CUBO, A Mathematical Journal Vol. 23, no. 02, pp. 245–264, August 2021 DOI: 10.4067/S0719-06462021000200245 Approximate solution of Abel integral equation in Daubechies wavelet basis Jyotirmoy Mouley 1 M. M. Panja 2 B. N. Mandal 3 1 Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata-700 009, India. jyoti87.cu.wavelet@gmail.com 2 Department of Mathematics,Visva-Bharati, Santiniketan, West Bengal, 731235, India madanpanja2005@yahoo.co.in 3 Physics and Applied Mathematics Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata-700108, India. bnm2006@rediffmail.com ABSTRACT This paper presents a new computational method for solv- ing Abel integral equation (both first kind and second kind). The numerical scheme is based on approximations in Daubechies wavelet basis. The properties of Daubechies scale functions are employed to reduce an integral equation to the solution of a system of algebraic equations. The error analysis associated with the method is given. The method is illustrated with some examples and the present method works nicely for low resolution. RESUMEN Este art́ıculo presenta un nuevo método computacional para resolver la ecuación integral de Abel (tanto de primer como de segundo tipo). El esquema numérico está basado en aproximaciones en la base de ondeletas de Daubechies. Se emplean las propiedades de las funciones de escala de Daubechies para reducir una ecuación integral a la solución de un sistema algebraico de ecuaciones. Se entrega el análisis de error asociado con el método. El método es ilustrado con algunos ejemplos donde el método presentado funciona bien en baja resolución. Keywords and Phrases: Abel integral equation, Daubechies scale function, Daubechies wavelet, Gauss-Daubechies quadrature rule. 2020 AMS Mathematics Subject Classification: 45D05. Accepted: 20 May, 2021 Received: 02 May, 2020 c©2021 J. Mouley et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000200245 https://orcid.org/0000-0001-8353-0813 https://orcid.org/0000-0002-3690-6395 https://orcid.org/0000-0001-9771-1287 246 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) 1 Introduction The theory of integral equations is a very important tool to deal with problems arising in math- ematical physics. Abel integral equation appears in many physical problems of water waves, astrophysics, solid mechanics and in many applied sciences (see [1, 2, 3, 4]). In the year 1823, Abel integral equation was derived directly from the tautochorone problem in physics. In fact this gave birth to the topic known as integral equation. Before 1930, the branch of mathematics which is related to wavelet began with Joseph Fourier with his theories of frequency analysis, now often referred to Fourier synthesis (see [5]). The concept of wavelet was first mentioned in an appendix of the thesis of A. Haar (see [6]), but the formulation of problems involving wavelets has been developed mostly over last 30 years. Grossman and Morelet [7] developed the continuous wavelet transform and the orthogonal one was developed by Lamarie and Meyer [8]. Daubechies (see [9, 10]) constructed a compactly supported orthogonal wavelet basis that can be generated from a single function with the aim to serve the multiresolution analysis (MRA of L2 (R)). Wavelets allow to represent variety of functions and operators very accurately. Furthermore, wavelets setup a connections with fast numerical algorithms [11]. Hence wavelets are used as an efficient tool to solve integral equations. In this paper we consider the Abel integral equations in the form First kind : ∫ x 0 y (t) dt (x − t)µ = f (x) , (1.1) Second kind : y(x) + λ ∫ x 0 y (t) dt (x − t)µ = f (x) . (1.2) Here 0<µ<1, 0 ≤ x ≤ 1 and the forcing term f(x) ∈ C[0,1] in order to confirm the existence and uniqueness of the solution y(x) ∈ C[0,1], the space of all continuous function defined on [0,1]. The Abel integral equation has been solved earlier analytically and numerically by various methods in the literature. For instance, Yousefi [12] constructed a numerical scheme based on Legendre multiwavelets to solve Abel integral equation. A system of generalized Abel integral