Numerical solution of parabolic partial differential

2nd Reading August 12, 2016 15:56 WSPC/246-AEJM

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Asian-European Journal of Mathematics Vol. 10, No. 1 (2017) 1750026 (11 pages) c World Scientific Publishing Company DOI: 10.1142/S1793557117500267

Asian-European J. Math. Downloaded from www.worldscientific.com by THE UNIVERSITY OF HONG KONG on 08/29/16. For personal use only.

Numerical solution of parabolic partial differential equations using adaptive gird Haar wavelet collocation method,

S. C. Shiralashetti∗,‡ , L. M. Angadi† , M. H. Kantli∗ and A. B. Deshi∗ ∗ Department

of Mathematics, Karnatak University, Dharwad–580003, India

† Department of Mathematics Govt. First Grade College, Chikodi–591201, India ‡ [email protected]

Communicated by B. K. Dass Received May 10, 2016 Accepted July 4, 2016 Published August 16, 2016

In this paper, we applied the adaptive grid Haar wavelet collocation method (AGHWCM) for the numerical solution of parabolic partial differential equations (PDEs). The approach of AGHWCM for the numerical solution of parabolic PDEs is mentioned, the obtained numerical results, error analysis are presented in figures and tables. This shows that, the AGHWCM gives better accuracy than the HWCM and FDM. Some of the test problems are taken for demonstrating the validity and applicability of the AGHWCM. Keywords: Adaptive grid; Haar wavelet collocation; parabolic PDEs; finite difference method. AMS Subject Classification: 65T60, 97N40, 49K20

1. Introduction Partial differential equations(PDEs) occur in many branches of applied mathematics, for example, in quantum mechanics, hydrodynamics, elasticity and electromagnetic theory. The analytical behavior of these equations is a rather involved process and requires applications of advanced mathematical methods. On the other hand, in recent years, the greater importance has been shifted from analytical techniques to numerical methods. The principal attraction of numerical methods is that solutions could be obtained for many problems which are not ready to analytical treatment.

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S. C. Shiralashetti et al.

The accessibility of modern digital computers paved the way for the development of efficient methods for solving PDEs. Beginning from 1980s, wavelets have been used for solution of PDEs. The good features of this approach are possibility to detect singularities, irregular structure and transient phenomena exhibited by the analyzed equations. The wavelet algorithms for solving PDEs are based on the Galerkin techniques or on the collocation method. One possibility for wavelet solutions for PDEs is to make use of the Haar wavelet family. Haar wavelets consist of piecewise constant functions and are the simplest orthonormal wavelets with a compact support. The Haar wavelet based techniques has been successfully used in various applications such as time–frequency analysis, signal de-noising, nonlinear approximation and solving different class of PDEs by various authors (Chen and Hsiao [12], Lepik [5–7], Bujurke et al. [1, 4, 9], Shiralashetti et al. [2]). In numerical simulation of highly nonlinear and singular equations, a high resolution is necessary to obtain a good accuracy. However, this type of problems is not reasonable to use a fine resolution with uniform grid across the whole domain, because it requires the large storage and very high computational cost. For avoiding this difficulty, the problem of numerical grid generation for the purpose of solving PDEs by numerical methods is of considerable current interest. The most vital problem at present is to different ways to incorporate the properties e.g. grid smoothness, orthogonality, etc in a grid generation scheme to minimize the truncation error as much as possible. Grid generation is an important first step in the solution of many computational problems, especially problems arising in scientific computing and computer graphics. For example, the accuracy of the numerical solution of PDEs depends on the quality of the grid used in the discretization of the problem, especially on the grid density, the distribution of the grid points, and the quality of the elements connecting these points. Also, when the solution has large variations like large gradients, peaks, or boundary layers in some parts of the solution domain, we need more grid points in such regions than in regions, where the solution changes smoothly. This type of mesh generation is known as adaptive grid generation [3]. The adaptive grid Haar wavelet collocation method (AGHWCM) is a latest method to construct an adaptive grid and it determines the grid collocation by analyzing the spatial singularities of the function using wavelet coefficients. The adaptive wavelet collocation method provided a mathematical foundation in the field of science and engineering [8, 10, 15]. The main objective of this paper is to present AGHWCM, an alternative method to existing ones for the numerical solution of parabolic PDEs. The present paper is organized as follows: Haar wavelet and its properties are presented in Sec. 2. Section 3 provides adaptive grid generation. In Sec. 4, method of solution is discussed. Section 5 deals with the numerical findings and error analysis. Finally, conclusion of the proposed work is discussed in Sec. 6.

