Chapter 2 Analytical Approximation Methods

Transcription

1 Chapter 2 Analytical Approximation Methods 2.1 Introduction As we mentioned in the previous chapter, most of the nonlinear ODEs have no explicit solutions, i.e., solutions, which are expressible in finite terms. Even if an explicit solution can be determined, it is often too complicated to analyze the principal features of this solution. Due to such difficulties, the study of nonlinear mathematical problems is the most time-consuming and difficult task for researchers dealing with nonlinear models in the natural sciences, engineering, and scientific computing. With the increasing interest in the development of nonlinear models, a variety of analytical asymptotic and approximation techniques have been developed in recent years to determine approximate solutions of partial and ordinary differential equations. Some of these techniques are the perturbation method, the variational iteration method, the homotopy perturbation method, the energy balance method, the variational approach method, the parameter-expansion method, the amplitude-frequency formulation, the iteration perturbation method, and the Adomian decomposition method. In this chapter, we present the variational iteration method and the Adomian decomposition method since these techniques have good convergence characteristics and can be used to treat strongly nonlinear ODEs. 2.2 The Variational Iteration Method The variational iteration method VIM was first proposed by He see e.g. [49, 5] and systematically elucidated in [51, 54, 126]. The method treats partial and ordinary differential equations without any need to postulate restrictive assumptions that may change the physical structure of the solutions. It has been shown that the VIM solves effectively, easily, and accurately a large class of nonlinear problems with approximations converging rapidly to accurate solutions, see e.g. [127]. Examples Springer India 216 M. Hermann and M. Saravi, Nonlinear Ordinary Differential Equations, DOI 1.17/ _2 33

2 34 2 Analytical Approximation Methods for such problems are the Fokker Planck equation, the Lane Emden equation, the Klein Gordon equation, the Cauchy reaction diffusion equation, and biological population models. To illustrate the basic idea of the VIM, we consider, the ODE Ly + Ny = f x, x I, 2.1 where L and N are linear and nonlinear differential operators, respectively, and f x is an given inhomogeneous term defined for all x I.IntheVIM,acorrection functional of the Eq. 2.1 is defined in the following form y n+1 x = y n x + λτ Ly n τ + Nỹ n τ f τ dτ, 2.2 where λτ is a general Lagrange multiplier, which can be identified using the variational theory [38]. Furthermore, y n x is the nth approximation of yx and ỹ n x is considered as a restricted variation, i.e., δỹ n x =. By imposing the variation and by considering the restricted variation, Eq. 2.2 is reduced to δy n+1 x = δy n x + δ [ τ = δy n x + λτ λτly n τdτ ] τ=x Lδy n ξdξ τ λ τ Lδy n ξdξ dτ. τ= 2.3 Obviously, in 2.3 we have used integration by parts, which is based on the following formula λτy n τdτ = λτy nτ λ τy n τdτ. 2.4 In the next sections, we will also use two other formulas for the integration by parts, namely λτy n τdτ = λτy n τ λ τy n τ + λ τy n τdτ, 2.5 and λτy n τdτ = λτy n τ λ τy n τ + λ τy n τ λ τy n τdτ. 2.6

3 2.2 The Variational Iteration Method 35 Now, by applying the stationary conditions for 2.3, the optimal value of the Lagrange multiplier λτ can be identified see e.g. [5], formula 2.13 and the next section. Once λτ is obtained, the solution of the Eq. 2.1 can be readily determined by calculating the successive approximations y n x, n =, 1,...,using the formula see Eq. 2.2 y n+1 x = y n x + λτ Ly n τ + Ny n τ f τ dτ, 2.7 where y x is a starting function, which has to be prescribed by the user. In the paper [5] it is shown, that the approximate solution y n x of the exact solution yx can be achieved using any selected function y x. Consequently, the approximate solution is given as the limit yx = lim n y n x. In other words, the correction functional 2.2 will give a sequence of approximations and the exact solution is obtained at the limit of the successive approximations. In general, it is difficult to calculate this limit. Consequently, an accurate solution can be obtained by considering a large value for n. This value depends on the interval I where a good approximation of the solution is desired. Let us consider, the following IVP The corresponding exact solution is Here, we have y x + yx 2 =, y = yx = 1 + x 1 = 1 x + x 2 x 3 + x 4 x 5 + x Ly = y, Ny = y 2, f x. To determine the Lagrange multiplier, we insert these expressions into 2.2 and obtain dyn τ y n+1 x = y n x + λτ +ỹ n τ dτ dτ Making the above correction functional stationary w.r.t. y n, noticing that δỹ n x = and δy n =, it follows with 2.3 δy n+1 x = δy n x + δ [ = δy n x + λτ λ τ τ λτ dy nτ dτ τ dτ ] τ=x d δy n ξ dξ dξ d δy n ξ dξ dτ dξ τ=