equations was solved using Fractional calculus by Mandal et al [13]. Liu and Tao [14] applied mechanical quadrature methods for solving first kind Abel integral equation. Numerical solution of Abel integral equation is obtained using orthogonal functions by Derili and Sohrabi [15]. Alipour and Rostamy [16] used Bernstein polynomials to solve Abel integral equations. Shahsavaram [17] used Haar wavelet as the basis function in the collocation method to solve Volterra integral equation with weakly singular kernel. In this paper, the unknown function in the integral equation is expanded by employing Daubechies wavelet basis with unknown coefficients. The integral equation is converted into a system algebraic equations utilizing the properties of Daubechies scale functions. After evaluating the unknown coefficients, the values of the unknown function in the integral equations can be determined at any CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 247 dyadic point in [0,1]. 2 Preliminary concept of Daubechies scale function Here some important properties of Daubechies scale function with a compact support are presented in a finite interval [a,b] ⊂ R , where a and b(> a) are integers. 2.1 Two-scale relations Daubechies constructed a whole new class of orthogonal wavelets that can be generated from a single function φ(x), known as Daubechies scale function. This scale function has some interesting features like compact support, fractal nature, and unknown structure at all resolutions. Daubechies -K (Dau-K) scale function(K ∈ N) has 2K filter coefficients and compact support [0,2K −1]. The two-scale relation of scale function is given by φ(·) = √ 2HT Φ(·), (2.1) where H = [h0,h1,h2, ...,h2K−1] T 2K×1 (2.2) and Φ(·) = [φ(2·),φ(2 · −1),φ(2 · −2), ...,φ(2 · −2K + 1)]T2K×1 (2.3) with the normalization condition ∫ R φ(x)dx = 1. (2.4) The elements hl (l = 0,1,2, ...,2K − 1) are known as filter coefficients or low pass filters. These filter coefficients satisfy the following algebraic relations 2K−1 ∑ l=0 hl = √ 2 ; 2K−1 ∑ l=0 hlhl−2m = δm0. (2.5) Here we define two operators, one is the translation operator T and other is the scale trans- formation operator D as T kφ(x) = φk(x) = φ(x − k) (2.6) and Djφ(x) = 2 j 2 φ(2jx). (2.7) For a specific value of resolution j, the translate of scaling functions are orthonormal to each other viz. ∫ R φjk1 (x)φjk2 (x)dx = δk1k2 (2.8) 248 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) where φjk(x) = 2 j 2 φ(2jx − k). (2.9) It is evident that all the properties of scaling functions are applicable on R. But in the finite interval [a,b] the translation property (2.6) does not hold good for all k ∈ Z as well as the orthogonalization condition (2.8) cannot be applied for φjk(x). So in order to apply the machinery of Dau-K scale function on a finite interval [a,b], we divide the translate of φ(x) for a specific resolution j into three classes (cf. Mouley et al. [18] and Panja et al. [19]) φLjk(·) = φjk(·)χk(x) if k ∈ { a2j − 2K + 2, ...,a2j − 1 } , φI jk (·) = φjk(·)χk(x) if k ∈ { a2j, ...,b2j − 2K + 1 } , φRjk(·) = φjk(·)χk(x) if k ∈ { b2j − 2K + 2, ...,b2j − 1 } . (2.10) Here χk(x) is the characteristic function assuming the value 1 or 0 according as x ∈ [a,b] or x 6∈ [a,b]. 2.2 Scale function at dyadic points A number of the form m 2n is known as a dyadic fraction or dyadic rational (m is an integer and n is a natural number). It has extensive application in measurement, the inch is normally subdivided in dyadic rather than decimal fraction. The ancient Egyptians also used dyadic fractions in mea- surement, with denominators up to 64 [20]. After knowing the value of scale function at integer points within support, it is possible to determine the scale function at any dyadic point with in the support [21] . Using the two-scale relation (2.1) the value of Dau-K scale function φ(x) at x = m 2n is calculated as φ (m 2n ) = 2K−1 ∑ l1=0 √ 2hl1φ ( m − 2n−1l1 2n−1 ) . (2.11) Again using the two-scale relation (2.1) we get φ (m 2n ) = 2K−1 ∑ l1=0 2K−1 ∑ l2=0 2hl1hl2φ ( m − 2n−1l1 − 2n−2l2 2n−2 ) . (2.12) Repeating the two-scale relation (2.1) n times, we get φ (m 2n ) = 2K−1 ∑ l1=0 2K−1 ∑ l2=0 ... 