International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 466 ISSN 2229-5518 Application of Haar -packets to the solution of linear and nonlinear Integral equations

Ruchi Agarwal, Khursheed Alam, C. S. Salimath

Abstract: The paper describes an innovative alternative to Walsh series and single term methods for the solution of linear and nonlinear integral equations using Haar wavelet packets. The underlying theory and properties of Haar wavelet packets are presented in some detail. The integral operators are expressed in wavelet packet bases resulting invariably into sparse matrices. The proposed methods, as a consequence, reduce the computation time and effort, concurrently achieving better accuracy in solving different kinds of integral equations. The error estimates are derived to establish the rapidity of convergence. A good agreement between computed results and the exact solutions is demonstrated using numerical tables and a computer generated graph. The methods are in general, conceptually simple, easy to implement, yield accurate results and more importantly, they have universal features for their applications to a wide range of problems, with higher order convergence rate. Keywords: Block-pulse solution, Haar wavelet packets, Multi-Resolution Analysis, operation matrix, Series and single term methods, Walsh functions, Wavelet decomposition.

Introduction There has been considerable revival of interest in solving Integral equations are well-known mathematical tools in differential and integral equations using techniques which the formulation of physical and mechanical problems. involve Walsh functions for quite some time (Hsiao and They arise in many branches of science, for example, in Chen (1979)). More recently, in the field of numerical potential theory, acoustics, elasticity, fluid mechanics, analysis, Walsh series methods such as, truncated series theory of population etc. Any integral equation to be and single term have been successfully used for the numerically solved can be reduced to a finite dimensional numerical solution of several classes of problems, problem or discretized. There are two approaches to the particularly linear and nonlinear integral equations (Sloss discretization of linear integral equations. In one, often and Blyth(2003), Sepehrian and Razzaghi (2005)). called the Galerkin (projection) method, expansions of The use of wavelet methods in numerical analysis has functions and kernels involved are truncated in some basis opened up floodgates to several areas of research in recent and the resulting system of algebraic equations is solved years. In general, wavelet methods havebeen successfully numerically. In the second, developed by Nystrom, the used in the numerical treatment of differential and integral integral operator is approximated yielding again a system equations for three main tasks: One, preconditioning large of equations to be solved numerically [1]. Projection systems arising from descretization of elliptic pde’s [17], methods have been around for a long time. They have been two, adaptive approximations of functions (operators) and modified in several ways. A more important modification finally sparse representation of initially dense matrices is the Sloan iteration. ProjectionIJSER methods have also been arising from the descretization of integral equations. The used to solve nonlinear integral equations. Nonlinear sparse representation of initially dense matrices arising in integral equations have also been solved with the discretization of integral equations via wavelet bases, generalization of the Nystrom method. But, these methods leads to new methods for solution of integral equations. of historical interest impose severe restrictions on the More precisely, integral operators when expressed in underlying functions and integral operators and differ wavelet basis result in sparse matrices containing only drastically in their requirements [2]. Projection methods for O(nlogn) non-negligible elements(Alpert (1993)).Various example, require mean convergence of the relevant wavelet bases have been employed in the numerical expansions and hence impose conditions of the classical treatment of integral equations. The ultimate aim is to see L2 -theory, whereas Nystrom method works on the that the convergence is as rapid as possible. In addition to assumption that the functions and kernels are continuous the conventional Daubechies , trigonometric (Beylkin (1992)). This ultimately limits their applicability to wavelets, linear B-splines have been used [18], [19], a large class of problems. Moreover, these methods [20].These solutions are often complicated due to inherent converge too slowly due to their low approximation order, difficulties involved in differentiation and integration of to be of much use in solving integral equations. This the underlying basis functions. Obviously, attempts to motivates search for suitable alternative is to solve integral simplify procedures based on the wavelet approach are equations. needed. In the present paper, rather than employing