4 36 2 Analytical Approximation Methods = δy n x + λτδy n τ τ=x λ τδy n τdτ. =. Thus, we obtain the equations 1 + λx = and λ x = Now, we substitute the solution λ = 1of2.11into2.1.It results the successive iteration formula dyn τ y n+1 x = y n x + y n τ dτ dτ We have to choose a starting function y x, which satisfies the given initial condition y = 1. Starting with y x 1, we compute the following successive approximations y x = 1, y 1 x = 1 x, y 2 x = 1 x + x x 3, y 3 x = 1 x + x 2 x x x x x 7, y 4 x = 1 x + x 2 x 3 + x x x 15, y 5 x = 1 x + x 2 x 3 + x 4 x x x 31. In Fig. 2.1 the first iterates y x,...,y 4 x are plotted. Comparing the iterates with the Taylor series of the exact solution see 2.9, we see that in y 5 x the first six terms are correct. The value of the exact solution at x = 1isy1 = 1/2 =.5. In Table 2.1, the corresponding value is given for the iterates y i x, i =,...,1. In the above example, the linear operator is L = assume that L = dm dx, m 1. m d. More generally, let us dx In [86], the corresponding optimal values of the Lagrange multipliers are given. It holds

5 2.2 The Variational Iteration Method 37 Fig. 2.1 The first successive iterates y i x for the IVP 2.8. The solid line represents the exact solution yx Table 2.1 The successive iterates at the right boundary for the IVP 2.8 i y i λ = 1, for m = 1, λ = τ x, for m = 2, λ = 1m m 1! τ xm 1, for m Substituting 2.13 into the correction functional 2.2, we get the following iteration formula 1 m y n+1 x = y n x + m 1! τ xm 1 Ly n τ + Ny n τ f τdτ, 2.14 where y x must be given by the user.

6 38 2 Analytical Approximation Methods 2.3 Application of the Variational Iteration Method In this section, we will consider some nonlinear ODEs and show how the VIM can be used to approximate the exact solution of these problems. Example 2.1 Solve the following IVP for the Riccati equation Solution. In2.1, we set y x + sinxyx = cosx + y 2, y =. Ly dy dx, Ny sinxy y2, f x cosx. Thus, the correction functional 2.2 is dyn τ y n+1 x = y n x + λτ + sinτỹ n τ ỹ n τ 2 cosτ dτ. dτ Since L is the first derivative, i.e., m = 1, a look at formula 2.13 shows that λ = 1 is the optimal value of the Langrange multiplier. The resulting successive iteration formula is y n+1 x = y n x y n τ + sinτy n τ y n τ 2 cosτ dτ Let us choose y x as starting function. Notice that y x satisfies the given initial condition. Now, with 2.15 we obtain the following successive approximations y 1 x = sinx, y 2 x = sinx,. y n x = sinx. Obviously, it holds lim n y n x = sinx. The exact solution is yx = sinx. Example 2.2 Determine with the VIM a solution of the following IVP for the second order ODE y x + ω 2 yx = gyx, y = a, y =. 2.16

