A numerical algorithm for the solution of telegraph equations

Applied Mathematics and Computation 190 (2007) 757–764 www.elsevier.com/locate/amc

A numerical algorithm for the solution of telegraph equations M.S. El-Azab *, Mohamed El-Gamel Mathematical and Physical Sciences Department, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt

Abstract In this paper, we present a new competitive numerical scheme to solve nonlinear telegraph equations. The method is based on Rothe’s approximation in time discretization and on the in the spatial discretization. The  Wavelet–Galerkin  approximate solutions converge in the space C ðð0; T Þ; L2 ðXÞÞ \ L2 ð0; T Þ; W 1;2 0 ðXÞ to the variational solution. A full error analysis is performed and a numerical experiment is given to illustrate the good convergence behavior of the approximate solution. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Hyperbolic partial differential equations; Rothe–Wavelet–Galerkin method; Error analysis

1. Introduction In the present work we are dealing with the numerical approximation of the following second order hyperbolic problem: o2 u ou þ þ Au ¼ f ðx; t; uÞ in Q  X  I; ot2 ot uðx; 0Þ ¼ u0 ðxÞ; ut ðx; 0Þ ¼ u1 ðxÞ in X; u¼0

ð1:1Þ ð1:2Þ

on R  oX  I: N

ð1:3Þ 0;1

Here X  R ðN ¼ 1; 2; 3Þ is a simply connected bounded domain with boundary oX 2 C , I  ð0; T Þ is a time interval and uðx; tÞ denotes the dependent variable representing the magnitude that is modeled. We introduce the elliptic differential operator A defined by Au :¼ r  ðAðxÞruÞ þ aðxÞu;

ð1:4Þ

where $ and $Æ denote the gradient and divergence operators, respectively and A(x) is a symmetric matrix with entries that are uniformly bounded and measurable. The functions f ; u0 ; u1 and A(x) are given data for the problem and will be assumed as regular as necessary.

*

Corresponding author. E-mail address: [email protected] (M.S. El-Azab).

0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.091

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Eq. (1.1), referred to as nonlinear telegraph equation in N-dimensional space with constant coefficients, models mixture between diffusion and wave propagation by introducing a term that accounts for effects of finite velocity to standard heat or mass transport equation. However, problem (1.1)–(1.4) is commonly used in signal analysis for transmission and propagation of electrical signals [10] and also has applications in other fields (see [14] and the references therein). In this study, we describe how wavelets may be used, in slightly different manner, for the discretization of telegraph equation. The wavelet method is similar to the finite element method but the weighting and trial functions are the Daubechies scaling functions. These functions are compactly supported and form an orthonormal basis for L2(R), and by tensor products, L2 ðRN Þ. Since these functions combine orthogonality and scaling properties it has been a natural idea to solutions to partial differential equations [2,4,5,16]. In these papers the differential equations are solved by projecting the solution onto the wavelet space to produce a system of ordinary differential equations. A major problem associated with the use of wavelets in time-dependent problems is that for very large scalar engineering problems, the resulting system of ordinary differential equations requires more computational effort to be solved. A second problem is the need, in many cases, for a nonuniform subdivision of the whole space-time domain in order to obtain satisfied degree of accuracy. A third problem arises in all packages of wavelets when using interpolation processes with the source term to express it as a product of its independent variables. For instance, in the linear case the term f ðx; tÞ, if it is not product of two functions, is approximated by a polynomial P ðx; tÞ in the form [15]: 2 30 1 c0;1    c0;p1 c0;0 1 6 c1;0 B C p1 X q1 c1;1    c1;p1 7 X 6 7B x C 7B . C f ðx; tÞ  P ðx; tÞ ð1:5Þ cr;s xr ts ¼ ð 1 t    tq1 Þ6 .. 6 .. 7B . C 4 . 5@ . A . r¼0 s¼0 xp1 cq1;0 cq1;1    cq1;p1 and so, another program is used to evaluate the unknown matrix. One of the suggested choices to address the aforementioned problems is to use Rothe’s method [7,13] to approximate the time dependent problem by a sequence of linear elliptic problems. Each time step results in an elliptic boundary value problem, which can be solved effectively by a Wavelet–Galerkin method without needing to use a further interpolation process in the time variable. By this means, we are able to develop a fast rate of convergent numerical scheme of order Oðs þ s2 þ 2J ðLþ1Þ Þ where j denotes the level of resolution of wavelet approximation, L is the genus number of the used Daubechies wavelets and s is the time step. Furthermore, an appropriate spatial mesh is introduced for each individual time-step. The outline of this paper is organized as follows: In Section 2 we present the general framework of the continuous problem. Section 3 is devoted to describe the Daubechies wavelet functions that are used as a basis for the solution space. In Section 4 a full discretization scheme of problem (1.1)–(1.4) is presented. We present some numerical results in Section 5 and give a conclusion in Section 6. 2. The continuous problem We will use the standard notations of functional analysis (see e.g. [8]) for the spaces L2 ðXÞ and V  W 1;2 0 ðXÞ. Moreover, (Æ,Æ) is used to define the inner product in L2 ðXÞ and the norms on L2 ðXÞ and V will be denoted by j  j and k  k, respectively. All the constants which occur in the course of this paper will be denoted by C (e is small and C e ¼ Cðe1 Þ). Finally, in order to define a generalized solution of problem (1.1)–(1.4) we introduce the bilinear form ð; ÞV  V corresponding to the differential operator A, which is given by ððu; vÞÞ ¼ ðAru; rvÞ þ ðau; vÞ 8u; v 2 V : ð2:1Þ For the sake of simplicity, we shall assume throughout this work the following hypotheses: (H1) f : X  I  R ! R is Lipschitz continuous in the sense jf ðx; t; sÞ  f ðx; t0 ; s0 Þj 6 cfjt  t0 jðjsj þ js0 jÞ þ js  s0 jg; 8t; t0 2 I; 8s; s0 2 R and satisfies the growth condition jf ðx; t; nÞj 6 Cð1 þ jnjÞ