2K−1 ∑ ln=0 2 m 2 hl1hl2...hlnφ(m − 2n−1l1 − 2n−2l2...2ln−1 − ln). (2.13) 3 Multiresolution analysis (MRA) and Daubechies wavelet Basic concepts of MRA and Daubechies wavelet are discussed in most of the texts on wavelets (see [9, 10, 18, 19, 21]). Why wavelet has started to dominate in different applications such as technology, engineering and applied mathematics, one serious reason behind it is MRA. A MRA CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 249 on R is defined as a sequence of nested subspaces Vj of function L 2 on R with scaling function φ(x) if the following properties hold, ∀j ∈ Z, Vj ⊆ Vj+1, (3.1) ClosL2 (∪j∈ZVj) = L2 (R) , (3.2) ∩j∈Z Vj = {0} , (3.3) φ(x) ∈ Vj ⇔ φ(2x) ∈ Vj+1, ∀j ∈ Z. (3.4) Here Vj’s are called approximation spaces. The scale function φ(x) belongs to V0 and the set {φ(x − k) : k ∈ Z} is a Riesz basis of V0. The scale function φ(x) satisfies the two-scale relation (2.1). Also the set {φjk(x) : k ∈ Z} given by (2.9) is a Riesz basis of Vj. From the property (3.1), it is evident that each element of Vj+1 can be uniquely written as the orthogonal sum of an element in Vj and an element in Wj that contains the complementary details i.e. Vj+1 = Vj ⊕ Wj = V0 ⊕ W0 ⊕ W1 ⊕ W2 ⊕ ... ⊕ Wj. (3.5) Let Wj be the span of ψjk(x) = 2 j 2 ψ(2jx − k), which is called wavelet function. The wavelet function ψ(x) satisfies the relation ψ(·) = √ 2GT Φ(·) (3.6) where G = [g0,g1,g2, ...,g2K−1] T 2K×1. (3.7) Here Φ(·) is given by (2.3) and gl (l = 0,1,2, ...,2K − 1) are known as high pass filter coefficients and are given by gl = (−1)lh2K−1−l. (3.8) 4 Method of approximation We approximate the unknown function of the integral equations (1.1) and (1.2) in the approxima- tion space Vj as y(x) ≈ yMSj (x) = 2j−1 ∑ k=0 cjkφjk(x) = 2j−2K+1 ∑ k=0 cIjkφ I jk(x) + 2j−1 ∑ k=2j −2K+2 cRjkφ R jk(x) = CT ~Φ(x). (4.1) 250 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) As the support of φ(x) is [0,2K − 1], so yMSj (x) always vanishes at x = 0. The value of y(x) at x = 0 for second kind Abel integral equation is obviously f(0) but for the first kind Abel integral equation y(x) cannot be evaluated at x = 0 but as y(x) can be evaluated at any dyadic point in (0,1], it can be evaluated very close to x = 0 by making the resolution fairly large. Here C and ~Φ(x) both are 2j × 1 vectors, given by C = [ cIj0,c I j1, ...,c I j2j −2K+1,c R j2j−2K+2, ...,c R j2j −1 ]T (4.2) and ~Φ(x) = [ φIj0(x),φ I j1(x), ...,φ I j2j −2K+1(x),φ R j2j −2K+2(x), ...,φ R j2j −1(x) ]T . (4.3) Using the approximate form of y(x) in (4.1) in both the first and second kind integral equations (1.1) and (1.2) we get, C T ∫ x 0 ~Φ(t)dt (x − t)µ = f (x) (4.4) and C T [ ~Φ(x) + λ ∫ x 0 ~Φ(t)dt (x − t)µ ] = f (x) . (4.5) We choose total 2j number of points by xjk′ = k ′ 2j (k ′ = 1,2,3, ...,2j) and substituting these points in both the equations (4.4) and (4.5) we get, C T B (k ′ ) = f ( k ′ 2j ) (4.6) and C T [ A (k ′ ) + λB(k ′ ) ] = f ( k ′ 2j ) (4.7) where A (k ′ ) = ~Φ ( k ′ 2j ) = [ φIj0 ( k ′ 2j ) ,φIj1 ( k ′ 2j ) , ...,φIj2j −2K+1 ( k ′ 2j ) ,φRj2j −2K+2 ( k ′ 2j ) , ...,φRj2j −1 ( k ′ 2j )]T (4.8) and B (k ′ ) =   ∫ k ′ 2j 0 φIj0 (t) dt (k ′ 2j − t)µ , ..., ∫ k ′ 2j 0 φI j2j −2K+1 (t) dt (k ′ 2j − t)µ , ∫ k ′ 2j 0 φR j2j −2K+2 (t) dt (k ′ 2j − t)µ , ..., ∫ k ′ 2j 0 φR j2j −1 (t) dt (k ′ 2j − t)µ   T . (4.9) As k = 0,1,2, ...,2j − 1 and k′ = 1,2,3, ....,2j, each of the equation (4.6) and (4.7) represents a system of 2j equations in 2j variables cI jk and cR jk . Solving these systems the unknown coefficients cIjk and c R jk are obtained. CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 251 In the last part of this section, we explain the procedure for calculating the matrix elements of the matrix B(k ′ ). We use the notation Iµ j(k ′ ,k) = ∫ k ′ 2j 0 φjk (t) dt (k ′ 2j − t)µ . (4.10) In the relation (4.10), for 0 ≤ k ≤ 2j −2K+1, φjk (t) means φIjk (t) and for 2j −2K+2 ≤ k ≤ 2j −1, φjk (t) means φ R jk (t) . Using (2.9) we find Iµ j(k ′ ,k) = 2 ( µ− 1 2 ) j Lµ(k ′ − k), (4.11) where Lµ(k) = ∫ k 0 φ(t) dt (k − t)µ . (4.12) As the support of