7 2.3 Application of the Variational Iteration Method 39 This problem is the prototype of nonlinear oscillator equations see, e.g., [3, 99]. The real number ω is the angular frequency of the oscillator and must be determined in advance. Moreover, g is a known discontinuous function. Solution. To apply the VIM, we set see formula 2.1 Ly y + ω 2 y, Ny gy, f x. The corresponding correction functional is d 2 y n τ y n+1 x = y n x + λτ, x + ω 2 y dτ 2 n τ gỹ n τ dτ. Before we identify an optimal λ, we apply the formula 2.5 for the following integration by parts λτ, xy n τdτ = λτ, xy n τ τ=x τ= + λτ, x τ 2 λτ, x y τ 2 n τdτ. y n τ Using this relation in the correction functional, imposing the variation, and making the correction functional stationary, we obtain δy n+1 x = δy n x + λτ, xδy n τ τ=x. =. Thus, the stationary conditions are λτ, x δy n τ τ 2 λτ, x + + ω 2 λτ, x δy τ 2 n τdτ τ=x τ= τ=x δy n : 2 λτ, x + ω 2 λτ, x =, τ 2 δy n : λτ, x τ=x =, δy n : 1 λτ, x τ =. τ=x 2.17 The solution of the Eqs. in 2.17 is λτ, x = 1 sinωx τ, 2.18 ω

8 4 2 Analytical Approximation Methods which leads to the following iteration formula y n+1 x = y n x + 1 ω d 2 y n τ sinωx τ + ω 2 y dτ 2 n τ gy n τ dτ Example 2.3 Let us consider the following IVP of the Emden-Lane-Fowler equation see e.g. [48] y + 2 x y + x k y μ =, y = 1, y =. This ODE is used to model the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules. Solve this equation for k = and μ = 5, which has a closed form solution. Solution. For the given parameters, the IVP to be solved is y + 2 x y + y 5 =, y = 1, y =. Obviously, there is a singularity at x =. To overcome this singularity, we set y z/x. Then, we get z + x 4 z 5 =, z =, z = 1. We set Lz d2 z dx 2, Nz x 4 z 5, f x. Thus, the correction functional 2.2 is d 2 z n τ z n+1 x = z n x + λτ + τ 4 z 5 dx 2 n τ dτ. Looking at formula 2.13, we obtain for m = 2 the Lagrange multiplier λ = τ x. Therefore, the corresponding iteration formula 2.14 is d 2 z n τ z n+1 x = z n x + τ x + τ 4 z 5 dx 2 n τ dτ.

9 2.3 Application of the Variational Iteration Method 41 Starting with z x = x, we obtain the following successive approximations z x = x, z 1 x = x x 3 6, z 2 x = x x x 5 24, z 3 x = x x x 5 24 x It is not difficult to show that lim z nx = zx = x 1 x 2 n 6 + x x = x 1 + x 2 1/2. 3 Thus yx = zx x = 1 + x 2 3 1/2 is the exact solution of the given IVP. Example 2.4 One of the problems that has been studied by several authors is Bratu s BVP see, e.g., [18, 71, 8, 87, 1, 18], which is given in one-dimensional planar coordinates by y = αe y, y =, y1 =, 2.2 where α > is a real parameter. This BVP plays an important role in the theory of the electric charge around a hot wire and in certain problems of solid mechanics. The exact solution of 2.2 is cosh.5x.5θ yx = 2ln, cosh.25θ where θ satisfies θ = 2α cosh.25θ. Bratu s problem has zero, one or two solutions when α > α c, α = α c, and α < α c, respectively, where the critical value α c satisfies 1 =.25 2α c sinh.25θ.

10 42 2 Analytical Approximation Methods In Chap. 5, thevalueα c is determined as UsetheVIMtosolvetheBVP2.2. α c = Solution. Let us expand e y and use three terms of this expansion. We obtain y + αe y = y + α i= y i i! y + α 1 + y + y2 2. Setting Ly d2 y dx 2, 1 Ny α + y + y2, f x, 2 the corresponding correction functional 2.2is d 2 y n τ y n+1 x = y n x + λτ + α dτ 2 1 +ỹ n τ + ỹnτ 2 2 dτ. Looking at formula 2.13, we obtain for m = 2 the Lagrange multiplier λ = τ x. Therefore, the corresponding iteration formula 2.14 is d 2 y n τ y n+1 x = y n x + τ x + α dτ y n τ + y nτ 2 Let us start with y x = kx, where k is a real number. The next iterate is Integrating by parts leads to y 1 x = kx + α τ x y 1 x = kx αx 2 2! 1 + kτ + k2 τ 2 αkx3 3! 2 λk2 x 4. 4! dτ. 2 dτ Substituting y 1 x into the right-hand side of 2.21, we obtain the next iterate y 2 x = kx αx 2 αkx3 αk2 x 4 2! 3! 4! + τ x ατ 2 2 2αkτ 3 + α α 5k2 τ 4 + αk 24 2α k2 τ 5 dτ. + 5α2 k 2 τ α2 k 3 τ α2 k 4 τ