8ðx; t; nÞ 2 X  I  R:

ð2:2Þ ð2:3Þ

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759

(H2) A(x) is symmetric and uniformly positive definite matrix in Q, i.e. 2

ðAn; nÞ P Cjnj :

ð2:4Þ

(H3) The coefficients A(x) is chosen in such a manner that the bilinear form (Æ) is symmetric, bounded and V-elliptic, i.e., jððu; vÞÞj 6 ckukkvk;

ððu; uÞÞ P ckuk2

8u; v 2 V :

ð2:5Þ

(H4) u0 2 V ;

u1 2 L2 ðXÞ:

Under these assumptions, we can now state the variational form of the telegraph problem (1.1)–(1.4), i.e., Problem (P) Find u : Q ! R such that u 2 H 1 ðI; L2 ðXÞÞ \ L2 ðI; V Þ and satisfies the initial conditions uðx; 0Þ ¼ u0 ðxÞ and ut ðx; 0Þ ¼ u1 ðxÞ as well as the integral relation Z Z Z Z  ðot u; ot vÞ  ðu1 ; vð; 0ÞÞ þ ðot u; vÞ þ ððu; vÞÞ ¼ ðf ; vÞ ð2:6Þ I

I

I

I

1

for all v 2 H ðI; L2 ðXÞÞ \ L2 ðI; V Þ with vðx; T Þ ¼ 0. The exact traveling wave solution for this problem is evaluated in [1] for the case f ðuÞ ¼ uða  uÞð1  uÞ; 0 6 a 6 1 (Nagumo telegraph equation). Generally, problem (P) with the assumed hypotheses admits a unique traveling wave solution (see [7, Theorem 4.1.7, p. 95]). 3. Daubechies wavelet bases and connection coefficients Our Galerkin procedure uses a class of compactly supported scaling functions introduced by Daubechies [3]. The scaling functions are determined by a genus index L and a set of filter coefficients P i ; i ¼ 0; 1; . . . ; L  1 that define the generator function u(x) through the scaling relation uðxÞ ¼

L1 X

P i uð2x  iÞ:

ð3:1Þ

i¼0

The function u(x) has a companion, the wavelet function w(x) which is derived in terms of u(x) from the second scaling relation wðxÞ ¼

1 X

i

ð1Þ P 1i uð2x  iÞ:

ð3:2Þ

i¼2L

However, the filter Daubechies wavelet coefficients P i ; i ¼ 0; 1; . . . ; L  1 appeared in (3.1) and (3.2) for L ¼ 4; 6; 8; 10 are listed in [6]. For each J P 0, the dilations and translations of u(x) and w(x) are defined by uJk ðxÞ ¼ 2J =2 uð2J x  kÞ; wJk ðxÞ ¼ 2