Dau-K scale function φ(t) is [0,2K−1], so if k ≤ 0 the range of the integration in (4.12) is completely outside of the support. In this case Lµ(k) vanishes. Again if k ≥ 2K,Lµ(k) has no singularity within the support [0,2K − 1]. Using Gauss-Daubechies quadrature rule involving Daubechies scale function [22], Lµ(k) is evaluated as Lµ(k) = M ∑ i=1 wi (k − ti)µ , (k ≥ 2K). (4.13) Here wi , ti are weights are nodes of Gauss-Daubechies quadrature rule involving Daubechies scale function [22]. For 0 < k ≤ 2K − 1, Lµ(k) has integrable singularity at the upper limit so that evaluation of such integrals by using the quadrature rule may not provide their approximate value with desired order of accuracy within less computational time. The two-scale relation (2.1) for φ(t), may be used to obtain a recurrence relation for Lµ(k) as Lµ(k) = 2µ− 1 2 2K−1 ∑ l=0 hlLµ(2k − l). (4.14) Using the symbols HK =           h1 h0 0 0 · · · 0 0 h3 h2 h1 h0 · · · 0 0 ... ... ... ... · · · ... ... 0 0 0 0 · · · h2K−2 h2K−3 0 0 0 0 · · · 0 h2K−1           (4.15) and bµ K =           0 0 ... ∑2K−4 l=0 hlLµ(4K − 4 − l) ∑2K−2 l=0 hlLµ(4K − 2 − l)           (4.16) 252 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) the relation (4.14) can be put in the form ( I − 2µ− 12 HK ) Lµ = bµ K. (4.17) So, the singular integrals in Lµ are found as Lµ = ( I − 2µ− 12 HK )−1 bµ K. (4.18) Thus, evaluation of Lµ(k) is summarized as L(k) =            0 k ≤ 0, solution obtained by (4.18) 1 ≤ k ≤ 2K − 1, ∑M i=0 wi (k − ti)µ k ≥ 2K. (4.19) Table 1: Values of L(k) k µ = 1 4 µ = 1 3 µ = 1 2 1 0.925995 1.098666 1.643812 2 1.064183 1.042183 0.954199 3 0.808341 0.759600 0.682604 4 0.748236 0.679445 0.560703 5 0.699178 0.620553 0.488824 In Table 1 the values of L(k) for k = 1,2, ...,5 are given taking Dau-3 scale function for µ = 1 4 , 1 3 , 1 2 . For other values of µ (0 < µ < 1) these can be easily calculated. Table 2: Accuracy of L(2K) for Dau-3 scale function µ Detemined by (4.13) Detemined by (4.18) 1/4 0.662722 0.662722 1/3 0.577792 0.577792 1/2 0.439182 0.439182 In Table 2 the values of L(2K) for Dau-3 scale function are presented for µ = 1 4 , 1 3 , 1 2 using the relations (4.13) and (4.18) separately. For the two methods the values of L(2K) are found to be same. The values of L(2K) establish the efficiency of the relation (4.18) in the determination of L(k) (k = 0,1,2, ...,2K − 1). CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 253 5 Error estimation In this section, the error of the proposed method is estimated in detail. For this we need the following definitions and theorems. Definition 5.1 ([23]). In a σ-finite measure space (X,F,µ∗) (X denotes underlying space, F is the σ-algebra of measurable sets and µ∗ is the measure) the Lp-norm (1 ≤ p < ∞) of a function f is defined by ‖f‖Lp(X, F, µ∗) = ( ∫ X |f(x)|pdµ∗(x) ) 1 p . The abbreviations ‖f‖Lp(X) , ‖f‖Lp , ‖f‖p are also used to mean Lp- norm. Definition 5.2 ([24]). The inner product of two functions f and g on a measure space X is defined by < f,g >= ∫ X fḡdµ. Theorem 5.3 (Minkowski [23]). If 1 ≤ p < ∞ and f,g ∈ Lp then f + g ∈ Lp and ‖f + g‖Lp ≤ ‖f‖Lp + ‖g‖Lp. Theorem 5.4. Let {φjk(x) : k ∈ Z} and {ψjk(x) : k ∈ Z} be the Riesz bases of approximation space Vj and detail space Wj. If N B j:k,k′ = ∫ b a φBjk(x)φ B jk′ (x)dx and T B j:k,k′ = ∫ b a ψBjk(x)ψ B jk′ (x)dx (B stands for L or R) then T Bj:k,k′ = 2K−1 ∑ l1=0 2K−1 ∑ l2=0 gl1gl2N B j+1:2k+l1,2k′+l2 . Proof. Here NBj:k,k′ = ∫ b a φBjk(x)φ B jk′ (x)dx. Now T Bj:k,k′ = ∫ b a ψBjk(x)ψ B jk′ (x)dx = 2j ∫ b a ψB(2jx − k)ψB(2jx − k′)dx (using expression of ψj,k(x)) = ∫ b2j a2j ψB(z − k)ψB(z − k′)dz = 2K−1 ∑ l1=0 2K−1 ∑ l2=0 gl1gl2 ∫ b2j+1 a2j+1 φB(z − 2k − l1)φB(z − 2k′ − l2)dz (using equation (3.6)) = 2K−1 ∑ l1=0 2K−1 ∑ l2=0 gl1gl2N B j+1:2k+l1,2k′+l2 . This completes the proof. 