11 2.3 Application of the Variational Iteration Method 43 Again, integration by parts yields y 2 x = kx αx 2 αkx3 3! 2! + 5α2 kk 2 2αx 7 7! αk2 αx 4 4! 25α3 k 2 x 8 8! + 4α2 kx 5 5! 35α3 k 3 x 9 9! + α2 5k 2 3αx 6 6! 35α3 k 4 x 1. 1! 2.22 The function y 2 x must satisfy the initial conditions see formula 2.2. For a given α, the equation y 2 1 = is a fourth degree polynomial in k. When an appropriate k is chosen from the corresponding four roots, the function y 2 x can be accepted as an approximation of the exact solution yx for x, 1. Let us consider Bratu s problem with α = 1. The polynomial in k is 35k 4 325k k k =. Solving this algebraic equation by a numerical method, the following approximated roots are obtained: k 1 = , k 2 = , k 3,4 = ± i. When we substitute k = k 1 and α = 1into2.22, the next iterate is determined. In Table 2.2, y 2 x is compared with yx for x =.1,.2,...,.9. Next, let us consider Bratu s problem with α = 2. The polynomial in k is 7k 4 29k k k =. Solving this algebraic equation by a numerical method, the following approximated roots are obtained: Table 2.2 Numerical results for Bratu s problem with α = 1; θ = x y 2 x yx Relative error

12 44 2 Analytical Approximation Methods Table 2.3 Numerical results for Bratu s problem with α = 2; θ = x y 2 x yx Relative error k 1 = , k 2 = , k 3,4 = ± i. As before, when we substitute k = k 1 and α = 2into2.22, the next iterate is determined. In Table 2.3, y 2 x is compared with yx for x =.1,.2,...,.9. The results in the Tables 2.2 and 2.3 show that the VIM is efficient and quite reliable. Only two iterations have lead to acceptable results. There is no doubt, if more terms of the expansion and/or more iterates are used, the VIM will generate far better results. 2.4 The Adomian Decomposition Method The Adomian decomposition method ADM is a semi-analytical technique for solving ODEs and PDEs. The method was developed by the Armenian-American mathematician George Adomian [7, 8, 9]. The ADM is based on a decomposition of the solution of nonlinear operator equations in appropriate function spaces into a series of functions. The method, which accurately computes the series solution, is of great interest to the applied sciences. The method provides the solution in a rapidly convergent series with components that are computed elegantly. The convergence of this method is studied in [1, 2, 73]. Let the general form of an ODE be Fy = f, where F is the nonlinear differential operator with linear and nonlinear terms. In the ADM, the linear term is decomposed as L + R, where L is an easily invertible operator and R is the remainder of the linear term. For convenience L is taken as the highest-order derivative. Thus the ODE may be written as

13 2.4 The Adomian Decomposition Method 45 Ly + Ry + Ny = f x, 2.23 where Ny corresponds to the nonlinear terms. Let L be a first-order differential operator defined by L d dx then the inverse operator L 1 is given by L 1 = dτ..ifl is invertible, Thus, L 1 Ly = yx y Similarly, if L 2 d2 dx, then the inverse operator 2 L 1 is regarded as a double integration operator given by L 1 = τ dt dτ. It follows L 1 Ly = yx xy We can use the same operations to find relations for higher-order differential operators. For example, if L 3 d3, then it is not difficult to show that dx3 L 1 Ly = yx y xy 1 2! x 2 y The basic idea of the ADM is to apply the operator L 1 formally to the expression Lyx = f x Ryx Nyx. This yields yx = Ψ x + gx L 1 Ryx L 1 Nyx, 2.27 where the function gx represents the terms, which result from the integration of f x, and y, y + xy, Ψ x y + xy + 1 2! x 2 y, y + xy + 1 2! x 2 y + 1 3! x 3 y, for L = d dx, for L 2 = d2 dx 2, for L 3 = d3 dx 3, for L 4 = d4 dx 4.