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Numerical solution of parabolic partial differential equations

2. Haar Wavelets and Its Properties


We used the simplest wavelet function i.e. Haar wavelet. We establish an operational matrix for integration via Haar wavelets. The scaling function h1 (x) for the family of the Haar wavelets is defined as 1, for x ∈ [0, 1) (2.1) h1 (x) = 0, Otherwise. The Haar wavelet family for x ∈ [0, 1) is defined as  k k + 0.5   1, for x ∈ ,   m m    k + 0.5 k + 1 hi (x) =  −1, for x ∈ ,   m m     0, Otherwise.

(2.2)

In the above definition, the integer m = 2l , l = 1, 2, . . . , J, indicates the level of resolution and integer k = 0, 1, 2, . . . , m − 1 is the translation parameter. Maximum level of resolution is J. The index i in Eq. (2.2) is calculated using i = m + k + 1. In case of minimal values m = 1, k = 0, then i = 2. The maximal value of i is N = 2J+1 . Let us define the collocation points xj = (j − 0.5)/N, j = 1, 2, . . . , N, discretize the Haar function hi (x), in this way we get Haar coefficient matrix H(i, j) = hi (xj ), which has the dimension N × N. The operational matrix of integration is obtained by integrating (2.2) as x hi (x)dx (2.3) P hi (x) = 0

Qhi (x) = and

0

Chi (x) =

These integrals can be evaluated  k   x− ,    m    P hi (x) = k + 1 − x,   m      0,

x

0

P hi (x)dx

(2.4)

P hi (x)dx.

(2.5)

1

by using Eq. (2.2) and they are given by k k + 0.5 for x ∈ , m m k + 0.5 k + 1 for x ∈ , m m Otherwise

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(2.6)


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 2  1 k   , x−   2 m      2     1 − 1 k+1 −x , 2 m Qhi (x) = 4m2     1   ,   2  4m      0, Asian-European J. Math. Downloaded from www.worldscientific.com by THE UNIVERSITY OF HONG KONG on 08/29/16. For personal use only.

and

 2  1 k   , 1 −   2 m      2     1 − 1 k+1 −1 , 2 m Chi (x) = 4m2     1   ,    4m2      0,

k k + 0.5 , for x ∈ m m k + 0.5 k + 1 , for x ∈ m m k+1 ,1 for x ∈ m

(2.7)

Otherwise for x ∈

k k + 0.5 , m m

k + 0.5 k + 1 , for x ∈ m m k+1 ,1 for x ∈ m

(2.8)

Otherwise

3. Adaptive Grids The reason to addition of adaptive grid is to increase nodes on the computational domain, where the numerical approximation required for more accuracy. This is clearly different from uniform nodes on the computational domain. Accordingly, this approach is suitable for various types of problems arising in fluid dynamics. The grid adaptations occur naturally in wavelet methods and are defined on closed interval. For the N collocation points are, {x1 < x2 < x3 < · · · < xk · · · < xN −1 < xN },

(k ∈ Z, N ∈ Z)

these collocation points are generated uniformly by the definition of xj , i.e. xj = (j − 0.5)/N in the direction of x, xk denotes the kth point of the collocation points after the grid generation. Now, we generate the adaptive grid points by using the formula xk =

xj + xj+1 , 2

k = 1, 2, . . . N − 1.