254 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) So to evaluate T Bj:k,k′ , we need to evaluate N B j+1:2k+l1,2k′+l2 (l1, l2 = 0,1,2, ...,2K − 1). The values of NB j:k,k′ are tabulated in Table 3 and Table 4 in [25]. In section 3 to find the approximate solution, the projection of the unknown function yMSj (x) is used in the approximation space (the linear span of φjk(x),k = 0,1,2, ....2 j − 1). To estimate the error of the unknown function y(x) ∈ L2([0,1]) satisfying both the integral equations (1.1) and (1.2), we employ the fact that the multiscale expansion of y(x) (the projection of y(x) into the approximation space Vj and detail space Wj) is y(x) = 2j−1 ∑ k=0 cjkφjk(x) + ∞ ∑ j′=j 2j ′ −1 ∑ k=0 dj′kψj′k(x) (5.1) where cjk ≈ ∫ 1 0 φjk(x)y(x)dx, (5.2) and djk ≈ ∫ 1 0 ψjk(x)y(x)dx. (5.3) Using the two-scale relation (2.1) and the equation (3.6), (5.2) and (5.3) are reduced to cjk = 2K−1 ∑ l=0 hlcj+1,2k+l , (5.4) djk = 2K−1 ∑ l=0 glcj+1,2k+l. (5.5) To evaluate cjk and djk,(k = 0,1,2, ...,2 j − 1) at level j, we need the values of cj+1,2k+l and dj+1,2k+l at level j +1. If 0 ≤ k ≤ 2j −2K +1, cjk and djk are denoted by cIjk and dIjk respectively. Again if 2j − 2K + 2 ≤ k ≤ 2j − 1, cjk and djk are denoted by cRjk and dRjk respectively. Now using the expression for yMSj (x) given by (4.1), (5.1) is reduced to y(x) = yMSj (x) + ∞ ∑ j′=j δyj′ (5.6) where δyj′ is given by δyj′ = 2j ′ −1 ∑ k=0 dj′kψj′k(x) = 2j ′ −2K+1 ∑ k=0 dIj′kψ I j′k(x) + 2j ′ −1 ∑ k=2j ′ −2K+2 dRj′kψ R j′k(x). (5.7) The error in the multiscale approximation is given by e(x) = y(x) − yMSj (x) = ∞ ∑ j′=j δyj′. (5.8) CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 255 Now ‖e(x)‖2L2[0,1] = ∥ ∥ ∥ ∥ ∥ ∥ ∞ ∑ j′=j δyj′ ∥ ∥ ∥ ∥ ∥ ∥ 2 L2[0,1] ≤ ∞ ∑ j′=j ‖δyj′‖2L2[0,1] = ‖δyj‖2L2[0,1] [ 1 + ‖δyj+1‖2L2[0,1] ‖δyj‖2L2[0,1] + ‖δyj+2‖2L2[0,1] ‖δyj‖2L2[0,1] + .... ] (5.9) We choose max η ‖δyj+η‖2L2[0,1] ‖δyj+η−1‖2L2[0,1] = τ for η = 1,2,3, ... and τ is found to satisfy the condition 0 < τ < 1, which is verified by taking a few examples of Abel first kind and second kind integral equations. The values of τ are different for different examples. Then the expression in (5.9) becomes ‖δyj‖2L2[0,1] [ 1 + ‖δyj+1‖2L2[0,1] ‖δyj‖2L2[0,1] + ‖δyj+2‖2L2[0,1] ‖δyj‖2L2[0,1] + .... ] ≤ ‖δyj‖2L2[0,1] [ 1 + τ + τ2 + τ3 + ... ] = ‖δyj‖2L2[0,1] 1 1 − τ . (5.10) The expression for ‖δyj‖2L2[0,1] is obtained by using orthonormality property of ψjk(x) within its support and Theorem 5.4 for the partial support of ψjk(x). This is given by ‖δyj‖2L2[0,1] = 〈 2j−1 ∑ k=0 djkψjk(x), 2j −1 ∑ k=0 djkψjk(x) 〉 = 2j−2K+1 ∑ k=0 2j−2K+1 ∑ k′=0 dIjkd I jk′δkk′ + 2j−1 ∑ k=2j −2K+2 2j−1 ∑ k′=2j−2K+2 dRjkd R jk′T R j:kk′. (5.11) As ∫ 1 0 ψR jk (x)ψI jk′ (x)dx and ∫ 1 0 ψI jk (x)ψR jk′ (x)dx vanish, so we neglect those terms in the expression (5.11) which contain these specific integrals. The bound of L2- norm of error ‖e(x)‖L2[0,1] can be estimated from the inequality (5.10). 6 Illustrative examples Example 1 Consider the first kind Abel integral equation ∫ x 0 y(t)dt (x − t)µ = B (1 − µ,1 + ν)x1+ν−µ, 0 < µ < 1, ν > 0 which has the exact solution y(x) = xν. Here B(m,n) is the beta function and defined by B(m,n) = ∫ 1 0 xm−1(1 − x)n−1dx, m > 0, n > 0. 256 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) Table 3 shows the exact and approximate solutions of the example 1 at the points x = i 8 for i = 1,2, ...,7 taking Dau-3 scale function and M = 5. In this table, four sets of values of µ and ν are considered taking both fraction and integer values of ν. Table 3: Comparison of exact and approximate solutions of Example 1 x Exact Solution Approximate solution j = 4 j = 6 j = 8 µ = 1 4 ,ν = 1 2 1/8 0.353553 0.309319 0.352867 0.353554 2/8 0.500000 0.486212 0.499995 0.500000 3/8 0.612372 0.608044 0.612374 0.612373 4/8 0.707107 0.705733 0.707108 0.707107 5/8 0.790569 0.790135 0.790570 0.790569 6/8 0.866025 0.865890 0.866026 0.866025 7/8 0.935414 0.935375 0.935415 0.935414 µ = 1 4 ,ν = 3 1/8 0.001953 0.001805 0.001951 0.001953 2/8 0.015625 0.015478 0.015623 0.015625 3/8 0.052734 0.052588 0.052732 0.052734 4/8 0.125000 0.124854 0.124998 0.125000 5/8 0.244141 0.243995 0.244138 0.244141 6/8 0.421875 0.421730 0.421873 0.421875 7/8 0.669922 0.669776 0.669920 0.669922 µ = 3 4 ,ν = 1 2 1/8 0.353553 0.358049 0.353775 0.353575 2/8 0.500000 0.500476 0.500099 0.500009 3/8 0.612372 0.613000 0.612433 0.612378 4/8 0.707107 0.707550 0.707149 0.707111 5/8 0.790569 0.790915 0.790602 0.790572 