J =2

J

wð2 x  kÞ;

J ; k 2 Z;

ð3:3Þ

J ; k 2 Z:

ð3:4Þ

The support of both of the scaling function u(x) and the corresponding wavelet w(x) is the interval ½0; L  1, while the support of uJk ðxÞ and wJk ðxÞ is the interval ½k2J ; ðk þ L  1Þ2J . We define the space V J  spanfuJk ; J > 0; k 2 Zg and WJ, its orthogonal complement in V J þ1 , by J W  spanfwJk ; J > 0; k 2 Zg. Every polynomial of degree 6 L=2 lies in the space V0, which is equivalent to w(x) having L/2 vanishing moments. The Daubechies class is distinguished by having this interpolation property and the smallest possible support. Thus, from the interpolation property, we see that u(x) has at least L/2 continuous derivatives.

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For the Galerkin approximation of partial differential equations, the field variables are projected into the space of trial functions belonging to VJ. When we use test functions from the same space, a system of differential equations in time for the coefficients of the field variable results when the inner products are evaluated and orthogonality among the elements of VJ is applied. In the case of wavelet bases, the inner products of the bases functions and their derivatives and translates are called the connection coefficients. However, the numerical approximation of these connection coefficients is unstable since the integrals are highly oscillatory. To overcome this difficulty, specific algorithms have been devised by [9,11] for the exact evaluation of them. Since the scaling functions and wavelets do not have explicit analytic expressions but are implicitly determined by the two scale relations (3.1) and (3.2), it is necessary to develop algorithm to compute several connection coefficients which occur in the application of the Wavelet– Galerkin procedure to differential and integral equations. For solving telegraph equations, we need to calculate the following two connection coefficients: M mJ;k ¼ ðxm ; uJk Þ; ;d 2 CdJ 1;k;l ¼

ð3:5Þ

ðd Þ ðd Þ ðuJk1 ; uJl 2 Þ:

ð3:6Þ

4. The suggested numerical scheme This Section is devoted to propose a new competitive numerical scheme based on Rothe’ s method in time discretization and on Wavelet–Galerkin method in spatial discretization. To this purpose, let us first introduce some notations and basic assumptions related to the discretizations. 4.1. Discretization in time We subdivide the time interval I by points ti ¼ is, s ¼ T =n, i ¼ 0; 1; 2; . . . ; n, where N is a positive integer. Let w1 and w0 be defined as w1 ðxÞ ¼ u0 ðxÞ  su1 ðxÞ and w0 ðxÞ ¼ u0 ðxÞ. A first discretization of problem (P) consists of the following problem. Problem ðP s Þ Find wi ffi uð; ti Þ 2 V ;

i ¼ 1; 2; . . . ; n such that the equations

ðd2 wi ; vÞ þ ðdwi ; vÞ þ ððwi ; vÞÞ ¼ ðfi ; vÞ

8v 2 V

ð4:1Þ

are satisfied. Here we use the notations dwi ¼

wi  wi1 ; s

d2 wi ¼

dwi  dwi1 ; s

f i ¼ f ðx; ti ; wi1 Þ:

Our assumptions permit a successive application of the Lax-Milgram theorem to the elliptic boundary value problem (4.1). This yields. Lemma 4.1. The discretized problem (4.1) is solvable. The Rothe function un : I ! V , intended to be an approximation of u, is introduced by un ðtÞ ¼ wi1 þ ðt  ti1 Þdwi ; n

2

8t 2 ½ti1 ; ti ; 1 6 i 6 n;

du ðtÞ ¼ dwi1 þ ðt  ti1 Þd wi ;

8t 2 ½ti1 ; ti ; 1 6 i 6 n:

Besides we will need to introduce the step functions  wi t 2 ½ti1 ; ti ; 1 6 i 6 n;  un ðtÞ ¼ u0 t 2 ½s; 0;  fi t 2 ½ti1 ; ti ; 1 6 i 6 n; f n ðtÞ ¼ f0 t ¼ 0:

ð4:2Þ ð4:3Þ

ð4:4Þ ð4:5Þ

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761

Let t be an arbitrary, but fixed element of ð0; T Þ and that v 2 H 1 ðI; L2 ðXÞÞ \ L2 ðI; V Þ, vðx; T Þ ¼ 0, the discretized problem (4.1) may be written in the form ðot dun ; vÞ þ ðot un ; vÞ þ ðð un ; vÞÞ ¼ ðf n ; vÞ:

ð4:6Þ

The integration of this equation over I yields Z Z Z Z n n n n  ðot u ; ot vÞ  ðdu ð0Þ; vð; 0ÞÞ þ ðot u ; vÞ þ ððu ; vÞÞ ¼ ðf n ; vÞ; I

I

I

ð4:7Þ

I

for all v 2 H 1 ðI; L2 ðXÞÞ \ L2 ðI; V Þ with vðx; T Þ ¼ 0. To prove the stability of the approximation scheme (4.1) we derive some a priori estimates for wi and dwi. In doing this, we use the following elementary relations: 2ab 6 ea2 þ C e b2 ; 2

ð4:8Þ 2

2

2aða  bÞ ¼ a  b þ ða  bÞ :

ð4:9Þ

Lemma 4.2. The estimates s X 2 2 jdws j 6 C; jdwi  dwi1 j 6 C; i¼1

s X

2

sjdwi j 6 C;

2

kws k 6 C;

i¼1

s X

2

kwi  wi1 k 6 C

ð4:10Þ

i¼1

hold uniformly for s. Proof. Take v ¼ sdwi as a test function in (4.1), and sum it over i ¼ 1; 2; . . . ; s, we obtain by the use of (2.2), (2.3), (4.8) and (4.9) s s s X X X 2 2 2 2 2 jdwi  dwi1 j þ sjdwi j þ kws k þ kwi  wi1 k jdws j þ i¼1

6Cþe

s X

i¼1 2

sjfi j þ C e

i¼1

s X

i¼1 2

sjdwi j 6 e 1 þ

i¼1

s X i1 X i¼1

r¼1

! 2 2

jdwi j s

þ Ce

s X

2

sjdwi j :

ð4:11Þ

i¼1

Choosing e sufficiently small and applying Gronwall’s lemma (see, e.g. [12]), we conclude the proof.

h

Corollary 4.1. There exists a constant C such that ot un kL2 ðI;L2 ðXÞÞ 6 C;

k un kL2 ðI;V Þ 6 C;

C ; n C C 2 2 un   un kL2 ðI;L2 ðXÞÞ 6 2 ; kun   uns kL2 ðI;L2 ðXÞÞ 6 2 ; n n C 2 dun  ot un kL2 ðI;L2 ðXÞÞ 6 ; n 2

kun   un kL2 ðI;V Þ 6

ð4:12Þ ð4:13Þ ð4:14Þ ð4:15Þ

ns ¼  where u un ð; t  sÞ. Proof. All the estimates of this corollary are a consequence of Lemma 4.2 and the definitions of un, un and dun . The estimates (4.12)1 and (4.14) are a consequence of (4.10)3 and the definition of uns , whereas (4.12)2 is a consequence of (4.10)4. The estimates (4.10)5 and (4.10)2 imply (4.13) and (4.15), respectively. Denoting by eu ¼ u  un and ef ¼ f  f n , we can prove the following theorem. h Theorem 4.1. Under the assumptions (H1)–(H4) we have 2

2

keu kCðI;L2 ðXÞÞ þ keu kL2 ðI;V Þ 6 Cðs2 þ sÞ:

ð4:16Þ

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Proof. Subtracting (4.7) from (2.6) and using v ¼ eu ðtÞ as a test function we obtain Z Z Z Z 1 1 kun  un k2 þ keu k2 þ Cs: jeu j2 þ keu k2 6 ekef k2L2 ðI;L2 ðXÞÞ þ C e jeu j2 þ 2 I 2 I I I Now we consider 2 kef kL2 ðI;L2 ðXÞÞ

Z

2

2

ð4:17Þ

2

ðjf ðt; uÞ  f ðt; un Þj þ jf ðt; un Þ  f ðt; uns Þj þ jf ðt; uns Þ  f ðti ; uns Þj Þ

6 I

Z 6

2

2

ðjeu j þ jun   uns j þ s2 Þ:

ð4:18Þ

I

Substituting from (4.18) in (4.17), using Corollary 4.1, choosing e sufficiently small and applying Gronwall’s lemma the proof completes. h 4.2. Discretization in space In this section, we will solve problem (Ps) by using the Wavelet–Galerkin method. To do that we project its real variables to the finite dimensional space V J  V such that V J ! V for J ! 1 in a canonical sense, i.e., 8v 2 V J 9vJ 2 V J such that vJ ! v for J ! 1 in V. In what follows, we project the semi-discrete real variable h, say, into VJ such that: ~ h¼

