An Integral Equation with Fixed Singularity

Integral equations occur naturally in many areas of mathematical physics. Many engineering and applied science problems arising in water waves, potential theory and electrostatics are reduced to solving integral equations. The problem of finding out the crack energy and distribution of stress in the vicinity of a cruciform crack leads to the integral equation

where

This is an integral equation of some special type since the kernel Kp(x,t) has singularity at (0,0) only. f{x) is a prescribed function relating to the internal pressure given by

Since the cracks are in the shape of a cross, the problem is known as the cruciform crack problem. Of interest here, is the stress intensity factor u(l) which is directly proportional to stress intensity at the crack tip. How the integral equation (5.0.0.1) occurs in the problem of cruciform crack is explained by Stallybrass (Stallybrass, 1970) who solved the integral equation in a closed form using Wiener-Hopf technique and provided the numerical results for the stress intensity factor. Rooke and Sneddon (Rooke and Sneddon, 1969) solved this integral equation approximately by using an expansion in terms of Legendre functions and obtained numerical results which are very close to those of Stallybrass (Stallybrass, 1970) although the convergence is slow. The two methods appear to be somewhat elaborate. The integral equation (5.0.0.1) has also been solved numerically by various other methods from time to time. For example, Elliot (Elliott, 1997) employed the method of Sigmoidal transformation to obtain approximate solution for the case f(x) = 1. It is not obvious if this method is useful for other forms of f(x). Tang and Li (Tang and Li, 2008) solved the integral equation approximately by employing Taylor series expansion for the unknown function and obtained very accurate numerical estimates for the stress intensity factor. They made use of Cramer’s rule in the mathematical analysis so that if one increases the number of terms in the approximation the calculation become unwieldy so as to make the method unattractive. Bhattacharya and Mandal (Bhattacharya and Mandal, 2010) solved the integral equation approximately by two different methods, one is based on expansion of the unknown function in terms of Bernstein polynomials and the other is based on expansion in terms of rationalized Haar functions. Singh and Mandal (Singh and Mandal, 2015) also solved it by using Legendre multiwavelets. All these methods provide numerical results for u(l) which are very close to exact results given by Stallybrass (Stallybrass, 1970). Expansion in terms of Bernstein polynomials, or Haar functions or Legendre multiwavelets suggest expansion in terms of other functions such as Daubechies scale functions since these provide a somewhat new tool in the numerical solution of integral equations. This has been carried out by Mouley et al. (Mouley et ah, 2019) and is given below.

Method Based on Scale Functions in Daubechies Family

Here, Daubechies scale functions are employed to expand the unknown function u(x). Daubechies scale function with К vanishing moments of their wavelets is employed to find approximate solution of integral equation taking К = 3. It may be noted that К = 1 corresponds to Haar wavelets. As the result can be improved taking larger values of K. so the results obtained by using Daubechies scale function are better than the results using the rationlized Haar functions. Though Legendre multiwavelet gives satisfactory results, Daubechies scale function with К vanishing moments of their wavelets (DauK) has some interesting features like compact support, fractal nature and no explicit form at all resolutions. Only the knowledge of the low pass filter coefficients in two scale relation is required throughout the calculation. For these reasons. Daubechies scale function is used as an efficient and new mathematical tool to solve integral equations. At x = 1, the expansion of u(x) reduces to a finite expansion because most of Daubechies scale functions vanish. Actually the integral equation (5.0.0.1) produces a system of linear equations in the unknown coefficients. After solving this linear system, the unknown function u(x) is evaluated at x = 1 so as to obtain numerically the value of the stress intensity factor. For different values of a in the expression of internal pressure f(x) given by (5.0.0.3), tt(l) is obtained and compared to known results available in the literature. It is found that the method is quite accurate as the approximate values of it(l) obtained by the present method are seen to differ negligibly from exact values.

Basic properties of Daubechies scale function and wavelets

Daubechies discovered a whole new class of compactly supported orthogonal wavelets, which is generated from a single function ф(х), known as Daubechies scale or refinable function. DauK scale function (К > 1) has 2К filter coefficients and has compact support [0. 2 К — 1]. The two scale relations of scale functions and the definition of mother wavelet functions are given by

and

where the low pass filter coefficients 1ц and the high pass filter coefficients gt are related by

A set of orthonormal basis in R is generated from the single scaling function ф(х) using the repeated application of two operators, translation operator T and the scale transformation D, defined as

and

The scaling coefficients hi (I = 0.1. 2,........., 2К — 1) for /v-Daubechies scale function are determined

using conditions (5.1.1.6), (5.1.1.7) and (5.1.1.8) given below.

Vanishing moment:

Orthonormal condition:

Normalization condition:

It is evident that all the properties of scaling function described above are applicable on R , some of which, viz., translation property (5.1.1.4) is not applicable on [o, 6] C R, where a and b are integers. In order to apply the machinery of A'-Daubechies scale function on a finite interval [a, 6], we have to modify the properties of scale function Ф_яп°' ^R for 2^sa — (2К — 2) < n < 2^sa — 1 and 2^sb — (2K — 2) < n < 2^sb — 1, whose supports overlap partially with the finite interval [a, 6]. In their explicit forms

Here X[_a.b] is the characteristic function for the set [а, Ь]. The superscripts L and R stand for the overlaps of support ip with left and right edges of the domain [a, fe] respectively. By ф[_п we mean the interior scale function ф_яп (x) whose supports are contained in [a, 6].