6/8 0.866025 0.866305 0.866051 0.866028 7/8 0.935414 0.935647 0.935436 0.935416 µ = 3 4 ,ν = 3 1/8 0.001953 0.001870 0.001951 0.001953 2/8 0.015625 0.015525 0.015623 0.015625 3/8 0.052734 0.052630 0.052732 0.052734 4/8 0.125000 0.124892 0.124998 0.125000 5/8 0.244141 0.244030 0.244139 0.244141 6/8 0.421875 0.421763 0.421873 0.421875 7/8 0.669922 0.669809 0.669920 0.669922 CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 257 Table 4: Values of ‖δyj‖2L2[0,1] for different resolution j j For dI jk For both dI jk and dR jk 4 5.75007 × 10−8 1.14422 × 10−3 5 1.43777 × 10−8 5.71189 × 10−4 µ = 1 4 ,ν = 1 2 6 3.59444 × 10−9 2.85378 × 10−4 7 8.9861 × 10−10 1.42637 × 10−4 8 2.24653 × 10−10 7.13055 × 10−5 9 5.61631 × 10−11 3.56496 × 10−5 4 3.92808 × 10−9 1.15222 × 10−3 5 7.16154 × 10−11 5.74173 × 10−4 µ = 1 4 ,ν = 3 6 1.19898 × 10−12 2.86245 × 10−4 7 1.93591 × 10−14 1.42869 × 10−4 8 3.07368 × 10−16 7.13653 × 10−5 9 4.84076 × 10−18 3.56647 × 10−5 4 1.28416 × 10−7 1.14442 × 10−3 5 3.21065 × 10−8 5.71226 × 10−4 µ = 3 4 ,ν = 1 2 6 2.85385 × 10−9 2.86245 × 10−4 7 2.00666 × 10−9 1.42638 × 10−4 8 5.01665 × 10−10 7.13058 × 10−5 9 1.25416 × 10−10 3.56496 × 10−5 4 3.92901 × 10−9 1.15223 × 10−3 5 7.16226 × 10−11 5.74174 × 10−4 µ = 3 4 ,ν = 3 6 1.9904 × 10−12 2.86245 × 10−4 7 1.93595 × 10−14 1.42869 × 10−4 8 3.07371 × 10−16 7.13653 × 10−5 9 4.84079 × 10−18 3.56648 × 10−5 Table 5: Comparison of Sup error and bound of L2-norm of error ‖e(x)‖L2[0,1] j Sup error Bound of ‖e(x)‖L2[0,1] taking dIjk taking d I jk and d R jk µ = 1 4 ,ν = 1 2 4 4.423400 × 10−2 2.768973 × 10−4 4.783764 × 10−2 6 6.867130 × 10−4 6.923054 × 10−5 3.389050 × 10−2 8 7.100990 × 10−7 1.730766 × 10−5 1.194198 × 10−2 258 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) µ = 1 4 ,ν = 3 4 1.48274 × 10−4 6.324620 × 10−5 4.800458 × 10−2 6 2.27344 × 10−6 1.104969 × 10−6 1.691878 × 10−2 8 3.55446 × 10−8 1.769186 × 10−8 1.194201 × 10−2 µ = 3 4 ,ν = 1 2 4 4.49596 × 10−3 4.165755 × 10−4 4.784182 × 10−2 6 2.21534 × 10−4 1.041480 × 10−4 2.389079 × 10−2 8 2.13190 × 10−5 2.603701 × 10−5 1.194201 × 10−2 µ = 3 4 ,ν = 3 4 8.32861 × 10−5 6.316013 × 10−5 4.800479 × 10−2 6 1.68850 × 10−6 1.105110 × 10−6 2.023927 × 10−2 8 2.90795 × 10−8 1.769375 × 10−8 1.194699 × 10−2 Example 2 Consider the second kind Abel integral equation [12] y (x) = x2 + 16 5 x 5 2 − ∫ x 0 y(t)dt√ x − t which has the exact solution y(x) = x2. Table 6 shows the exact and approximate solutions of the example 2 at the points x = i 8 for i = 0,1,2, ...,7 taking Dau-3 scale function and M = 5. Table 6: Comparison of exact and approximate solutions of example 2 x Exact Solution Approximate solution j = 4 j = 6 j = 8 0 0 0 0 0 1/8 0.015625 0.015508 0.015624 0.015625 2/8 0.062500 0.062463 0.062499 0.062500 3/8 0.140625 0.140603 0.140625 0.140625 4/8 0.250000 0.249984 0.250000 0.250000 5/8 0.390625 0.390613 0.390625 0.390625 6/8 0.562500 0.562490 0.562500 0.562500 7/8 0.765625 0.765617 0.765625 0.765625 CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 259 Table 7: Values of ‖δyj‖2L2[0,1] for different resolution j j For dI jk For both dI jk and dR jk 4 1.45784 × 10−12 1.15094 × 10−3 5 4.66357 × 10−14 5.73210 × 10−4 6 1.46128 × 10−15 2.85926 × 10−4 7 4.53812 × 10−17 1.42779 × 10−4 8 1.40555 × 10−18 7.13417 × 10−5 9 4.35409 × 10−20 3.56587 × 10−5 Table 8: Comparison of Sup error and bound of L2- norm of error ‖e(x)‖L2[0,1] j Sup error Bound of ‖e(x)‖L2[0,1] taking dIjk taking d I jk and d R jk 4 1.16723 × 10−4 1.22720 × 10−6 4.79779 × 10−2 6 9.56852 × 10−7 3.88534 × 10−8 2.39134 × 10−2 8 1.34341 × 10−8 1.20500 × 10−9 1.19450 × 10−2 Example 3 Consider the second kind Abel integral equation [17] y (x) = 1√ x + 1 + π 8 − 1 4 sin−1 ( 1 − x 1 + x ) − 1 4 ∫ x 0 y(t)dt√ x − t which has the exact solution y(x) = 1√ x + 1 . Table 9 shows the exact and approximate solutions of the example 3 at the points x = i 8 for i = 0,1,2, ...,7 taking Dau-3 scale function and M = 5. 