14 46 2 Analytical Approximation Methods Now, we write yx = y n x and Nyx = A n x, n= n= where A n x A n y x, y 1 x,...,y n 1 x are known as the Adomian polynomials. Substituting these two infinite series into 2.27, we obtain y n x = Ψ x + gx L 1 R y n x L 1 n= n= n= A n x Identifying the zeroth component y x by Ψ x + gx, the remaining components y k x, k 1, can be determined by using the recurrence relation y x = Ψ x + gx, y k x = L 1 Ry k 1 x L 1 A k 1 x, k = 1, 2, Obviously, when some of the components y k x are determined, the solution yx can be approximated in form of a series. Under appropriate assumptions, it holds yx = lim n k= n y k x. The polynomials A k x are generated for each nonlinearity so that A depends only on y, A 1 depends only on y and y 1, A 2 depends on y, y 1, y 2, etc. [7]. An appropriate strategy to determine the Adomian polynomials is A = Ny, A 1 = y 1 N y, A 2 = y 2 N y + 1 2! y2 1 N y, A 3 = y 3 N y + y 1 y 2 N y + 1 3! y3 1 N 3 y, 1 A 4 = y 4 N y + 2! y2 2 + y 1y 3 N y + 1 2! y2 1 y 2N 3 y ! y4 1 N 4 y, where N k y dk dy k Ny.

15 2.4 The Adomian Decomposition Method 47 The general formula is k A k = 1 k! λ k [ ] N y n λ n n= λ=, k =, 1, 2, A Mathematica program that generates the polynomials A k automatically can be found in [16]. Moreover, in [2] a simple algorithm for calculating Adomian polynomials is presented. According to this algorithm the following formulas for the Adomian polynomials result: A = Ny, A 1 = y 1 N y, A 2 = y 2 N y y2 1 N y, A 3 = y 3 N y + y 1 y 2 N y y3 1 N 3 y, A 4 = y 4 N y + y 1 y y2 2 N y y2 1 y 2N 3 y y4 1 N 4 y Before we highlight a few examples and show how the ADM can be used to solve concrete ODEs, let us list the Adomian polynomials for some classes of nonlinearity. 1. Ny = expy: A = expy, A 1 = y 1 expy, A 2 = y ! y2 1 expy, A 3 = y 3 + y 1 y ! y3 1 expy ; Ny = lny, y > : A = lny, A 1 = y 1 y, A 2 = y 2 y 1 2 y1 2 y 2, A 3 = y 3 y 1y 2 y y y1 3 y 2 ; Ny = y 2 : A = y 2, A 1 = 2y y 1, A 2 = 2y y 2 + y1 2, A 3 = 2y y 3 + 2y 1 y 2 ; 2.34

16 48 2 Analytical Approximation Methods 4. Ny = y 3 : 5. Ny = yy : A = y 3, A 1 = 3y 2 y 1, A 2 = 3y 2 y 2 + 3y y 2 1, A 3 = 3y 2 y 3 + 6y y 1 y 2 + y 3 1 ; 2.35 A = y y, A 1 = y y 1 + y y 1, A 2 = y y 2 + y 1 y 1 + y 2 y, A 3 = y y 3 + y 1 y 2 + y 2 y 1 + y 3 y ; Ny = y 2 : A = y 2, A 1 = 2y y 1, A 2 = 2y y 2 + y 1 2, A 3 = 2y y 3 + 2y 1 y 2 ; Ny = cosy: A = cosy, A 1 = y 1 siny, A 2 = y 2 siny 1 2 y2 1 cosy, A 3 = y 3 siny y 1 y 2 cosy 8. Ny = siny: y3 1 siny ; 2.38 A = siny, A 1 = y 1 cosy, A 2 = y 2 cosy 1 2 y2 1 siny, A 3 = y 3 cosy y 1 y 2 siny y3 1 cosy. 2.5 Application of the Adomian Decomposition Method In this section, we consider some IVPs for first-order and second-order ODEs. Example 2.5 Solve the IVP y x = 1 x 2 yx + yx 3, y =. Solution. This is Abel s equation and its exact solution is yx = x. We apply the ADM to solve it. First, let us look at formula We have Ly y, Ry x 2 y, Ny y 3, f x 1.