(3.1)

By adding these grid points in the above, we get new grid space. For instance J = 2 ⇒ N = 8, then we have grid space with interval length

1 3 5 7 9 11 13 15 , , , , , , , . 16 16 16 16 16 16 16 16 1750026-4

1 8

is


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Numerical solution of parabolic partial differential equations 2 4 6 8 10 12 14 From these points, we get mid-points as 16 , 16 , 16 , 16 , 16 , 16 , 16 . By adding these midpoints in the above grid space, we get new grid space as

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 , , , , , , , , , , , , , , (3.2) 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 1 . with interval length is 16 Now, we get the numerical solution on these points [10, 11, 15], whose method of solution and numerical implementation are given by the following sections.


4. Method of Solution Consider the general second-order parabolic PDE of the form, ut = αuxx + βu + φ(x, t)

(4.1)

with the initial conditions 0≤x 0,

(4.3)

where φ(x, t), f (x), g0 (t), g1 (t), α, β are functions of independent or dependent variables or constants. Let us assume that u˙ (x, t) =

N

ai hi (x),

(4.4)

i=1

where · and are differentiation w.r.t. t and x respectively and ai s, i = 1, 2, . . . , N are Haar coefficients to be determined. Now integrating Eq. (4.4) once w.r.t. t from ts to t u (x, t) = (t − ts )

N

ai hi (x) + u (x, ts ),

(4.5)

i=1

where ts is the initial time and ∆t = t − ts is the time interval. Also integrating (4.5) w.r.t. x from 0 to x, we get

u (x, t) = ∆t

N

ai P hi (x) + u (x, ts ) + u (0, t) − u (0, ts )

(4.6)

i=1

and again integrating Eq. (4.6) w.r.t. x u(x, t) = ∆t

N

ai Qhi (x) + u(x, ts ) + u(0, t) − u(0, ts ) + x(u (0, t) − u (0, ts )).

i=1

(4.7) 1750026-5


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Put x = 1 in (4.7) and by using Eq. (4.3) (i.e. boundary conditions), we get u(1, t) = ∆t

N

ai Chi (x) + u(1, ts ) + u(0, t) − u(0, ts ) + (u (0, t) − u (0, ts ))

i=1

⇒ u (0, t) − u (0, ts ) = g1 (t) − ∆t

N

ai Chi (x) − g1 (ts ) − g0 (t) + g0 (ts ).

i=1

(4.8)


Substituting Eq. (4.8) in (4.7), then Eq. (4.7) becomes u(x, t) = ∆t

N

ai Qhi (x) + u(x, ts ) + g0 (t) − g0 (ts )

i=1

+ x g1 (t) − ∆t

N

ai Chi (x) − g1 (ts ) − g0 (t) + g0 (ts ) .

(4.9)

Differentiating Eq. (4.9) w. r. t. t then we have

N N u(x, ˙ t) = ai Qhi (x) + g˙ 0 (t) + x g˙ 1 (t) − ai Chi (x) − g˙ 0 (t) .

(4.10)

i=1

i=1

i=1

Substituting (4.5), (4.9), (4.10) in (4.1), we get

N N ai Qhi (x) + g˙ 0 (t) + x g˙ 1 (t) − ai Chi (x) − g˙ 0 (t) i=1

i=1

= α ∆t

N

ai hi (x) + u (x, ts )

i=1





N

ai Qhi (x) + u(x, ts ) + g0 (t) − g0 (ts ) ∆t     i=1  ×β

 + φ(x, t). N     + x g1 (t) − ∆t ai Chi (x) − g1 (ts ) − g0 (t) + g0 (ts ) i=1

(4.11) By solving (4.11) using MATLAB, we obtain the Haar wavelet coefficients ai s. Substituting these values of ai s in (4.9), we get the HWCM based numerical solution of the given problem (4.1). To improve the accuracy of the HWCM based solution, we interpolate obtained solution at the adaptive grid points as explained in Sec. 3, which gives the required AGHWCM solution. In order to know the accuracy of AGHWCM for the test problems, we use the error measure, i.e. maximum absolute error. The maximum absolute error will be calculated by Emax = max |u(x, t)e − u(x, t)a |, where u(x, t)e and u(x, t)a are exact and approximate solutions respectively. 1750026-6


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5. Test Problems In this section, we apply AGHWCM discussed in Sec. 4 to some of the parabolic PDEs. 5.1. Test problem Consider the nonhomogeneous parabolic PDE [14], ut = uxx + cos x

(5.1)

u(x, 0) = 0,

0≤x 0.