260 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) Table 9: Comparison of exact and approximate solutions of Example 3 x Exact Solution Approximate solution j = 4 j = 6 j = 8 0 1 1 1 1 1/8 0.942809 0.964541 0.947179 0.943883 2/8 0.894427 0.905166 0.897201 0.895110 3/8 0.852803 0.861371 0.854894 0.853318 4/8 0.816497 0.823468 0.818192 0.816914 5/8 0.784465 0.790355 0.785898 0.784818 6/8 0.755929 0.761042 0.757173 0.756236 7/8 0.730297 0.734819 0.731398 0.730568 Table 10: Values of ‖δyj‖2L2[0,1] for different resolution j j For dIjk For both d I jk and d R jk 4 4.60915 × 10−6 5.80526 × 10−4 5 2.39761 × 10−6 2.88967 × 10−4 6 1.22773 × 10−6 1.44416 × 10−4 7 6.23369 × 10−7 7.20037 × 10−5 8 3.14886 × 10−7 3.59832 × 10−5 9 1.58536 × 10−7 1.79872 × 10−5 Table 11: Comparison of Sup error and bound of L2- norm of error ‖e(x)‖L2[0,1] j Sup error Bound of ‖e(x)‖L2[0,1] taking dI jk taking dI jk and dR jk 4 2.17315 × 10−2 3.09877 × 10−3 3.40742 × 10−2 6 4.37017 × 10−3 1.59930 × 10−3 1.69951 × 10−2 8 1.07361 × 10−3 8.09946 × 10−4 8.48330 × 10−3 We present in Tables 4, 7 and 10 the values of ‖δyj‖2L2[0,1] (j = 4,5,6, .....,9) given by equation (5.11) for the examples 1, 2 and 3 respectively. Second column of all tables present the values ‖δyj‖2L2[0,1] taking only d I jk i.e. taking only first term of (5.11), whereas third column presents the values ‖δyj‖2L2[0,1] taking both dIjk and dRjk. From these tables it appears that the values of ‖δyj‖2L2[0,1] gradually decrease if the resolution j increases. The presence of a few d R jk in (5.11) makes a lot of difference in calculating ‖δyj‖2L2[0,1] taking only dIjk and taking both dIjk and dRjk. CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 261 In Tables 5, 8 and 11, the Sup errors are compared with the bound of L2-norm of error ‖e(x)‖L2[0,1] taking dI jk and taking both dI jk and dR jk for examples 1, 2 and 3 respectively. To evaluate bound of L2- norm of error ‖e(x)‖L2[0,1], τ = 0.250044,τ = 0.50; τ = 0.250044,τ = 0.50; τ = 0.250044,τ = 0.50 and τ = 0.250044,τ = 0.50 are used for the four sets of values of µ and ν taking only dI jk and taking both dIjk and d R jk for example 1. Also to evaluate bound of L 2 norm of error ‖e(x)‖L2[0,1], τ = 0.032, τ = 0.50 and τ = 0.52, τ = 0.50 are used for Examples 2 and 3 respectively. Sup errors are calculated taking maximum absolute difference of exact and approximate solutions from Tables 3, 6 and 9. Figures 1 to 6 display the exact and approximate solutions of examples 1, 2 and 3 for different resolutions (j = 4,6,8). We observe from these figures that as j increases, an approximate solution becomes closer to exact solution. This demonstrates efficiency of the proposed method. j=8 j=6 j=4 Exact 0.0 0.2 0.4 0.6 0.8 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x y H x L Figure 1: Example 1 (µ = 1 4 ,ν = 1 2 ) j=8 j=6 j=4 Exact 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x y H x L Figure 2: Example 1 (µ = 1 4 ,ν = 3) j=8 j=6 j=4 Exact 0.0 0.2 0.4 0.6 0.8 0.4 0.5 0.6 0.7 0.8 0.9 x y H x L Figure 3: Example 1 (µ = 3 4 ,ν = 1 2 ) j=8 j=6 j=4 Exact 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 x y H x L Figure 4: Example 1 (µ = 3 4 ,ν = 3) 262 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) j=8 j=6 j=4 Exact 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x y H x L Figure 5: Example 2 j=8 j=6 j=4 Exact 0.0 0.2 0.4 0.6 0.8 0.75 0.80 0.85 0.90 0.95 1.00 x y H x L Figure 6: Example 3 7 Conclusion The purpose of the present work is to develop an efficient and accurate numerical scheme based on Daubechies wavelet basis to solve Abel integral equation. As wavelets are orthogonal systems, they have different resolution capabilities. The detail error estimation shows that the bound of L2-norm of error ‖e(x)‖L2[0,1] depends on resolution j. From Tables 3, 6 and 9 it appears that the present numerical scheme works nicely for low resolution (j = 4,6,8). The results can be further improved by taking larger resolution j. Acknowledgement JM acknowledges financial support from University Grants Commission, New Delhi, for the award of research fellowship (File no. 16-9(june2017/2018(NET/CSIR))). CUBO 23, 2 (2021) Approximate solution of Abel integral equation in Daubechies.... 