CHAPTER III HAAR WAVELET METHOD FOR SOLVING FISHER S EQUATION A version of this chapter has been published as Haar Wavelet Method for solving Fisher s equation, Appl. Math.Comput.,(ELSEVIER) 211 (2009) 284-292.
Chapter 3 Haar Wavelet Method for solving Fisher s equation 3.1 Introduction The problems of the propagation of nonlinear waves have fascinated scientists for over two hundred 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 20 th 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 [[117],[200]]. 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: 68
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 [[162],[190],[191]]. 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 [111]. Traveling wave fronts are an important and much studied solution form for reaction-diffusion equations, with important applications to chemistry, biology and medicine [166]. Such solutions were first studied in the 1930s by Fisher for the scalar equation U t = 2 U + U (1 U) (3.1) x2 A traveling wave solution U (x, t) = U (ξ = x ct) propagating with a speed c, is restricted to be positive and bounded [28]. Therefore the boundary conditions for the traveling wave solution are usually U (ξ ) 1, U (ξ ) 0 (3.2) In addition 0 U (x, t) 1 and c > 0 is the wave speed. Both Fisher and KPP found that Eq. (3.1) has an infinite number of traveling wave solutions of characteristic speeds c 2. 69
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 [[25],[61],[191],[190], [188],[198],[192],[195], [200],[211]] The well-known Fisher s equation combines diffusion with logistic nonlinearity. Fisher proposed equation (3.1) as a model for the propagation of a mutantgene, with U denoting the density of an advantageous. This equation is encountered in chemical kinetics [131] 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 [28], 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 [28]. 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 [30] employed a least-squares finite element method. A collocation approach based on Whittaker s sinc interpolation function [26] was also considered in [11]. The work in [70] considered a nonlocal form of FE. The exact solitary wave solution of equation (3.1) was not obtained until 1979. Noting the Painleve property of equation (3.1) and using the Laurent series [187], Ablowitz and Zepetella [4] looked for a solution of the form 70
U (z) = 6 z 2 + a 1 z + a 0a 1 z +..., (3.3) where z = x ct. The solution can then be found by solving nontrivial recursion relations, where the exact solution U (x, t) = { [ ( 1 1 2 tanh 2 x 5 ) t + b ] + 1 2 (3.4) 6 6 2 2} is readily obtained, where b is a constant. Wang [187] recovered the same solution by introducing the transformation U = w 2/α and used the equation dw dz = aw (1 w) (3.5) to obtain an exact solution for the generalized Fisher s equation U t = 2 U x 2 + U (1 U α ) (3.6) The exact solution of equation (3.6) obtained by Wang [187] was given by { [ 1 U (x, t) = 2 tanh α 2 2α + 4 ( x α + 4 ) t + b ] + 1 2/α (3.7) 2α + 4 2 2} where b is a constant. Malfliet [131] showed that traveling wave solutions of complicated nonlinear wave equations were found with the aid of tanh functions. With the tanh method, Wang [187] formally derived U (x, t) = ( ) { [( ) ( 1 1 1 tanh 4 2 x 5 2 t)]} (3.8) 6 6 as a solution to Fisher s equation (3.1) which represents a shock waves structures. 71