(5.3)

Solving (5.1) by the method explained in Sec. 4, we get the Haar wavelet coefficients ai s using MATLAB. Substituting these ai s in (4.9), we obtain the HWCM based numerical solution of the problem. To improve the accuracy of the HWCM solution, we interpolate obtained solution at the adaptive grid points as explained in method of solution, which gives the required AGHWCM solution. This solution is compared with the exact solution u(x, t) = cos x × (1 − e−t ) in Fig. 1 for N = 8 and the error analysis for higher values of N is given in Table 1 with ∆t = N1 .

0.12 AGHWCM EXACT 0.11

0.1

u


with initial condition:

0.09

0.08

0.07

0.06 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 1.

Comparison of AGHWCM with exact solutions for N = 8 of the test problem 5.1. 1750026-7


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S. C. Shiralashetti et al. Table 1.

Error analysis of the test problem 5.1.

N

Emax (FDM)

Emax (HWCM)

Emax (AGHWCM)

4 8 16 32 64

7.7262e−03 3.3537e−03 1.1947e−03 3.7099e−04 1.0431e−04

5.9886e−04 2.8294e−04 1.0919e−04 3.7447e−05 1.1756e−05

5.7025e−04 1.8637e−04 4.3689e−05 8.7566e−06 2.3859e−06

5.2. Test problem ut = uxx − 2u

(5.4)

with initial condition: 0≤x 0.

(5.6)

Solving (5.4) by the method explained in Sec. 4, we get the Haar wavelet coefficients ai s using MATLAB. Substituting these ai s in (4.9), we obtain the HWCM based numerical solution of the problem. To improve the accuracy of the HWCM solution, we interpolate obtained solution at the adaptive grid points as explained in method of solution, which gives the required AGHWCM solution. This solution is compared with the exact solution u(x, t) = sinh x × e−t in Fig. 2 for N = 16 and the error analysis for higher values of N is given in Table 2 with ∆t = N1 . 1.4 AGHWCM EXACT

1.2 1 0.8 u


Next consider homogeneous parabolic PDE [14],

0.6 0.4 0.2 0 0.1

0.2

0.3

0.4

0.6

0.5

0.7

0.8

0.9

1

x

Fig. 2.

Comparison of AGHWCM with exact solutions for N = 16 of the test problem 5.2. 1750026-8


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Numerical solution of parabolic partial differential equations Table 2.


N

Emax (FDM)

Emax (HWCM)

Emax (AGHWCM)

4 8 16 32 64

1.6398e−02 6.7120e−03 2.4811e−03 8.0987e−04 2.4493e−04

2.5704e−03 1.3042e−03 4.9747e−04 1.7017e−04 5.2762e−05

1.8419e−04 4.1012e−05 2.4393e−05 7.1017e−06 5.8251e−07

5.3. Test problem ut = uxx + u(1 − u)

(5.7)

with initial condition: u(x, 0) = 0,

0≤x 0.

(5.9)

Solving (5.7) by the method explained in Sec. 4, we get the Haar wavelet coefficients ai s using MATLAB. Substituting these ai s in (4.9), we obtain the HWCM based numerical solution of the problem. To improve the accuracy of the HWCM solution, we interpolate obtained solution at the adaptive grid points as explained in method of solution, which gives the required AGHWCM solution. This solution is compared xe−t with the exact solution u(x, t) = (1−x+xe −t ) in Fig. 3 for N = 32 and the error analysis for higher values of N is given in Table 3 with ∆t = N1 . 1.4 AGHWCM EXACT

1.2 1 0.8

u


Finally, consider the nonlinear parabolic PDE, i.e. Fisher’s equation [13],

0.6 0.4 0.2 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 3.