263 References [1] S. B. Healy, J. Haase, O. Lesne, “Abel transform inversion of radio occulation measurement made with a receiver inside the earth’s atmosphere”, Ann. Geophys., vol. 20, no. 8, pp. 1253- 1256, 2002. [2] R. N. Bracewell, A. C. Riddle, “Inversion of Fan-Beam scans in radio astronomy”, Astrophys- ical Journal, vol. 150, pp. 427-434, 1967. [3] Lj. M. Ignjatovic and A. A. Mihajlov, “The realization of Abel’s inversion in the case of discharge with undetermined radius”, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 72, no. 5, pp. 677-689, 2002. [4] S. De, B. N. Mandal and A. Chakrabarti, “Water wave scattering by two submerged plane vertical barriers–Abel integral-equation approach”, J. Eng. Math., vol. 65, no. 1, pp. 75-87, 2009. [5] J. Fourier, Théorie Analytique de la chaleur, Firmin Didot, United Kingdom: Cambridge University Press, ISBN 978-1-108-00180-9, 2009. [6] A. Graps, “An introduction to wavelets”, IEEE Computing in Science and Engineering, vol. 2, no.2, pp. 50-61, 1995. [7] A. Grossman and J. Morlet, “Decomposition of Hardy functions into square integrables wavelets of constant shape”, SIAM J. Math. Anal., vol. 15, no. 4, pp. 723-736, 1984. [8] P. G. Lamarie and Y. Meyer, “Ondelettes et bases hilbertiennes”, Rev. Mat. Iberoam., vol. 2, no. 1, pp. 1-18, 1986. [9] I. Daubechies, “Orthonormal bases of compactly supported wavelets”, Comm. Pure Appl. Math., vol. 41, no.7, pp. 909-996, 1988. [10] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Philadelphia, PA: SIAM, 1992. [11] G. Beylkin, R. Coifman and V. Rokhlin, “Fast wavelet transforms and numerical algorithms I”, Comm. Pure Appl. Math, vol. 44, no. 2, pp. 141-183, 1991. [12] S. A. Yousefi, “Numerical solution of Abel’s integral equation by using Legendre wavelets”, Appl. Math. Comput., vol. 175, no.1, pp. 574-580, 2006. [13] N. Mandal, A. Chakrabarti and B. N. Mandal, “Solution of a system of generalized Abel integral equations using fractional calculus”, Appl. Math. Lett., vol.9, no. 5, pp. 1-4, 1996. 264 J. Mouley, M. M. Panja & B. N. Mandal CUBO 23, 2 (2021) [14] Y. Liu and L. Tao, “Mechanical quadrature methods and their extrapolation for solving first kind Abel integral equations”, J. Comput. Appl. Math, vol. 201, no.1, pp. 300-313, 2007. [15] H. Derili and S. Sohrabi, “Numerical solution of singular integral equations using orthogonal functions”, Math. Sci. (QJMS), vol. 3, pp. 261-272, 2008. [16] M. Alipour and D. Rostamy, “Bernstein polynomials for solving Abel’s integral equation”, J. Math. Comput. Sci., vol. 3, no. 4, pp. 403-412, 2011. [17] A. Shahsavaram, “Haar Wavelet Method to Solve Volterra Integral Equations with Weakly Singular Kernel by Collocation Method”, Appl. Math. Sci., vol. 5, pp. 3201-3210, 2011. [18] J. Mouley, M. M. Panja and B. N. Mandal, “Numerical solution of an integral equation arising in the problem of cruciform crack using Daubechies scale function”, Math. Sci., vol. 14, no. 1, pp. 21-27, 2020. [19] M. M. Panja and B. N. Mandal, “Solution of second kind integral equation with Cauchy type kernel using Daubechies scale function”, J. Comput. Appl. Math., vol. 241, pp. 130-142, 2013. [20] L. J. Curtis, “Concept of the exponential law prior to 1900”, Amer. J. Phys., vol. 46, no. 9, pp. 896-906, 1978. [21] B. M. Kessler, G. L. Payne, W. W. Polyzou, “Notes on Wavelets”, 2003. . [22] M. M. Panja and B. N. Mandal, “Gauss-type quadrature rule with complex nodes and Weights for integrals involving Daubechies scale functions and wavelets”, J. Comput. Appl. Math., vol. 290, pp. 609-632, 2015. [23] E. M. Stein, R. Shakarchi, Functional Analysis: Introduction to Further topics in Analysis’, Princeton Lectures in Analysis, Princeton: Princeton University Press, ISBN-978-0-691-11387- 6, 2011. [24] A. Wang, “Lebesgue measure and L2 space”, Mathematics department, University of Chicago, 2011. [25] M. M. Panja and B. N. Mandal, “Evaluation of singular integrals using Daubechies scale function”, Adv. Comput. Math. Appl., vol. 1, pp. 64-75, 2012. Introduction Preliminary concept of Daubechies scale function Two-scale relations Scale function at dyadic points Multiresolution analysis (MRA) and Daubechies wavelet Method of approximation Error estimation Illustrative examples Conclusion