Brazhnik and Tyson [28] 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 [28]. Mansour [142] showed that traveling wave solutions of a nonlinear reaction-diffusion-chemotaxis model for bacterial pattern formation. Daniel Olmos and Bernie D.Shizgal [148] have shown that a pseudospectral method of solution of Fisher s equation. Wazwaz [198] 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 (3.1) and for the generalized Fisher s equation (3.6). For more details about these investigations, the reader is advised to see refs. [[28],[111],[166],[187]] and the references therein. This chapter 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 (3.1) with the initial and boundary conditions (3.2), 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 1980 s, wavelets have been used for solution of partial differential equations 72
(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 (for example, B-splines or Deslaurier-Dabuc interpolating wavelets). This approach has been applied by Cattani [34], 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 [37]. Lepik [123] had solved higher order as well as nonlinear ODEs by using Haar wavelet method. There are discussions by other researchers[[77],[94],[108]]. 3.2 Method of solution of Fisher s equation Consider the Fisher s equation u t = 2 u + u (1 u) (3.9) x2 73
with the initial condition u (x, 0) = f (x), 0 x 1 and the boundary conditions u (0, t) = g 0 (t), u (1, t) = g 1 (t), 0 < t T. Let us divide the interval (0,1] into N equal parts of length = (0, 1]/N and denote t s = (s 1) t, s = 1, 2,, N. We assume that u (x, t) can be expanded interms of Haar wavelets as formula u (x, t) = m 1 n=0 c s (n) h n (x) = c T (m)h (m) (x), (3.10) where. and means differentiation with respect to t and x respectively, the row vector c T (m) is constant in the subinterval t (t s, t s+1 ]. Integrating formula (3.10) with respect to t from t s to t and twice with respect to x from 0 to x, we obtain u (x, t) = (t t s ) c T (m)h (m) (x) + u (x, t s ), (3.11) [ ] u (x, t) = (t t s ) c T (m)q (m) h (m) (x)+u (x, t s ) u (0, t s )+x u (0, t) u (0, t s ) +u (0, t) (3.12) u (x, t) = c T (m)q (m) h (m) (x) + x u (0, t) + u (0, t) (3.13) By the boundary conditions, we obtain u (0, t s ) = g 0 (t s ), u (1, t s ) = g 1 (t s ), u (0, t) = g 0 (t), u (1, t) = g 1 (t). 74
Putting x = 1 in formulae (3.12) and (3.13), we have u (0, t) u (0, t s ) = (t t s ) c T (m)q (m) h (m) + g 1 (t) g 0 (t) g 1 (t s ) + g 0 (t s ), (3.14) u (0, t) = g 1 (t) c T (m)q (m) h (m) g 0 (t) (3.15) Substituting formulae (3.14) and (3.15) into formulae (3.11)-(3.13), and discretizising the results by assuming x x l, t t s+1 we obtain u (x l, t s+1 ) = (t s+1 t s ) c T (m)q (m) h (m) (x l ) + u (x l, t s ) (3.16) u (x l, t s+1 ) = (t s+1 t s ) c T (m) Q (m)h (m) (x l ) + u (x l, t s ) g 0 (t s ) + g 0 (t s+1 ) +x l [ (ts+1 t s ) c T (m)p (m) f + g l (t s+1 ) g 0 (t s+1 ) g 1 (t s ) + g 0 (t s ) ], (3.17) ] u(x l, t s+1 ) = c T (m)q (m) h (m) (x l )+g 0 (t s+1 )+x l [ C (m)p T (m) f + g 1 (t s+1 ) g 0 (t s+1 ) where the vector f is defined asf = 1, 0,, 0 }{{} (m 1)elements T (3.18) In the following the scheme u(x l, t s+1 ) = u (x l, t s+1 ) + u (x l, t s+1 ) (1 u (x l, t s+1 )) (3.19) which leads us from the time layer t s to t s+1 is used. Substituting equations (3.16)-(3.18) into the equation (3.19), we gain c T (m) Q (m)h (m) (x l ) +x l [ c T (m) p (m)f +g 1 (t s+1 ) g 0 (t s+1 )] +g 0 (t s+1 ) = u (x l, t s+1 ) + 75