Comparison of AGHWCM with exact solutions for N = 32 test problem 5.3. 1750026-9


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S. C. Shiralashetti et al. Table 3.


N

Emax (FDM)

Emax (HWCM)

Emax (AGHWCM)

4 8 16 32 64

4.3528e−02 1.6847e−02 5.7329e−03 1.7228e−03 4.7047e−04

4.2103e−02 1.6056e−02 5.3416e−03 1.5918e−03 4.3929e−04

3.0191e−02 7.5575e−03 1.6333e−03 3.2302e−04 7.9248e−05


6. Conclusions In this paper, we applied the AGHWCM for the numerical solution of parabolic PDEs, which has been justified through the test problems. The numerical solutions are presented in figures and from which we observed that AGHWCM gives comparable results with exact solution. Subsequently error analysis is presented, this shows that the error is reduced by increasing the level of adaptive grids (i.e. for the higher values of N ), which demonstrate that the convergence of the AGHWCM is better than HWCM and FDM. Hence the proposed method is easy to implement and very effective for solving PDEs. References 1. N. M. Bujurke, S. C. Shiralashetti and C. S. Salimath, An application of single-term haar wavelet series in the solution of nonlinear oscillator equations, J. Comput. Appl. Math. 227 (2010) 234–244. 2. S. C. Shiralashetti, L. M. Angadi, A. B. Deshi and M. H. Kantli, Haar wavelet method for the numerical solution of Klein–Gordan equations, Asian-Eur. J. Math. 9(1) (2016) 14 1650012. 3. A. M. Khodier and A. Y. Hassan, One-dimensional adaptive grid generation, Int. J. Math. Math. Sci. 20(3) (1997) 577–584. 4. N. M. Bujurke, S. C. Shiralashetti and C. S. Salimath, Computation of eigenvalues and solutions of regular Sturm–Liouville problems using haar wavelets, J. Comput. Appl. Math. 219 (2008) 90–101. 5. U. Lepik, Numerical solution of evolution equations by the haar wavelet method, Appl. Math. Comput. 185 (2007) 695–704. 6. U. Lepik, Numerical solution of differential equations using haar wavelets, Math. Comput. Simulation 68 (2005) 127–143. 7. U. Lepik, Application of the haar wavelet transform to solving integral and differential equations, Proc. Est. Acad. Sci. Phys. Math. 56(1) (2007) 28–46. 8. O. V. Vasilyev, S. Palucci and M. Sen, A multilevel wavelet collocation method for solving partial differential equations in a finite domain, J. Comput. Phys. 120 (1996) 33–47. 9. N. M. Bujurke, C. S. Salimath and S. C. Shiralashetti, Numerical solution of stiff systems from nonlinear dynamics using single-term haar wavelet series, Nonlinear Dynam. 51 (2008) 595–605. 10. S. Bertoluzza and G. Naldi, A wavelet collocation method for the numerical solution of partial differential equations, Appl. Comput. Harmon. Anal. 3 (1996) 1–9. 11. H. Wenyu, W. Rougsheng and F. Juan, The adaptive wavelet collocation method and its application in front simulation, Adv. Atmos. Sci. 27(3) (2010) 594–604. 1750026-10


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12. C. F. Chen and C. H. Hsiao, Haar wavelet method for solving lumped and distributedparameter systems, IEEE Proc. Control Theory Appl. 144(1) (1997) 87–94. 13. M. Matinfar and M. Ghanbari, Solving the Fisher’s equation by means of variational iteration method, Int. J. Contemp. Math. Sci. 4 (2009) 343–348. 14. A. M. Wazwaz, The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Comput. Math. Appl. 54 (2007) 933–939. 15. O. V. Vasilyev and S. Palucci, A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain, J. Comput. Phys. 125 (1996) 498–512.

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