wavelet basis for L2 ()R we use Haar wavelet packets that ———————————————— transform the dense matrices resulting from the * • Corresponding Author: Ruchi Agarwal, Department of Mathematics, discretization of 2nd kind integral equations into sparse School of Basic Sciences and Research, Sharda University, Greater Noida, India matrices, a fact which enables the corresponding integral • [email protected], [email protected], equations to be solved rapidly thereby, providing [email protected] substantial improvement over classical and current methods. The main focus of the paper is to show how the IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 467 ISSN 2229-5518 use of Haar wavelet-packet series (HWPS) leads to Clearly setting n = 0, we get 0 = φ ttw )()( , the father wavelet sparsification of smooth integral operators on a finite and n = 1 yields 1 = ψ ()( ttw ), the mother wavelet. Various interval, eventually leading to high order convergence. The combinations of these and their translations and dilations paper is organized as follows. In section 2, a mathematical can give rise to variety of bases for the function spaces. So, construction of new bases i.e. wavelet-packet bases is we have a whole collection of orthonormal bases generated explained. In section 3, the basic properties of Haar from {}tw .)( We call this collection “library of wavelet wavelet packet series are discussed in some detail; n j followed by abrief explanation of how Haar wavelet- packets”, and functions of the form wnjk = n {}22 − ktw packet basis can be used to solve both Fredholm and are called wavelet packets. Volterra linear integral equations. Section 4, describes the As a particular case, we look at the wavelet packets method of solution of nonlinear Volterra Hammerstein generated from the Haar integral equation using single term Haar wavelet-packet 1 series (STHWPS). Section 5, is devoted to convergence and filters. Since the Haar filter has hh 10 == and using 2 error analysis. Numerical examples purely for illustrative k 1 purposes are discussed in Sec. 6. gk −= )1( h1−k , g g10 =−= , we have 2 −+= 2. Wavelet packets and wavelet packet 2 nn n twtww )12()2( and −−= = χ Transform 2n +1 n n twtww )12()2( with 0 tw )( ]1,0[ the Haar A simple but powerful generalization of wavelets and the scaling function and 1 tw )( −= χχ ]1,5.0[]5.0,0[ , the Haar associated multiresolution analysis is wavelet packets. By wavelet. It indeed, turns out that {}wnjk are the well- generalizing the methods of multiresolution analysis, it is possible to construct orthonormal wavelet packets which known Walsh functions. Walsh proved that {}wnjk forms a complete orthonormal set (Fine, (1946). The full collection provide a family of of L2 (R ), which are of Haar wavelet packets consists of translated and dilated related in some way to the classical Walsh functions. The Walsh functions and can be represented by set of Walsh functions is, in fact, the prototype of wavelet- 1 2 j wnjk = n ()22 − ktw where wn is the nth . packet basis of L2 (R ), just as the Haar wavelet system is The wavelet packets transform generalizes the the prototype of wavelet basis on L (R ). Wavelet packets 2 discrete and provides a more flexible being particular linear combinations of wavelets; they tool for the time-scale analysis of functions. The wavelet retain many of the properties such as orthonormality, transform is actually a subset of a far more versatile smoothness and localization of their parent wavelets. An transformation, the wavelet packet transform. One step in interesting observation is that, wavelet packets generated the wavelet transform is that it calculates a low pass from Haar wavelets coincideIJSER with Walsh functions. To fix (scaling function) result and a high pass (wavelet function) ideas and notations, we know that, given the bases result. The wavelet transform applies the wavelet functions {}φ ,1 k()t of V1 from multiresolution analysis transform step only to the low pass result. The wavelet (MRA), ( ){}ψφ (−− ktkt ){} form orthonormal bases for V0 packet transform applies the transform step to both the low and W0 respectively and ⊕= WVV 001 , here pass and the high pass results. The wavelet packet method is a generalization of wavelet decomposition that offers a φ = φ − ψ = ψ − t ∑ k kth )2(2)( and (t) ∑ k ktg )2(2 richer range of possibilities for signal analysis and k k synthesis.. In wavelet analysis, a signal is split into an This splitting trick can be used to decompose W spaces as approximation and a detail. The approximation is then well. For example, if we analogously define: itself split into a second-level approximation and detail, = ψ − 2 tw ∑ k kth )2(2)( and the process is repeated. For n-level decomposition, k there are n+1 possible ways to decompose or encode the = ψ − − − signal. We can visualize this from Fig.1. and 3 tw ∑ k ktg )2(2)( then {}2 ,)( {}3 ktwktw )( k form orthonormal bases for the two subspaces whose direct sum is W1 . In general, for n = 0,1, . . , we define a sequence of functions as follows: = − 2n tw ∑ k n ktwh )2(2)( k

