. Beginning from 1980s, wavelets have been used for solution of partial differential equations (PDE). The good features of this approach are possibility to detect singularities, irregular structure and transient phenomena exhibited by the analyzed equations. Most of the wavelet algorithms can handle exactly periodic boundary conditions. The wavelet algorithms for solving PDE are based on the Galerkin techniques or on the collocation method. Evidently all attempts to simplify the wavelet solutions for PDE are welcome one possibility for this is to make use of the Haar wavelet family. Haar wavelets (which are Daubechies of order 1) consists of piecewise constant functions and are therefore the simplest orthonormal wavelets with a compact support. A drawback of the Haar wavelets is their discontinuity. Since the derivatives do not exist in the breaking points it is not possible to apply the Haar wavelets for solving PDE directly. There are two possibilities for getting out of this situation. One way is to regularize the Haar wavelets with interpolating splines (e.g. B-splines or Deslaurier-Dabuc interpolating wavelets). This approach has been applied by Cattani [7], but the regularization process considerably complicates the solution and the main advantage of the Haar wavelets – the simplicity gets to some extent lost. The other way is to make use of the integral method, which was proposed by Chen and Hsiao [8]. Lepik [16–18] had solved higher order as well as nonlinear ODEs by using Haar wavelet method. There are discussions by other researchers [12,13]. The paper is organized the following way. For completeness sake the Haar wavelet method is presented in Section 2. Function approximation is presented in Section 3. The method of solution the PDE is proposed in Section 4. Some numerical examples are presented in Section 5. Concluding remarks are given in Section 6. 2. Haar wavelets Haar functions have been used from 1910 when they were introduced by the Hungarian mathematician Haar [11]. The Haar transform is one of the earliest examples of what is known now as a compact, dyadic, orthonormal wavelet transform. The Haar function, being an odd rectangular pulse pair, is the simplest and oldest orthonormal wavelet with compact support. In the mean time, several definitions of the Haar functions and various generalizations have been published and used. They were intended to adopt this concept to some practical applications as well as to extend its in applications to different classes of signals. Haar functions appear very attractive in many applications as for example, image coding, edge extraction, and binary logic design. Recently, Haar wavelets have been applied extensively for signal processing in communications and physics research, and have proved to be a wonderful mathematical tool. After discretizing the differential equations in a conventional way like the finite difference approximation, wavelets can be used for algebraic manipulations in the system of equations obtained which lead to better condition number of the resulting system. The previous work in system analysis via Haar wavelets was led by Chen and Hsiao [8], who first derived a Haar operational matrix for the integrals of the Haar function vector and put the application for the Haar analysis into the dynamical systems. Then, the pioneer work in state analysis of linear time delayed systems via Haar wavelets was laid down by Hsiao [12], who first proposed a Haar product matrix and a coefficient matrix. Hsiao and Wang proposed a key idea to transform the time-varying function and its product with states into a Haar product matrix. Kalpana and Raja Balachandar [13] presented Haar wavelet based method of analysis for observer design in the generalized state space or singular system of transistor circuits. The orthogonal set of Haar function hi(t) is shown in Fig. 1. This is a group of square waves with magnitudes of ±1 in certain intervals and zeros elsewhere. For applications of the Haar transform in logic design, efficient ways of calculating the Haar spectrum from reduced forms of Boolean functions are needed. The Haar wavelet family for t 2 [0, 1] is defined as follows:
8
for t 2 mk ; kþ0:5 ; > < 1; kþ0:5m kþ1
hi ðtÞ ¼ 1; for t 2 m ; m ; > : 0; elsewhere:
ð9Þ
Integer m = 2j (j = 0, 1, 2, . . . , J) indicates the level of the wavelet; k = 0, 1, 2, . . . , m 1 is the translation parameter. Maximal level of resolution is J. The index i is calculated according the formula i = m + k + 1; in the case of minimal values. m = 1, k = 0
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S.No.
Integrals of Haar functions
Haar functions
1.
h0(t)
h1(t)
2.
1
1 0 1
t
1 1
0
h2(t)
0
1
1
0
t
-1
h3(t)
4.
1
0
t
-1
h4(t)
5.
1
0
t
-1
h5(t)
6.
1
1
0
t
-1
h6(t)
7.
1
1
0
t
-1
h7(t)
8.
1
1
0 -1
t
1
0.25 0
1
1
t
t
0.25 0
1
1
t
-1
3.
0 0.5
0.125 0
0.125 0
0.125 0
0.125 0
1
1
1
1
1
t
t
t
t
t
t
Fig. 1. First eight Haar functions and their integrals.
we have i = 2, the maximal value of i is i = 2M = 2J+1. It is assumed that the value i = 1 corresponds to the scaling function for which h1 1 in [0, 1]. Let us define the collocation points tl = (l 0.5)/2M, (l = 1, 2, . . . , 2M) and discretise the Haar function hi(t); in this way we get the coefficient matrix H(i, l) = (hi(tl)), which has the dimension 2M 2M. The operational matrix of integration P, which is a 2M square matrix, is defined by the equation
ðPHÞil ¼
Z
tl
0
Z
ðQHÞil ¼
tl
hi ðtÞdt; Z t dt hi ðtÞdt:
0
ð10Þ ð11Þ
0
The elements of the matrices H, P and Q can be evaluated according to 8, 10 and 11.