17 2.5 Application of the Adomian Decomposition Method 49 Using 2.27, we obtain yx = Ψ x + gx L 1 Ryx L 1 Nyx = y + x Now, applying formula 2.28, we get n= x y n x = x τ 2 n= τ 2 yτdτ + y n τ dτ yτ 3 dτ. A n τ dτ. The Adomian polynomials, which belong to the nonlinearity Ny = y 3,aregiven in Setting y x = Ψ x + gx = y + x = x, the recurrence relation 2.29 yields y 1 x = y 2 x =. τ 3 dτ τ 2 y 1 τdτ + 3 A τdτ = n= y τ 2 y 1 τdτ =, τ 3 dτ + τ 3 dτ =, Hence yx = y k x = x = x. k= Example 2.6 Solve the IVP y x + xe yx =, y =. Solution. Here, we have Ly y, Ry, Ny xe y, f x. Using 2.27, we obtain yx = Ψ x + gx L 1 Ryx L 1 Nyx = y τe yτ dτ = τe yτ dτ.

18 5 2 Analytical Approximation Methods Now, applying formula 2.28, we get n= x y n x = τ A n τ dτ. n= The Adomian polynomials, which belong to the nonlinearity expy, are given in Setting the recurrence relation 2.29 yields y 1 x = y 2 x = y 3 x = y x = Ψ x + gx = y =, τ 1 dτ = x 2 2, τ τ 2 2 dτ = τ 3 2 dτ = x 4 8, τ 4 τ τ dτ = 2 2 = x 6 24,. τ 5 4 dτ Thus, y n x = τ A n 1 τdτ = 1 n x 2 n, n = 1, 2,..., 2 and it holds yx = n= y n x = 1 2 x x x 6 + = ln 1 + x 2 2. Example 2.7 Solve the IVP y x + 2yxy x =, y =, y = 1. Solution. Here, we have Ly y, Ry, Ny 2yy, f x.

19 2.5 Application of the Adomian Decomposition Method 51 Using 2.27, we obtain yx = Ψ x + gx L 1 Ryx L 1 Nyx = y + xy 2 Now, applying formula 2.28, we get n= τ yty tdtdτ. τ y n x = x 2 A n t dtdτ. n= The Adomian polynomials, which belong to the nonlinearity yy,aregivenin2.36. Setting y x = Ψ x + gx = y + xy = x, the recurrence relation 2.29 yields τ x y 1 x = 2 y ty t dtdτ = 2 = 2 y 2 x = 2 = 2 = 2 3 y 3 x = 2 = Obviously, it holds τ 2 2 τ τ τ τ = yx = τ dτ = τ 2 dτ = x 3 3, y ty 1 t + y ty 1 t dtdτ 1 t 33 + t t 2 dtdτ = tdtdτ τ t 3 dtdτ τ 4 dτ = 2 15 x 5, y ty 2 t + y 1 ty 1t + y 2 ty t dtdτ t 5 + t 2 t t 4 t dtdτ τ t 5 dtdτ = y n x = x x 3 n= τ 6 dτ = x x x 7 +

20 52 2 Analytical Approximation Methods Example 2.8 Solve the IVP Solution. Here, we have We write Using 2.27, we obtain y x y x 2 + yx 2 = e x, y = y = 1. Ly y, Ry, Ny y 2 + y 2, f x e x. Ny = N 1 y + N 2 y, N 1 y y 2, N 2 y y 2. yx = Ψ x + gx L 1 N 1 yx L 1 N 2 yx = y + xy + τ τ yt 2 dtdτ Now, applying formula 2.28, we get n= e t dtdτ + y n x = 1 + x + e x x 1 + τ τ B n t dtdτ. n= τ y t 2 dtdτ A n t dtdτ The Adomian polynomials A n, which belong to the nonlinearity y 2,aregivenin 2.37, and for the nonlinearity y 2, the Adomian polynomials B n are given in Setting y x = Ψ x + gx = e x, the recurrence relation 2.29 yields τ y 1 x = A tdtdτ τ = e t 2 dtdτ τ τ n= B tdtdτ e t 2 dtdτ =.