Method of solution

Here the interval [a, 6] is [0,1]. To find the approximate solution of Eq. (5.0.0.1), we approximate the unknown function u(x) defined in [0.1] in terms of Daubechies scale functions in the form

Here ij>_sn{x) is defined in (5.1.1.5) with a sufficiently fine scale s.

Using (5.1.2.1) in the integral equation (5.0.0.1), we obtain

Using the orthonormal condition of 4>_sm for different values of m, the equation (5.1.2.2) is reduced to the form

with

Here the range of m and n depends on scales s and K. As the support of ф (x) is [0, 2К — 1] and K(> 1) is a positive integer so m, n € [—(2К — 2), 2^s — 1].

After solving the system (5.1.2.3) the unknown constants c_sn are obtained. Now from (5.1.2.1) we get

As the support of ф (x) is [0, 2K — 1], the values of n in (5.1.2.7) satisfy the range 1—2A'+2^S < n < 2^s. If we take Dau3 scale function, we need values of ф{р) for p = 5,4,3,2,1,0. The detailed tricks for calculation of integrals appearing in (5.1.2.4), (5.1.2.5) and (5.1.2.6) are described below.

Using the explicit form of f(x) in (5.0.0.3) and the two scale relation (5.1.1.1), the expression in (5.1.2.4) reduces to the form

Now using the N-point Gauss-type quadrature rule with complex nodes and weights for integrals involving Daubechies scale function (cf. Panja and Mandal (Panja and Mandal, 2015)), we obtain

where

The determination of the nodes Xi and weights Wj are described by Panja and Mandal (Panja and Mandal, 2015).

The basic trick for the calculation of the integral in (5.1.2.5) is described by Kessler et al. (Kessler et ah, 2003b), Panja and Mandal (Panja and Mandal, 2012). If ф(х) is the scale function with compact support [0,2К — 1] (К > 1) then it produces a system of orthonormal basis ф_зп given by (5.1.1.5) in E. From (5.1.2.5) we get

If m,n = — (2K — 2),—{2K — 3),......., — 1 we denote N_rnn by N^_n and ф_т by ф^. Again if

m, n = 2^s — (2K — 2),2^S — (2К — 3),...., 2^s — 1 we denote N_mn by N^_n and ф_т by фThe two-scale relation (5.1.1.1) produces

From (5.1.2.11) one gets the recursion relation

Here hi (l = 0,1, 2,.........2К — 1) are the low pass filters . The recursion relation (5.1.2.13) together

with Nfr_in = 0 when morn< —(2K — 1) or | m — n |> 2К — 1 and Nf*_n = 0 when in or n > 2^s or

| m — n |> 2К — 1 gives all the values of N^°^T for m.n = —(2К — 2), —(2K — 3),......., —1 and

2^s — (2K — 2), 2^s — (2K — 3),......, 2^s — 1. Here the superscripts L and R stand for left edge and right

edge of [0,1] respectively. Again if 0 < m, n < 2^s — (2К — 1), we mean N_mn by N_mn = Nj_nn = 8_mn. For (К = 3)-Daubechies scale function, the numerical values of N^°^{r R} and the corresponding values of the inverse of the matrix formed by the elements A^r,('_lr“^{r R} are tabulated in different tables by Panja and Mandal (Panja and Mandal, 2012).

Now the calculation of integral in (5.1.2.6) is described. Using the definition (5.1.1.5) in the right side of (5.1.2.6) we obtain

where I*_{n n} is given by the relation

Due to the finite support of ([0, 2A' — 1] here), the integral _n can be put in the form

for m,n < 2^s — 2K + 1. With the help of two scale relation (5.1.2.12), the expression for /,), _n in Eq. (5.1.2.15) becomes

where

Use of the similar transformations in (5.1.2.16) provides an equivalent relation

Whenever the kernel Kf(x, t) in the integrals present in (5.1.2.15) and (5.1.2.16) is regular in the support of m(x)4>n(t) and 4>(x)(t) respectively, their approximate values can be evaluated efficiently with the aid of Gauss quadrature rule

involving the Daubechies scale function discussed in Chapter 3.

