Wavelet Galerkin method for solving singular integral equations

Wavelet Galerkin method for solving singular integral equations

K. Maleknejad^*; M. Nosrati; E. Najafi

School of Mathematics, Iran University of Science & Technology Narmak Tehran 16846, 13114, Iran. E-mails: [e-mail] / [e-mail] / [e-mail]

ABSTRACT

An effective technique upon linear B-spline wavelets has been developed for solving weakly singular Fredholm integral equations. Properties of these wavelets and some operational matrices are first presented. These properties are then used to reduce the computation of integral equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.

Mathematical subject classification: 45A05, 32A55, 34A25, 65T60.

Key words: integral equation, weakly singular, operational matrices, linear B-spline wavelets, thresholding parameter.

1 Introduction

The aim of this study is to present a high order computational method for solving a special case of singular Fredholm integral equations of the second kind namely Abel's integral equation defined as follows:

where f(x) and K(x, t) are known functions and y(x) is the unknown function that to be determined.

Abel's equation is one of the integral equations derived directly from a concrete problem of mechanics or physics (without passing through a differential equation). Historically, Abel's problem is the first one that led to the study of integral equations. The generalized Abel's integral equations on a finite segment appeared in the paper of Zeilon [1] for the first time.

A comprehensive reference on Abel-type equations, including an extensive list of applications, can be found in [2]-[5].

The construction of high order methods for the equations is, however, not an easy task because of the singularity in the weakly singular kernel. In fact, in this case the solution y is generally not differentiable at the endpoints of the interval [6]-[9], and due to this, to the best of the authors' knowledge the best convergence rate ever achieved remains only at polynomial order. For example, if we set uniform meshes with n + 1 grid points and apply the spline method so for order m, then the convergence rate is only O(n^-2P) at most [10]-[11], and it can not be improved by increasing m. One way of remedying this is to introduce graded meshes [10]-[12]. Then the rate is improved to O(n^-m) [12] which now depends on m, but still at polynomial order. Fettis [13] proposed a numerical form of the solution to Abel equation by using the Gauss-Jacobi quadrature rule. Piessens and Verbaeten [14] and Piessens [15] developed an approximate solution to Abel equation by means of the Chebyshev polynomials of the first kind. Numerical solutions of weakly singular Volterra integral equations were introduced in [16]-[21]. Yanzhao et al. [22] applied hybrid collocation methods for solving these equations. Rashit Ishik [23] used Bernstein series solution for solving linear integro-differential equations with weakly singular kernels. In [24] wavelet method is applied to solve noisy Abel-type equations. Wazwaz [25] studied on singular initial value problems in the second-order ordinary differential equations.

An algorithm for solving nonlinear singular perturbation problems is discussed in [26].

In this work we assume that the K(x, t) ∈ [a, b] × [a, b] and satisfies in Lipschitz condition, that is:

and L_s is the Lipschitz constant. In this paper, we use the semiorthogonal linear B-spline wavelets for solving weakly singular integral equations. Our method consists of reducing the given weakly singular integral equation to a set of algebraic equations by expanding the unknown function by B-spline wavelets with unknown coefficients. Galerkin method is utilized to evaluate the unknown coefficients. Because of semiorthogonality, compact support and having vanishing moments properties of these wavelets, the operational matrix is very sparse. Without loss of generality, we may consider [a, b] = [0, 1].

The structure of this paper is arranged as follows. The main problem and brief history of some presented methods are expressed in Introduction 1. Linear B-spline scaling and wavelet functions on bounded interval are introduced in Section . Section is devoted to function approximation by using B-spline wavelets and respective theorems. In Section , linear B-spline wavelets are applied as testing and weighting functions of Galerkin method for efficient solution of equation 1. In Section sparsity of the operational matrix and thresholding parameter is discussed. In Section 6, we report our numerical founds and compare them with other methods in solving these integral equations, and Section 7 contains our conclusion.

2 Linear B-spline scaling and wavelet functions

Basic definitions and concepts of wavelets is given in [27]-[33].