21 2.5 Application of the Adomian Decomposition Method 53 This implies y n x, n = 1, 2,..., and we obtain the exact solution of the given problem: yx = e x =e x. The convergence of the ADM can be accelerated if the so-called noise terms phenomenon occurs in the given problem see, e.g., [9, 1]. The noise terms are the identical terms with opposite sign that appear within the components y x and y 1 x. They only exist in specific types of nonhomogeneous equations. If noise terms indeed exist in the y x and y 1 x components, then, in general, the solution can be obtained after two successive iterations. By canceling the noise terms in y x and y 1 x, the remaining non-canceled terms of y x give the exact solution. It has been proved that a necessary condition for the existence of noise terms is that the exact solution is part of y x. Example 2.9 Solve the IVP y x y x 2 + yx 2 = 1, y = 1, y =. Solution. As in the Example 2.8, weset Ly y, Ry, N 1 y y 2, N 2 y y 2, f x 1. Using 2.27, we obtain yx = 1 + x τ Now, applying formula 2.28, we get τ y t 2 dtdτ yt 2 dtdτ y n x = 1 + x 2 τ 2 + A n t dtdτ n= τ B n t dtdτ. n= The Adomian polynomials A n, which belong to the nonlinearity y 2,aregivenin 2.37, and for the nonlinearity y 2, the Adomian polynomials B n are given in Setting y x = x 2, n=

22 54 2 Analytical Approximation Methods the recurrence relation 2.29 yields τ τ y 1 x = t 2 dtdτ [ t 3 ] τ [ 1 = dτ 3 2 t t 3 + t τ 3 1 = 3 dτ 2 τ τ 3 + τ = 1 12 x x x x 2 = 1 12 x x t dtdτ ] τ dτ Comparing y x with y 1 x, we see that there is the noise term 1 2 x 2. Therefore, we can conclude that the solution of the given IVP is yx = 1. Several authors have proposed a variety of modifications of the AMD see, e.g., [12] by which the convergence of the iteration 2.29 can be accelerated. Wazwaz [124, 126] suggests the following reliable modification which is based on the assumption that the function hx Ψ x + gx in formula 2.27 can be divided into two parts, i.e., hx Ψ x + gx = h x + h 1 x. The idea is that only the part h x is assigned to the component y x, whereas the remaining part h 1 x is combined with other terms given in It results the modified recurrence relation dτ y x = h x, y 1 x = h 1 x L 1 Ry x L 1 A x, y k x = L 1 Ry k 1 x L 1 A k 1 x, k = 2, 3, Example 2.1 Solve the IVP y x yx 2 = 2 x 4, y = y =. Solution. Here, we have Ly y, Ry, Ny y 2, f x 2 x 4.

23 2.5 Application of the Adomian Decomposition Method 55 Using 2.27, we obtain yx = Ψ x + gx L 1 Ryx L 1 Nyx = x 2 1 τ 3 x 6 + yt 2 dtdτ. Now, applying formula 2.28, we get n= y n x = x 2 1 τ 3 x 6 + A n t dtdτ. n= The Adomian polynomials, which belong to the nonlinearity y 2,aregivenin2.34. Dividing hx = x x 6 into h x x 2 and h 1 x 1 3 x 6, and starting with y x = x 2, the recurrence relation 2.29 yields y 1 x = 1 τ 3 x 6 + t 2 2 dtdτ = 1 3 x 6 τ dτ = 1 3 x x 6 =. This implies y k x =, k = 1, 2,... Thus, we can conclude that the exact solution of the given IVP is yx = x 2. Let us compare the modified ADM with the standard method. The ADM is based on the recurrence relation Here, we have to set y x = x x 6. Now, the recurrence relation 2.29 yields y 1 x = τ t t 6 dtdτ = = 1 3 x x x τ τ τ 5 dτ

24 56 2 Analytical Approximation Methods τ y 2 x = 2 t t x x x 6 dtdτ 1 = τ τ τ τ 9 dτ = x x x x 22. Since yx = y x + y 1 x + y 2 x +, we see that the terms in x 6 and x 1 cancel each other. The cancelation of terms is continued when further components y k, k 3, are added. This is an impressive example of how fast the modified ADM generates the exact solution yx = x 2, compared with the standard method. Many problems in the mathematical physics can be formulated as ODEs of Emden- Fowler type see, e.g., [26, 3] defined in the form y x + α x y x + β f xgy =, α, y = a, y =, 2.41 where f and g are given functions of x and y, respectively. The standard Emden- Lane-Fowler ODE results when we set f x 1 and gy y n. Obviously, a difficulty in the analysis of 2.41 is the singularity behavior that occurs at x =. Before the ADM can be applied, a slight change of the problem is necessary to overcome this difficulty. The strategy is to define the differential operator L in terms of the two derivatives, y + α/xy, which are contained in the ODE. First, the ODE 2.41 is rewritten as Lyx = β f xgy, 2.42 with L x α d x α d. dx dx The corresponding inverse operator L 1 is L 1 = τ τ α t α dtdτ.