For — (2К — 2) < m.n < 0, both the integrals I,_{n n} in (5.1.2.15) and /_{m n} in (5.1.2.16) have singularity at (0,0). For these values of m.n the values of Ц_пп cannot be determined using the formula (5.1.2.20). Instead, values of these integrals are obtained by solving a system of equations. For the derivation of such equations, it is convenient to recast the recursion relation for Ц_{п n} as

In this relation the integrals in the first term in the right hand side are singular and the remaining three have no singularities. So, their values can be evaluated with the aid of suitable quadrature rules. There are two different types of integrals present in the right hand side. Some of these integrals contain interior scale functions and ip(t) and other integrals contain one scale function having partial support, e.g., [1, 2К — 1], • • •, [2К — 2,2К — 1] within the the domain of integration. For the evaluation of these integrals we have used appropriate nodes and weights available in Table 6.8

of (Panja and Mandal, 2015). In case of 2^s — 2К + 2 < m, or n the integrand in the integral /f_nn or I_m,n are well behaved. Their values have been calculated by using quadrature rules with nodes and weights corresponding to the scale function with partial supports [0,1], • • •, [0, 2K — 2] available in the same table. Using the Gauss quadrature rule involving the Daubechies scale function, the integral in (5.1.2.21) is reduced to the form

Here hi_} for j = 1.2 (lj = 0,1,2.......,2К — 1) are the low-pass filters. Basic trick for calculating

weights w_i. , for j = 1,2 and nodes x_it, x_i} and t_i2 , f^with a program in MATHEMATIC A has been discussed by Panja and Mandal (Panja and Mandal, 2015). Also /;)_ln = 0 for m or n < —(2K — 1) or m or n > 2^s. We present here the numerical values of I*_{n n} for Dau3 scale functions taking the resolution s = 3 for those values of m and n for which ©_mn,q (₂ i^x,t) has singularity at (0,0) . Table 5.1 shows the values of If_{n n} for N = 5, whereas Table 5.2 shows the values of /;), _n for N = 7.

Table 5.1: Numerical values of lf_{n n} for К = 3, N = 5.

1m,n	-4	-3	-2	-1	0
-4	6.20189(—7)	6.17887(—6)	-2.42358(—4)	1.32994(—3)	8.65630(—4)
-3	1.75110(—5)	5.53604(—4)	-1.32760(—3)	8.53732(—3)	7.54718(—2)
-2	-7.09342(—5)	-1.8897Ц—3)	-8.75770(—3)	-4.55777(—2)	-1.24792(—2)
-1	3.23710(—4)	1.26285(—2)	-7.43785(—2)	2.79172(—1)	1.64992(—1)
0	9.93316(—7)	4.0792Ц-4)	-6.06907(—3)	4.73939(—2)	3.27281 (— 1)

Table 5.2: Numerical values of Ff_{n n} for К = 3, N = 7.

1m,n	-4	-3	-2	-1	0
-4	6.20146(—7)	6.18006(—6)	-2.42335(—4)	1.32979(—3)	8.65506(—4)
-3	1.75097(—5)	5.53569(—4)	-1.32778(—3)	8.53718(—3)	7.54618(—2)
-2	-7.09290(—5)	-1.88961(—3)	-8.75638(—3)	-4.55772(—2)	-1.24797(—2)
-1	3.23683(—4)	1.26275(—2)	-7.43729(—2)	2.79151(—1)	1.64981 (—1)
0	9.91086(—7)	4.07828(—4)	-6.06822(—3)	4.73891(—2)	3.27262(—1)

Numerical results

Numerical values for B_m, N_mn, J_{s mn} obtained by using the formulae (5.1.2.9), (5.1.2.13), (5.1.2.14) respectively have been used in Ecj. (5.1.2.3) and solved for the unknown coefficients c₄„to get the approximate solution u_approx from (5.1.2.1). The values of c_sn have been used in the formula (5.1.2.7) for the evaluation of u,_m,_rox (1). A comparison between the numerical values of u(l) obtained here by using Daubechies scale functions and exact results of Stallybrass (Stallybrass, 1970) is given in the Table 5.3 for different values of <7 = 1,2,3,......,10.

Table 5.3: Approximate values of «(1) for different o.

(7	Exact	Approx. N = 5	Rel. Error	Approx. N = 7	Rel. Error
l	0.86354	0.863656	0.000134	0.863660	0.000139
2	0.57547	0.575551	0.000141	0.575463	0.000012
3	0.46350	0.463562	0.000134	0.463424	0.000164
4	0.39961	0.399170	0.001101	0.398996	0.001536
5	0.35681	0.355325	0.004162	0.355123	0.004728
6	0.32549	0.322496	0.009198	0.322270	0.009893
7	0.30125	0.296382	0.016159	0.296137	0.016973
8	0.28176	0.274740	0.025057	0.274476	0.025852
9	0.26564	0.256272	0.035266	0.255994	0.036312
10	0.25201	0.240179	0.046946	0.239886	0.048109

The table shows the exact values of u(l) (a = 1, 2,....., 10) according to Stallybrass (Stallybrass,

1970) and the results obtained by the present method with their relative errors.

The numerical results are displayed in Fig. 5.1. For the sake of clarity five figures are drawn wherein, the stress intensity factor u(l) is depicted against the parameter for different integral values. In each figure m(1) obtained from Stallybrass’s (Stallybrass, 1970) exact result is denoted by the symbol and u(l) obtained from other approximate methods are shown. The figures are self

explanatory. However, as the result obtained by Sigmoidal transformation method is only for a = 1, this is not shown here. From these figures, it is obvious that all the methods including the present method provide very accurate results.

Figure 5.1: Stress intensity factor x(l)(= -u(l)) against exponent a.