u (x l, t s+1 ) [1 u (x l, t s+1 )]. From the above formula the wavelet coefficients C T (m) can be successively calculated. 3.3 Test problems Problem 1. Consider the following problem u t = 2 u x 2 + u (1 u), 0 < x < 1. subject to a constant initial condition u (x, 0) = λ ] c T (m) Q (m)h (m) (x l ) + x l [ c T (m) p (m)f + g 1 (t s+1 ) g 0 (t s+1 ) + g 0 (t s+1 ) = u (x l, t s+1 ) u (x l, t s+1 ) [1 u (x l, t s+1 )]. (3.20) From formula (3.21) the wavelet coefficients c T (m) can be successively calculated. Using Adomian decomposion method, the exact solution in a closed form is given by u (x, t) = λet 1 λ+λe t which is in full agreement with the results in [187]. Computer simulation was carried out in the cases m=32 and m=64,the computed results were compared with the exact solution, more accurate results can be obtained by using a larger m. ( See figs. 3.1-3.3) Problem 2. In this case we will examine the Fisher s equation Subject to the initial condition u = 2 u + u (1 u), 0 < x < 1. t x 2 76
u (x, 0) = 1 (1+e x ) 2. From the above formula, the wavelet coefficients C T (m) can be successively calculated. This process is started with u (x l, t s ) = 1 (1+e x ) 2, u (x l, t s ) = u (x l, t s ) = 2 Problem 3. Consider the Fisher s equatiion 2ex, (1+e x ) 3 [ ] e x 2e 2x. (1+e x ) 4 u = 2 u + 6u (1 u), 0 < x < 1. t x 2 subject to the initial condition u (x, 0) = 1 (1+e x ) 2. The exact solution in a closed form is given by This process is started with u (x, t) = 1 (1+e x 5t ) 2. u (x l, t s ) = 1 (1+e x ) 2, u (x l, t s ) = u (x l, t s ) = 2 2ex, (1+e x ) 3 [ ] e x 2e 2x. (1+e x ) 4 It is worth noting that applying the scheme proposed above for the Fisher s equation u t = u xx + αu (1 u), 77
the solution is u (x, t) = 1 1+e ( α 6 )x 5 6 αt! 2 can be compared with the Haar solution. Problem 4. Consider the Fisher s equation u = 2 u + (1 u), 1 < x < 1, t > 0 t x 2 with thw 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 { } u (x, t) = 1 coshx 16 cosh1 π Σ ( 1) n cos[(2n 1)(πx/2)] n=1 exp [1 + (2n 1) 2 π2 ]t (2n 1)[(2n 1) 2 π 2 +4] 4 Problem 5. Consider the generalised Fisher s equation u = 2 u + u (1 u 6 ), t x 2 subject to the initial condition u (x, 0) = 1 (1+e (3/2)x ) 1/3. The solution in a closed form is given by u (x, t) = { 1 tanh[ 3(x 5t)] + } 1 1/3 2 4 2 2 In a similar manner, we can show that for the generalised Fisher s equation u t = u xx + u(1 u α ), the solution by Wang [187] is given by u(x, t) = [ 1 2 tanh[ α 2 2α+4 (x α+4 2α+4 t) + b 2 ] + 1 2 ] 2 α 78
Table 3.1: Comparison of the analytical and the Haar solutions of Fisher s equation at t=0.25 Haar solution x Exactsolution m = 16 m = 32 m = 64 0.0 16.0 16.2 16.4 1.2 0.0625 0.0113 0.0018 0.0014 0.0013 0.1875 0.0037 0.0045 0.0038 0.0037 0.3125 0.0054 0.0061 0.0055 0.0054 0.4375 0.0061 0.0065 0.0059 0.0061 0.5625 0.0059 0.0067 0.0060 0.0059 0.6875 0.0048 0.0053 0.0047 0.0048 0.8125 0.0031 0.0039 0.0033 0.0032 0.9375 0.0010 0.0018 0.0011 0.0010 1.0 0.0 0.006 0.0 0.0 Table 3.2: Comparison of the analytical and the Haar solutions of Fisher s equation at t=0.48. Haar solution x Exactsolution m = 16 m = 32 m = 64 0.0 0.0 0.0 0.0 0.0 0.0625 0.0012 0.0020 0.00014 0.00013 0.1875 0.0008 0.0079 0.0072 0.00078 0.3125 0.00160 0.00146 0.00140 0.00150 0.4375 0.00210 0.00207 0.00200 0.00210 0.5625 0.00230 0.00216 0.00210 0.00230 0.6875 0.00200 0.00197 0.00190 0.00200 0.8125 0.00140 0.00136 0.00130 0.00140 0.9375 0.00057 0.00058 0.00052 0.00056 1.0 0.0 0.006 0.0 0.0 3.4 Features 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 79
Figure 3.1: Comparison between exact and Haar solution of Fisher s equation x = 20 and k= 12.5. 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 80
Figure 3.2: Comparison between exact and Haar solution of Fisher s equation x = 10 and k = 12.5. comparison with existing numerical schemes used to solve Fisher s equation, the scheme in this chapter is an improvement over other methods in terms of accuracy. Another benefit of our method is that the scheme presented here, with 81
Figure 3.3: Comparison between exact and Haar solution of Fisher s equation x = 5 and k = 12.5. 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. 82