Definition 2.1. Let m and n be two positive integers and

be an equally spaced knots sequence. The functions

and

are called cardinal B-spline functions of order m > 2 for the knot sequence , and Supp [B_m,j,X(x) = [x_j, x_{j + m}] ∩ [c, d].

For the sake of simplicity, suppose [c, d] = [0, n] and x_k = k, k = 0,..., n. The B_m,j,X = B_m(x - j), j = 0,..., n - m, are interior B-spline functions, while the remaining B_m,j,X, j = - m + 1,..., -1 and j = n - m + 1,..., n - 1 are boundary B-spline functions, for the bounded interval [0, n]. Since the boundary B-spline functions at 0 are symmetric reflections of those at n, it is sufficient to construct only the first half functions by simply replacing x with n - x.

By considering the interval [c, d] = [0, 1], at any level j ∈ Z⁺, the discretization step is 2^{- j}, and this generates n = 2^j number of segments in [0, 1] with knots sequence

Let j₀ be the level for which 2^j0> 2m - 1; for each level j > j₀ the scaling functions of order m can be defined as follows in [33]:

and the two-scale relation for the m-order semiorthogonal compactly supported B-wavelet functions are defined as follows:

where q_i,k = q_{k - 2i}.

Hence, there are 2(m - 1) boundary wavelets and (n - 2m + 2) inner wavelets in the boundary interval [c, d]. Finally by considering the level j with j > j₀ , the B-wavelet functions in [0, 1] can be expressed as follows:

The scaling functions φ_m,j,i(x), occupy m segments and the wavelet functions ψ_m,j,i(x) occupy 2m - 1 segments.

Therefore the condition 2^j > 2m - 1, must be satisfied in order to have at least one inner wavelet. In the following, the scaling functions and wavelet functions used in the paper, for j₀ = j = 2 and m = 2, are reported in [35]:

Boundary scalings

Inner scalings

Boundary wavelets

Inner wavelets

Some of the important properties relevant to the present work are given below:

1) Vanishing moments: A wavelet ψ(x) is said to have a vanishing moments of order m if

All wavelets must satisfy the above condition for p = 0. Linear B-spline wavelet has 2 vanishing moments. That is

For a good approximation and data compression, vanishing moments property is necessary condition.

2) Semiorthogonality: The wavelets ψ_j,k form a semiorthogonal basis if

Linear B-spline wavelet are semiorthogonal.

3 Function approximation

A function f(x) defined over [0, 1] may be approximated by B-spline wavelets as [34]:

where and ψ_j,k are scaling and wavelets functions, respectively. If the infinite series in equation 17 is truncated, then it can be written as:

where C and Ψ are column vectors given by:

with

where and are dual functions of ,i = - 1,..., and ψ_j,k, j = j₀,..., j_u, respectively. These can be obtained by linear combinations of and ψ_j,k.

Theorem 3.1.We assume that f ∈ C²[0, 1] is represented by linear B-spline wavelets as equation 18, where ψ has 2 vanishing moments, then

where α = max|f⁽²⁾(t)|,t ∈ [0,1] and β = . Moreover if e_j(x) be the approximation error in the subspace V_j, then:

Proof. ([35])

4 Description of the numerical method

In this section, we solve the singular integral equation of the form 1 by using B-spline wavelets. For this purpose the unknown function of the equation 1 is expanded by linear B-spline wavelets as equation 18. The integral term in equation 1 can be written as:

and

thus we have

now as x → t,

On the other hand

so, as x → t

Now we introduce the function H(x, t) as:

So the integral term of equation 1 can be written as:

and we note that the new kernel function is not singular in [0, 1]. Thus the integral equation 1 can be rewritten as follows:

Substituting function approximation 18 in current equation and employing Galerkin method, the following set of linear system of order + 1 is generated. Linear B-spline scaling and wavelet functions are used in testing and weighting functions of Galerkin method.

where

And the subscripts i, r, k, j, l and s assume values as given below:

It can be shown that the total number of unknowns in 28, does not depend on j₀ as given below:

The limits of integrations in 31-36 range from zero to one, the actual integration limits are much smaller because of the finite supports of the semi orthogonal scaling functions and wavelets. Moreover, a lot of the integrals in (28) become zero due to the semi orthogonality and vanishing moments properties of the wavelet functions.