1 H2 ¼
1
! ;
1 1 2
1
1
2 1
1 P2 ¼ 4 1
1
; 1
0
1
0
3
2
7 6 6 1 1 1 1 7 7 6 7; H4 ¼ 6 7 6 6 1 1 0 0 7 5 4 0
!
8 4 2 2
6 64 1 6 6 P4 ¼ 16 6 61 4
0
2
1
0
1 1
1
0
32 16 8 8 4 4 4 4 6 16 0 8 8 4 4 4 4 7 7 6 7 6 6 4 4 0 0 4 4 0 0 7 7 6 4 0 0 4 4 0 0 7 1 6 7 6 4 P8 ¼ 7: 6 7 64 6 1 1 2 0 0 0 0 0 7 6 6 1 1 2 0 0 0 0 0 7 7 6 7 6 4 1 1 0 2 0 0 0 0 5 1
1
0
2
0
0
0
0
7 2 7 7 7; 7 0 7 5 0
3
2
3
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Chen and Hsiao [8] showed that the following matrix equation for calculating the matrix P of order m holds
PðmÞ ¼
1 2m
2mPðm=2Þ
Hðm=2Þ
H1 ðm=2Þ
O
where O is a null matrix of order
m 2
! ;
m2 ,
HmXm D½hm ðt 0 Þhm ðt1 Þ hm ðtm1 Þ
ð12Þ
T 1 and mi 6 t < i þ m1 and H1 mxm ¼ m H mxm diagðrÞ It should be noted that calculations for P(m) and H(m) must be carried out only once; after that they will be applicable for solving whatever differential equations. Since H and H1 contain many zeros, this phenomenon makes the Haar transform must faster than the Fourier transform, and it is even faster than the Walsh transform. This is one of the reason for rapid convergence of the Haar wavelet series.
3. Function approximation Any function y(x) 2 L2[0, 1) can be decomposed as
yðxÞ ¼
1 X
cn hn ðxÞ;
ð13Þ
n¼0
where the coefficients cn are determined by
c n ¼ 2j
Z
1
yðxÞhn ðxÞdx;
ð14Þ
0
R1
where n = 2j + k, j P 0, 0 6 k < 2j. Specially c0 ¼ 0 yðxÞdx. The series expansion of y(x) contains an infinite terms. If y(x) is piecewise constant by itself, or may be approximated as piecewise constant during each subinterval, then y(x) will be terminated at finite terms, that is
yðxÞ ¼
m1 X
cn hn ðxÞ ¼ cTðmÞ hðmÞ ðxÞ;
ð15Þ
n¼0
where the coefficients cTðmÞ and the Haar function vector h(m)(x) are defined as
cTðmÞ ¼ ½c0 ; c1 ; . . . ; cm1 and h(m)(x) = [h0(x), h1(x), . . . , hm1(x)]T where ‘T’ means transpose and m = 2j. 4. Method of solution of Fisher’s equation Consider the Fisher’s equation
ou o2 u þ uð1 uÞ ¼ ot ox2
ð16Þ
with the initial condition u(x, 0) = f(x), 0 6 x 6 1 and the boundary conditions u(0, t) = g0(t), u(1, t) = g1(t), 0 < t 6 T. Let us divide the interval (0, 1] into N equal parts of length Dt = (0, 1]/N and denote ts = (s 1)Dt, s = 1, 2, . . . , N. We assume that u_ 00 ðx; tÞ can be expanded in terms of Haar wavelets as formula
u_ 00 ðx; tÞ ¼
m1 X
cs ðnÞhn ðxÞ ¼ cTðmÞ hðmÞ ðxÞ;
ð17Þ
n¼0
where and 0 means differentiation with respect to t and x, respectively, the row vector cTðmÞ is constant in the subinterval t 2 (ts, ts+1]. Integrating formula (17) with respect to t from ts to t and twice with respect to x from 0 to x, we obtain
u00 ðx; tÞ ¼ ðt ts ÞcTðmÞ hðmÞ ðxÞ þ u00 ðx; ts Þ; uðx; tÞ ¼ ðt
t s ÞcTðmÞ Q ðmÞ hðmÞ ðxÞ
ð18Þ 0
0
þ uðx; t s Þ uð0; t s Þ þ x½u ð0; tÞ u ð0; t s Þ þ uð0; tÞ;
_ tÞ ¼ cTðmÞ Q ðmÞ hðmÞ ðxÞ þ xu_ 0 ð0; tÞ þ uð0; _ uðx; tÞ: By the boundary conditions, we obtain
uð0; ts Þ ¼ g 0 ðts Þ; uð1; t s Þ ¼ g 1 ðt s Þ; _ _ uð0; tÞ ¼ g 00 ðtÞ; uð1; tÞ ¼ g 01 ðtÞ:
ð19Þ ð20Þ
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Putting x = 1 in formulae (19) and (20), we have
u0 ð0; tÞ u0 ð0; ts Þ ¼ ðt t s ÞcTðmÞ Q ðmÞ hðmÞ ðxÞ þ g 1 ðtÞ g 0 ðtÞ g 1 ðts Þ þ g 0 ðt s Þ;