25 2.5 Application of the Adomian Decomposition Method 57 Applying L 1 to the first two terms of 2.41, we obtain L 1 y x + α x y x τ = τ α t α y t + α t y t dtdτ [ τ τ ] = τ α y τ αt α 1 y tdt + αt α 1 y tdt dτ = y τdτ = yx y = yx a. Now, operating with L 1 on 2.41, we find yx = a βl 1 f xgy It is interesting, that only the first initial condition is sufficient to represent the solution yx in this form. The second initial condition can be used to show that the obtained solution satisfies this condition. Let us come back to the Adomian decomposition method. As before, the solution yx is represented by an infinite series of components yx = y n x n= In addition, the given nonlinear function gy is represented by an infinite series of Adomian polynomials as we have done it for Ny gyx = A n x, 2.45 n= where A n x A n y x, y 1 x,...,y n 1 x. Substituting 2.44 and 2.45 into2.43 gives y n x = a βl 1 f x A n x n= Now, the components y n x are determined recursively. The corresponding recurrence relation is n= y x = a, y k x = βl 1 f xa k 1 x, k = 1, 2,...,

26 58 2 Analytical Approximation Methods or equivalently y x = a, τ y k x = β τ α t α f ta k 1 xdtdτ, k = 1, 2, Example 2.11 Solve the IVP y x + 2 x y x + e yx =, y = y =. Solution. This problem is a special case of 2.41, where α = 2, β = 1, f x 1, gy = expy and a =. A particular solution of the ODE is 2 yx = ln. x 2 Obviously, this solution does not satisfy the initial conditions. We will see that the ADM can be used to determine a solution which satisfies the ODE as well as the initial conditions. For the nonlinearity gy = expy, the Adomian polynomials are given in Eq Using the recurrence relation 2.47, we obtain y x =, y 1 x = τ 2 τ t 2 1 dtdτ = = 1 6 x 2, τ t y 2 x = τ 2 t 2 2 dtdτ = 6 = τ 2 τ 5 3 dτ = = 1 12 x 4, τ y 3 x = τ 2 = = x 6, t t 2 4 τ 2 τ 7 dτ = 315 τ 3 3 dτ 12 + t 4 72 τ τ 2 τ 3 τ dτ = 3 3 dτ τ τ 2 t 4 6 dtdτ τ dtdτ = τ 2 τ dτ t 6 45 dtdτ

27 2.5 Application of the Adomian Decomposition Method 59 y 4 x = = τ 2 τ t τ τ 9 61 = x 8. Thus, we have 189 t t t dτ = τ 7 dτ t 6 dtdτ yx = 1 6 x x x x 8 + In [125] further variants of the general Emden-Fowler equation are discussed, and solved by the ADM. 2.6 Exercises Exercise 2.1 Solve following ODEs by VIM or ADM: 1 y x yx = yx 2, y = 1, 2 y x = 1 + yx yx 2 +, y =, x x 3 y x + 5 yx x y x + expyx + 2exp =, y = y =, 2 4 y x + 2yxy x yx = sinh2x, y =, y = 1, 5 y x + cosyx =, y =, y = π 2, 6 y x + 8 x y x + 2yx = 4yx lnyx, y = 1, y =, 7 y x + y x 2 + y x 2 = 1 cosx, y = y =, y = 1. Exercise 2.2 Given the following IVP for the Emden-Fowler ODE y x + 2 x y x + αx m yx μ =, y = 1, y =. Approximate the solution of this IVP by the VIM and the ADM. Exercise 2.3 Given the following IVP for the Emden-Fowler ODE y x + 2 x y x + αx m e yx =, y = y =. Approximate the solution of this IVP by the VIM and the ADM.

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