and Fig .1: Wavelet decomposition = − )2(2)( 2n +1 tw ∑ k n ktwg )2(2)( k

IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 468 ISSN 2229-5518 In wavelet packet analysis, the details as well as the  1 . . ...   2  approximations can be split. This yields more than − 2  . . ()m ()mI ...8  −   2n 1 . . . − () ()mIm ..41 2 different ways to encode the signal. This can be E =   ×mm  2 () () − 2 () visualized from Fig 2. This is an example of a ()m mI .8 mO .8 .. ()m mI 2  () () ()  representation that is not possible with ordinary wavelet  mIm .41. mO ..4   1 () ()  analysis (Andre Quinquis (1998)).  . . ()2m mI ..2 mO 2 

The Haar wavelet-packet transform matrix is defined as H = 1[)1( ], 1  11  H(2) =  , 2  1-1 

1 H k−1 H k−1)2()2(  k = H(2 ) k  k−1 k−1  2 H − H )2()2(  − ⊗= HH 2()2( k 1), 2 k ∈≤ N This is the matrix which corresponds to the Fast Haar wavelet-packet transform. After developing the above computational tools, we shall proceed to explain the technique of solving linear integral equations. A linear Fredholm integral equation of second Fig. 2: Wavelet packet decomposition kind is of the form b 3. Properties of Haar wavelet packet series += ∫ tftxKxgxf )(),()()( dt (3.5) (HWPS) a A function ∈ Lf 2([ 1,0 ]) , may be approximated using here K and g are given functions. 22 2 wavelet packets as The kernel is in baK ],[L and and bagf ],[L We use the symbol К to denote the integral operator of (3.5), which is )( = i i twctf )( (3.1) ∑ given by the formula i b From the property of wavelet 1 (Кf)(x) = ∫ tftxK dt ,)(),( = a packets, we have ci ∫ ()()wtf i t dt here i tw )( is ith 0 IJSER∈∀ 2 baf ],[L ∈ bax ],[ (3.6) , wavelet packet and ci is the corresponding Now we pass on to the solutions based on the Haar coefficient. In practice, only the first m terms are wavelet-packets. Since the Haar wavelet-packets are considered, where m is the integral power of 2. defined only for the interval [0,1], we must normalize Then from (3.1), we get equations (3.5) and (3.6). This can be done by the change of − m 1 ∗ () ()−−= = = T variable abatt ∑ i i twctf m wc m t)()()( , 0 To solve (3.5), using HWPS approach; we represent f and g here by their HWP series truncated to m terms: m−1 m−1 T cm = c ,( ,c . . . , 1-m10 ,)c (3.2) = = )( ∑ ii xwfxf )( and g( x ∑ i i xw )(g) T i=0 i=0 w m = (()( ), 10 (w ttwt ), . . ., 1-m (w t)) . (3.3) There are several methods by which we can approximate a The integration of a vector w m t)( defined above can be kernel. In the proposed method, we expand the kernel approximated by K(x,t) by a double HWP series (Shih and Han (1978)). t ∞ ∞ w m ττ = w m()( tEd ), (3.4) ≈ ∫ txK ),( ∑∑ ij ji twxwk )()( (3.7) 0 0 0 where E ×mm is the operational matrix for This series is convergent in the L2 mean and the coefficients 1 are given by integration (Chen and Hsiao (1975)) with E × = 11 2 1 1 = and ij ∫∫ ji twxwtxKk dxdt .)()(),( 0 0

IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 469 ISSN 2229-5518 An obvious choice of approximating kernel is the truncated and τ expression i−1 1 i 1 1 m−1 n −1 ττ += + λλλτλλλτ gf ii )()( ∑ )(),( dzk iiii )(),( dzk iiii txK ),( ≈ twxwk )()( m ∫ m∫ ∑∑ ij ji j=1 0 0 0 0 (4.4) Because the series (3.7) is convergent, the coefficients kij are τ λτ guaranteed to converge to zero as m, n increase. Let g i )( and ii ),k( be expressed by STHWPS i)( Now it is convenient to pass to a matrix-vector formulation of as τ i = wFg 0()( τ i ), i = 1,2,. . . , m. (4.5) (3.1). Let K be the matrix of the average values of K(x,t) on ji ),( mn k λτ ii ),( = wwK 00 ()( λτ ii ), i = 1,2, . . ., m, j = 1, . . ., i, all sub-squares of [0,1]× [0,1], a straightforward sampling of (4.6) K(x,t). This is a full matrix heavy to store, to multiply or to / mi invert. i)( = ()() τττ Here, ∫ wgF 0 ii d i and Next, our approach differs from the earlier work in that; it − /1 mi uses 2D-Haar wavelet packet transform (HWPT) and / mi / mj operational matrix of integration to approximate the integral ji ),( = mK 2 stk ),( dtds (4.7) operator: ∫ ∫ We first approximate K(x,t)using −/1 mi − /1 mj T Similarly, sf )( and z(s) are expanded by STHWPS as (), ≈ m mn HKHtxK m (3.8) i)( This is the double wavelet packet series λi = wYf 0 λi )()( , (4.8) approximation of the kernel. Secondly, / mi T where i)( = ()()wfY dτττ (Кf)(x) ≈ ×mm m mn m i EfHKHE ×mm (3.9) ∫ i 0 i i This is how a Haar wavelet-packet transform can be used − /1 mi i)( to “sparsify” or “compress” operators in integral = λλ ii wYhz 0(,()( λi )) (4.9) equations. Let z λi )( be expressed by STHWPS as It is easy to see that, the original integral equation (3.5) can λ = i)( λ be approximated by a system of algebraic equations i wZz 0 i )()( , (4.10) (Murlan Corrington (1973)), / mi i)( = ()() λλλ =− gfKI .)( (3.10) Where ∫ wzZ 0 ii d i Thus a Fredholm integral equation of 2nd kind is reduced to − /1 mi a finite matrix equation. Using E = ½ (Hsiao and Chen (1979)), and (4.6) It may be noted that, a Volterra integral equation with and (4.10), we get τ i convolutional kernel can be rewritten in Fredholm form. 1 λλλτ = iii )(),( τ 4. Single-term Haar IJSERWavelet packet series )(),( iiii wZKdzk 0( i ), i=1,. . .,m. (4.11) ∫ 2 (STHWPS) 0 In this section we extend single term approach developed Using (4.3) and (4.11), the Block-pulse value Y )1( in the by Sepehrian and Razzaghi (2005), to Haar wavelet-packets first interval is given by for the numerical solution of nonlinear Volterra- 1 FY )1()1( += ZK )1()1,1( , (4.12) Hammerstein integral equation of the form 2m t And similarly, += ∈ i −1 (,(),()()( sfshstkxgxf ))ds, x [0,1), (4.1) 1 1 ∫ FY iI )()( += ZK jji )(),( + ZK iii )(),( , (4.13) 0 m ∑ 2m Let = (,()( )), ssfshsz ∈[0,1). (4.2) j =1 In order to solve (4.1) using STHWPS, we first divide the which is a system of nonlinear equations for Y i)( , interval [0,1) into m equal subintervals, where m∈ N. We Then by using then stretch each interval i)( ifYif −−= )1(2)( =  (4.14) , i ,1 2, , ,//1 =≤≤− 1,2,..., mimitmi , to [0, 1) by using transforma gives Block-pulse and discrete values of solution function τ −−= λ −−= tions. i mt i )1( and i ms i )1( f(x). we then have 5. Numerical Examples τ1 1 We choose integral equations that can be solved gf ττ )()( += dzk λλλτ ;)(),( (4.3) 11 m ∫ 1111 analytically, so that accuracy and efficiency of the method 0 can be checked easily. We consider Fredholm, Volterra transformed into Fredholm form, Fredholm-Hammerstein and Volterra-Hammerstein type of integral equations.

IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 470 ISSN 2229-5518 1 original equation is obtained. Similar expressions can be ex+1 − 1 5.1 Consider, , )( exf x −= + ()()sfsxk ds x,, ∈[0,1] obtained when m = 16. x + 1 ∫ 0 Results obtained for m=8 and 16 are given in Table a Fredholm type linear integral equation, with a symmetric No. 1 and 2 respectively. xs s kernel, ),( = esxk f (s) = e . , Table. 1. Comparison of HWPS solution with exact The exact solution is f () = x .ex solution of Ex. 5.1, for m = 8 This is solved by the method explained in section 4.3, for m = 8 and 16.

Now, K88× the matrix of the average values of K(x,s) on all sub-squares of [0,1]× [0,1] is 1.00391 1.01179 1.01972 1.02772 1.03578 1.04391 1.05209 1.06034  1.01179 1.03578 1.06034 1.08549 1.11123 1.13758 1.16456 1.19218 1.01972 1.06034 1.10258 1.14651 1.19218 1.23967 1.28905 1.34041  1.02772 1.08549 1.14651 1.21095 1.27902 1.35091 1.42685 1.50705 1.03578 1.00023 1.19218 1.27902 1.37219 1.47214 1.57938 1.69443  1.04391 1.03758 1.23967 1.35091 1.47214 1.60425 1.74821 1.90509 1.05209 1.16456 1.28905 1.42685 1.57938 1.74821 1.35509 2.14195  1.06034 1.19218 1.34041 1.50705 1.69443 1.90509 2.14195 2.40826 xs From equation (3.8), e ≈ HKH ××× 888888 this in expanded form is  0.0129 0.0137 0.0147 0.0157 0.0168 0.0180 0.0184 0.0208   −0.0002 −− 0.0006 0.0011 − 0.0017 0.0023 0.0031 0.0030 0.0048 0 0.0001− 0.00004 −−−− 0.00009 0.0002 0.0003 0.0005 0.0006  −−0.0001 0.0003 0.00057 0.00085 0.00117− 0.0015 0.0010 0.0024 0 0− 0.00001 0.00002 0.00004−− 0.0008 0.0010 0.0001  Table. 2. Comparison of HWPS solution with exact 0 0 0 0 0−− 0.00001 0.0009 0.0003 solution of Ex.5.1, for m=16. 0 0 0.00002 −− 0.00005 0.00008 0.00013 0.0011 0.0003  −−0.00005 0.0001 0.000287 0.00043 − 0.00058 0.00078 0.0019 0.0012

From equation (3.9), b ∫ ()(), tftxK dt ≈ a IJSER 0.006311− 0.000067 − 0.000181 − 0.000113 − 0.0003999 −− 0.00041 0.000440 -0.000462  −0.003202 0.000039 0.000122 0.000130 0.0002023 0.000212 0.000221 0.000232 −0.001591 0.000017 0.000053 0.000054 0.000102 0.00010 0.000112 0.000121  −0.000028 0 −−0.000033 0.000043 0 0 0 -0.000151 . f −0.000798 0 0.000291 0.000026 0.000050 0.000053 0.000057 0.000061    0.000016 0 −−0.000131 0.000023 0 0 0 0   0 0 0 0 0 0 0 0   x  0 0 0− 0.000011 0 0 0 0  5.2 Consider )( si n ()()−+= tftxxxf dt a Volterra ∫ 0 We would like to draw attention to the above matrix. It is equation with convolutional but nonsymmetrical observed that, most of the entries are zero (or near to kernel. zero).Here the matrix contains only O(nlogn) non-negligible 1 elements proving the sparsity of the integral operator The exact solution is )( += xxxy 3 К(Alpert (1993)).This is one of the reasons for rapid 6 This can be rewritten in Fredholm form (Blyth convergence of the proposed method. 1