ð21Þ
u_ 0 ð0; tÞ ¼ g 01 ðtÞ cTðmÞ Q ðmÞ hðmÞ ðxÞ g 00 ðtÞ:
ð22Þ
Substituting formulae (21) and (22) into formulae (18)–(20), and discretizising the results by assuming x ? xl, t ? ts+1 we obtain
u00 ðxl ; t sþ1 Þ ¼ ðt sþ1 t s ÞcTðmÞ hðmÞ ðxl Þ þ u00 ðxl ; ts Þ;
ð23Þ
uðxl ; tsþ1 Þ ¼ ðt sþ1 ts ÞcTðmÞ Q ðmÞ hðmÞ ðxl Þ þ uðxl ; t s Þ g 0 ðt s Þ þ g 0 ðtsþ1 Þ þ xl ½ðt sþ1 t s ÞcTðmÞ PðmÞ f þ g l ðt sþ1 Þ g 0 ðt sþ1 Þ g 1 ðt s Þ þ g 0 ðt s Þ; _ l ; tsþ1 Þ ¼ cTðmÞ Q ðmÞ hðmÞ ðxÞ þ g 00 ðt sþ1 Þ þ xl ½cTðmÞ PðmÞ f þ g 01 ðtsþ1 Þ g 00 ðtsþ1 Þ; uðx
ð24Þ ð25Þ
where the vector f is defined as
f ¼ ½1; 0; . . . ; 0 T : |fflfflfflffl{zfflfflfflffl} ðm1Þelements
In the following the scheme
_ l ; tsþ1 Þ ¼ u00 ðxl ; t sþ1 Þ þ uðxl ; tsþ1 Þð1 uðxl ; t sþ1 ÞÞ uðx
ð26Þ
which leads us from the time layer ts to ts+1 is used. Substituting Eqs. (23)–(25) into Eq. (26), we gain
cTðmÞ Q ðmÞ hðmÞ ðxl Þ þ xl ½cTðmÞ PðmÞ f þ g 01 ðt sþ1 Þ g 00 ðt sþ1 Þ þ g 00 ðt sþ1 Þ ¼ u00 ðxl ; tsþ1 Þ þ uðxl ; tsþ1 Þ½1 uðxl ; tsþ1 Þ: From formula (27) the wavelet coefficients
cTðmÞ
ð27Þ
can be successively calculated.
5. Test problems Problem 1. Consider the following problem:
ou o2 u þ uð1 uÞ; ¼ ot ox2
0<x 0
with the data u(1, t) = u(1, t) = 0 and the initial condition u(x, 0) = 0. The exact solution of the model problem is given by
uðx; tÞ ¼ 1
1 2 cosh x 16 X ð1Þn cos½ð2n 1Þðpx=2Þ 2p exp 1 þ ð2n 1Þ t : cosh 1 p n¼1 ð2n 1Þ½ð2n 1Þ2 p2 þ 4 4
Problem 5. Consider the generalized Fisher’s equation
ou o2 u þ uð1 u6 Þ ¼ ot ox2 1 subject to the initial condition uðx; 0Þ ¼ ð1þeð3=2Þx . Þ1=3
The solution in a closed form is given by
uðx; tÞ ¼
1=3 1 3 5 1 tanh x t þ : 2 4 2 2
In a similar manner, we can show that for the generalized Fisher’s equation
ut ¼ uxx þ uð1 ua Þ;
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the solution by Wang [7] is given by
uðx; tÞ ¼
2=a 1 a aþ4 b 1 : tanh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t þ þ 2 2 2 2a þ 4 2 2a þ 4
6. Conclusion The theoretical elegance of the Haar wavelet approach can be appreciated from the simple mathematical relations and their compact derivations and proofs. It has been well demonstrated that in applying the nice properties of Haar wavelets, the partial differential equations can be solved conveniently and accurately by using Haar wavelet method systematically. Haar wavelets approach for the Fisher’s equation and the generalized Fisher’s equation is proposed. In comparison with existing numerical schemes used to solve Fisher’s equation, the scheme in this paper is an improvement over other methods in terms of accuracy. Another benefit of our method is that the scheme presented here, with some modifications, seems to be easily extended to solve model equations including more mechanical, physical or biophysical effects, such as nonlinear convection, reaction, linear diffusion and dispersion. Acknowledgement The authors of this paper wish to thank the reviewers for their useful comments towards the improvements of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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