J 2X 1

ð4:19Þ

hJk uJk ;

k¼2L

where uJk are of resolution 2J and genus L. Here the integer j is used to control the smoothness of the solution. The larger integer j is used, the more accurate a solution can be obtained. Moreover, for h 2 C P jh  ~ hj 6 C2J ðP þ1=2Þ ; ð4:20Þ where C is a constant independent upon the space VJ [16]. Now we consider the spatial discretization of problem (Ps) using the Wavelet–Galerkin method. Problem (P Js ) ~i ¼ Find w

P2J 1 j¼1

wji 2 V J ;

i ¼ 1; 2; . . . ; n0 such that

~ i ; vÞ þ ðd~ wi ; vÞ þ ðð~ wi ; vÞÞ ¼ ðf~ i ; vÞ 8v 2 V J ; ðd2 w J ~ 0 ¼ P ðu0 Þ: w

ð4:21Þ ð4:22Þ

By the same argument as in Section 4.1 we obtain the existence of wij for j ¼ 1; 2; . . . ; 2J  1 and a priori estimates (see Lemma 4.2) j j j X X X 2 2 2 2 2 jdwjs j þ kdwji  dwji1 k þ skdwji j þ kwjs k þ kwji  wji1 k 6 C; ð4:23Þ i¼1

i¼1

i¼1

which hold uniformly for s and j. Let wj0 2 V J such that wj0 ! u0 for j ! 1. By means of wij, i ¼ 1; 2; . . . ; n; j ¼ 1; 2; . . . ; 2J  1, we define the functions ua and  ua , a ¼ ðs; 2J Þ, in the same way as the functions un and un have been constructed by means of j wi . We are now in a position to formulate the main contribution of this work by combining (4.16) and (4.20) in the following theorem. Theorem 4.2. Let the assumptions (H1)–(H4) be satisfied and for J 2 N , P J ðu0 Þ ¼ V J such that P J ðu0 Þ ! u0 in V for J ! 1. Then ua ! u in u 2 H 1 ðI; L2 ðXÞÞ \ L2 ðI; V Þ for a ! 0 where u is the variational solution of problem (P) and ua is constructed from the solution of problem ðP Js Þ. Moreover, we have the error estimates 2

ku  ua kL2 ðI;L2 ðXÞÞ 6 Cðs þ s2 þ 2J ðLþ1Þ Þ; where j is the resolution level and L is the genus number of the used Daubechies wavelets.

ð4:24Þ

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763

5. Numerical results Here we shall present the results of a simple numerical experiments with (1.1)–(1.4). We have only examined the case when X is an interval (Our assumptions for RN hold for R1). So we take X  ½0; 1, AðxÞ ¼ aðxÞ ¼ 1, u0 ¼ u1 ¼ 0 and we proceed as in Section 4.2. The Wavelet–Galerkin solution of ðP Js Þ can be derived as follows: Let ~i ¼ w

J 2X 1

wiJ ;k uJk ðxÞ;

ð5:1Þ

k¼2L

and substitute with this series in (4.21) with v ¼ uJl ðxÞ; l ¼ 1; 2; . . . ; 2J  1 as a test function we obtain the linear system of algebraic equations J 2X 1

0;0 2 wiþ1 J ;k CJ ;k;l  ð2  s  s Þ

k¼2L

J 2X 1

k¼2L

wiJ ;k C0;0 J ;k;l þ ð1  sÞ

J 2X 1

0;0 2 wJi1 ;k CJ ;k;l þ s

k¼2L

J 2X 1

2 m wiJ ;k C1;1 J ;k;l ¼ s M J ;k :

ð5:2Þ

k¼2L

The initial data for this scheme is w0J ;k ¼ P J ðu0 Þ;

ð5:3Þ

w1J ;k

ð5:4Þ

J

¼ P ðu0  su1 Þ:

Eq. (5.3) is a linear system of ð2J þ L  2Þ equations in ð2J þ L  2Þ unknown coefficients. This system may be easily solved at each time step by a variety of methods. In this paper we use Q–R method to obtain wiJ ;k and thus we can formulate the approximate solution of the telegraph equation by the use of Rothe–Wavelet method. The case study reported in this section was selected from a large collection of problems to which Rothe– Wavelet method could be applied. For purposes of comparison, contrast and performance, an example with known solutions was chosen. In the Rothe–Wavelet solutions, Daubechies 6 wavelets are used because they give better results than those of lower degree wavelets the computations were run for a sequence of J ¼ 2; 3; 4; 5; 6; 7; 9. We use absolute error which is defined as ~ i j: Absolute error ¼ kuexact  w Case study Solving problem (1.1)–(1.4) with the suggested numerical scheme taking f ðx; t; uÞ ¼ 2et ðx  x2 þ t2 Þþ ðt  2Þu and the initial and the Dirichlit boundary conditions specified so that the exact solution is uðx; tÞ ¼ ðx  x2 Þt2 et . The following table gives the comparison of the Rothe–Wavelet and exact solutions at time t = 1. In the numerical computations, we used the Daubechies scaling function with L = 6, J = 9. x

Exact solution

Rothe–Wavelet solution

Absolute error 1.0E04

0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000

0.00 0.04023 0.06897 0.08622 0.09196 0.08622 0.06897 0.04023 0.00

0.00 0.04021 0.06895 0.08620 0.09196 0.08622 0.06899 0.04026 0.00

0.00 0.15 0.22 0.23 0.00 0.00 0.24 0.36 0.00

All the computations were performed by using MATLAB.

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6. Conclusion In this paper we have investigated the application of Rothe–Wavelet method to the solution of telegraph equations. We have used the family of compactly supported orthonormal wavelets constructed recently by Daubechies as the solution bases for the reason that it offers several properties. As it has been shown in the paper, the proposed method converts the problem of solving partial differential equation to solving a linear system, for unknown wavelet series coefficients. The numerical example given in this work has demonstrated the potential of this method in solving telegraph equations and similar hyperbolic partial differential equations. References [1] H.A. Abdusalam, Analytic and approximate solutions for Nagumo telegraph reaction diffusion equation, Appl. Math. Comput. 157 (2004) 515–522. [2] K. Amaratunga, J.R. Wiliams, S. Qian, J. Weiss, Wavelet–Galerkin solution for one-dimensional partial differential equations, Int. J. Numer. Meth. Eng. 37 (1994) 2705–2716. [3] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [4] R. Glowiniski, W.M. Lawton, M. Ravachol, E. Tenenbaum, Wavelet solutions of linear and nonlinear elliptic, parabolic, and hyperbolic problems in one space dimension, Comput. Meth. Appl. Sci. Eng. (1990) 55–120 (Chapter 1). [5] S. L Ho, S.Y. Yang, Wavelet–Galerkin method for solving parabolic equations in finite domains, Finite Element Anal. Des. 37 (2001) 1023–1037. [6] F. Jin, T.Q. Ye, Instability analysis of prismatic members by Wavelet–Galerkin method, Adv. Eng. Software 30 (1999) 361–367. [7] J. Kacˇur, Method of Rothe in evolution equations, Teubner-Texte zur Mathematik, BSB Teubner Verlagsges, Leipzig, 1985. [8] A. Kufner, O. John, S. Fucˇik, Function Spaces, Nordhoff, Leyden, 1997. [9] A. Latto, H.L. Resniko, E. Tenenbaum, The evaluation of connection coefficients of compactly supported wavelets, in: Proc. French– USA Workshop on Wavelets and Turbulence, Princeton Univ., June 1991, Springer, New York, 1992. [10] A.C. Metaxas, R.J. Meredith, Industrial Microwave, Heating, Peter Peregrinus, London, 1993. [11] C. Ming-quayer et al., The computation of Wavelet–Galerkin approximation on a bounded interval, Int. J. Numer. Meth. Eng. 39 (1996) 2921–2944. [12] V. Pluschke Rothe’s method for parabolic problems with nonlinear degenerating coefficient, Report No. 14, des FB Mathematik und Informatik, 1996. [13] K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Reidel Publ. Comp., Dortrecht-BostonLondon, 1982. [14] G. Roussy, J.A. Pearcy, Foundations and Industrial Applications of Microwaves and Radio Frequency Fields, John Wiley, New York, 1995. [15] R. Vudu, U.C. Barkeley, A wavelet collection methods for solving partial differential equations, Mathematical report Matyh. 228B, 2001. [16] J.R. Williams, K. Amaratunga, Introduction to Wavelets in engineering, Int. J. Numer. Meth. Eng. 37 (1994) 2365–2388.