In fact the entries with significant magnitude are in the ‹Hϕ, ϕ› - ‹ϕ, ϕ› and ‹Hψ, ψ› - ‹ψ, ψ› sub matrices which are of order ( + 1) and respectively.

5 Matrix sparsity and thresholding error

Because of the local supports and vanishing moments properties of B-spline wavelets, many of the matrix elements in equation 18 are very small compared to the largest element, and hence we can set to zero with an opportune threshold technique without significantly affecting the solution. Typically one thresholds the elements of a wavelet matrix by setting to zero all elements that are less than some small positive number multiplied by the largest matrix element, we show by δ. The matrix sparsity S_δ defined by

where N_e is the total number of elements and N_δ is the number of nonzero elements remaining after thresholding. The relative error caused by thresholding the wavelet matrix is defined by

6 Illustrative examples

In this section, to show the accuracy and efficiency of the described method we present some numerical examples then we compare the results of our method with the results of some other methods. The effects of different thresholding parameters on the error and grayscale plots of the moment matrix elements are shown in figures. The matrix sizes for the B-spline wavelets in j₀ = 2 were 17 ×17 and 33×33, respectively, for j_u = 3 and j_u = 4. In grayscale plots of matrices, a darker colour on an element indicates a larger magnitude. Because of the vanishing moments and semiorthogonality of B-spline wavelets, we expect that the matrix elements in ‹Hϕ, ψ› - ‹ϕ, ψ› and ‹Hψ, ϕ› - ‹ψ, ϕ› were very small, and hence can be set to zero without significantly affecting the solution.

Example 1. Consider the singular integral equation [36]

with

The exact solution y(x) = x²(1 - x)². The solution for y(x) is obtained bythe method in Section 4 at the octave level j₀ = 2 and at the levels j_u = 3 and 4. The results without thresholding and for different thresholding parameters and diverse scales are shown in Tables 1 and 2. In Table 1, we present exact and approximate solutions of Example 1 in some arbitrary points. As proved in Theorem 1, the error at the level j_u = 4 is smaller than the error at j_u = 3, moreover errors in our method are smaller than those in other methods. Moreover, because of semiorthogonality and having vanishing moments of B-spline wavelets, matrices in our method are sparse, thus we do not need large memory requirement and a high computational time.

Figure 1 shows the grayscale plot of the matrix obtained by setting the threshold to 10^-5 at the level j_u = 3. Table 2 shows a comparison of sparsity and relative error for semi orthogonal B-spline wavelets in different scales and threshold parameter. It is interesting that at the scale j_u = 4 with threshold parameter 10^-4, the number of matrix element 1089 decrease to 262.

Example 2. Consider the problem [10], [11]

with the exact solution . The solution for y(x) is obtainedby the method in Section 4 at the octave level j₀ = 2 and at the levels j_u = 3 and 4. The results without thresholding and for different thresholding parameters and diverse scales are shown in Tables 3 and 4. Figure 2 shows the grayscale plot of the matrix obtained by setting the threshold to 10^-4 at the level j_u = 4.

7 Conclusions

In this paper, we proposed an advanced numerical model in solving weakly singular Fredholm integral equation of the second kind by means of semi orthogonal compactly supported spline wavelets. The wavelet MOM used via the Galerkin procedure. Based on the consideration reported in figures and tables, the method presented in this paper determines a strong reduction of the computation time and memory requirement in inverting the matrix. The approach can be extended to nonlinear singular integral and integro-differential equations with little additional work. Further research along these lines is under progress and will be reported in due time.

Received: 06/IX/11.

Accepted: 05/IV/12.

#CAM-409/11.

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Corresponding author.

Publication Dates

• Publication in this collection
  05 Dec 2012
• Date of issue
  2012

History

• Received
  06 Sept 2011
• Accepted
  05 Apr 2012