() += ()() Also, (2003)) as )( ∫ , tftxKxgxf dt x+1 0 x e − 1  ()1 16 ()3, 16 ()5, 16 gggg ()7, 16 ,  )( exg −= ≈  ⋅ H × + ()9 16 , ()1611 , ()1613 , gggg ()1615 88 si n () 0, ≤−  xttx x 1   here, )( = xxg and (),txK =  ={-0.648092,-0.056142, 0.018611,-0.0296445, 0.004797,-  ,0 ≤ tx  1 0.0015815, 0.0093635,-0.015025}. The solution obtained by solving, Equation (3.10) is in wavelet- This is solved by the method explained in section 3. packet domain. By inverse transformation, the solution of the Results obtained for m=8 and 16 are given in Table No. 3. The agreement between the exact solution

IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 471 ISSN 2229-5518 and the computed solution is impressive, even for The results obtained are plotted graphically. The graph taking small values for m. Numerical findings shows that plots of STHWPS solution and exact solution suggest that, the accuracy will improve are almost indistinguishable, even for small values of m. dramatically by taking m large enough.

HWPS-DISC-m=40 Table. 3. Comparison of HWPS solution with exact HWPS-Cont-m=40 HWPS-DISC-m=60 solution of Ex-5.2, for m = 8 and =16. 1.0 HWPS-Cont-m=60 EXACT

0.9

0.8 y

0.7

0.6

0.5 0.0 0.2 0.4 0.6 0.8 1.0 t

Fig. 3 Comparison of HWPS solution of example 5.4 with exact solution for m = 40 and m = 60. 5.3 Consider 1 6. Convergence and Error analysis ex−1 − (,(),()( sfsgsxkxf ))ds 1−= (si n 2 − cos 2) An important characteristic of any numerical method is not ∫ 2 only its ability to guarantee convergence but equally 0 important is the rate at which convergence is achieved. ex − ()cos1− si n ,1 Approximation with Haar wavelet-packets is equivalent to 2 the approximation with piecewise constant functions. If f, g x ∈[]1,0 , a Fredholm-Hammerstein nonlinear integral and K in (3.5) are sufficiently smooth, then the convergence equation with convolutional kernel, rate for piecewise constant functions is () = esxk −sx ,, (), () cos()+= ()sfxsfsg here the exact solution O(m2). This property can be transferred to HWPS approach is f(x) = 1. [20]. The efficiency of the method is demonstrated by some This is solved by the method explained in section 4. numerical examples; for getting error estimates, for which Results obtained with m=10 and 20 are given in the exact solution yex(x) is known are considered. The Table No. 4. IJSERaccuracy of the results is estimated by the error Table. 4. Comparison of error estimates for m=10 function ε J = max ( ()()i − ex xyxy i ) 1 ≤≤ mi and 20 of Ex-5.3. So it is evidenced numerically that in the case of our solution, by halving the step size (doubling the resolution from m=8, to m=16) the error function roughly decreases quadratically. This theoretical estimation in general holds for the numerical finding in Tables 1-4. Results from Tables 1 and 2, show that the method yields rapid convergence with convergence rate ≈ 2 . Thus second order convergence is observed, as predicted by the theoretical considerations. The results of Table 4 show that STHWPS approach applied to example 3 is behaving as an order two scheme as expected from error analysis. All calculations are done using mathematica programs. Conclusions 5.4Next consider In this paper, the application of Haar wavelet-packet basis t has been used for the solution of variety of second-kind tf 1)( si n 2 3)( si n( −−+= 2 sfstt ds t ∈ 1,0[,)() ), integral equations in which integral operators are ∫ represented as sparse matrices which involves solution of 0 A Volterra-Hammerstein integral equation which is ‘more’ the corresponding integral equations to be solved rapidly nonlinear than Example 5.3. This has the exact and accurately. These bases are very effective for the fast solution )( = cos ttf This equation is solved using the solution of wide class of such problems. The basic idea behind series and single-term approaches is to reduce the procedure explained in section 4.

IJSER © 2017 http://www.ijser.org International Journal of Scientific & Engineering Research Volume 8, Issue 6, June-2017 472 ISSN 2229-5518 problem to solving a system of linear or nonlinear 9. W.F.Blyth and V.Uljanov, Sloan Iteration and algebraic equations. HWPS approach does not involve any Richardson Extrapolation for Walsh Series Solutions of integration since operational matrix of integration Integral Equations ANZIAM J. Vol. 46(E) 690-703. transforms integration into matrix-vector multiplication. (2005) By contrast, STHWPS approach does not require 10. Murlan S. Corrington, Solution of Differential and operational matrices of integration avoiding the possibility Integral Equations with Walsh Functions IEEE Trans. of computing and storing matrices of enormously large On circuit theory, Vol. 20, NO.5 470-476. (1973) sizes. Furthermore, there is no restriction on m as in the 11. Yen-Ping Shih and Joung-Yi Han Double Walsh case of series approach. Another major advantage of these Series Solution of First Order Partial Differential new approaches is that besides meeting accuracy Equations Int. J. Systems Sci., Vol. 9, No. 5, 569-578. requirements, they lead to higher order convergence. (1978) These approaches have the additional advantage of O(N) 12. Jens Olav Nygaard and John Grue Wavelet Methods complexity, in the sense that the number of required for the Solution of Wave-Body Problems J, engg. Math. operations is of the same order as the number of computed Vol. 38, 323-354. (2000) values. Our experience dictates that, many other classes of 13. B. Alpert et al. Wavelet-Like Base for the Fast Solution integral equations can be solved efficiently using these of Second-Kind Integral Equations SIAM J. SCI. techniques. COMPUT. Vol.14. no. 1, 159-184. (1993) 2 14. B.Alpert A class of Bases in L for the Sparse References Representation of Integral Operators SIAM J.MATH. 1. Kendall E. Atkinson The Numerical Solution of ANAL Vol.21, No.1,pp. 246-262. (1993) Integral Equations of the Second Kind Cambridge 15. G.Beylkin On the Representation of Operators in University Press (1997) Bases of Compactly Supported Wavelets SIAM 2. L.M.Delves and J.L.Mohmed Computation Methods J.NUMER.ANAL. Vol.6, No.6, pp. 1716-1740. for Integral Equations Camridge University Press (1992) (1985) 16. G.Beylkin et al. Fast Wavelet Transforms and 3. C.H.Hasio and C.F.Chen Solving Integral Equations Numerical Algorithms I Communication on Pure via Walsh functions Comput. & Elec Engng Vol. 6. and Applied Mathematics, Vol. XLIV, 141-183. pp 279-292. (1979) (1991) 4. C.F.Chen and C.H.Hasio A Walsh Series Direct 17. Vainikko, G. Fast Solvers of Integral Equations of the Method for Solving Variational Problems Jl. Franklin Second Kind: Wavelet Methods. J. Complexity, 21, Inst. Vol. 4. pp 265-280. (1975) 243-273. (2005) 5. B.Sepehrian and M.Razzaghi Solution of Nonlinear 18. Chen, W.S. and Lin, W. Galerkin Trigonometric Volterra-Hammerstein Integral Equations via Single- Wavelet Methods for the Natural Boundary Integral Term Walsh Series MethodIJSER Math. Prob. Engng. Vol. Equations. Appl. Math. Comput. 121, 79-92, (2005) 5, 547-554. (2005) 19. Anioutine, A.P. and Kyurkchan, A.G. Application of 6. B.G.Sloss and W.F.Blyth A Walsh Function Wavelet Technique to the Integral Equations of the Method for a Non-linear Volterra Integral Method of Auxiliary Currents. J. Quant. Spectrosc. Equation Jl. Franklin Inst. Vol. 340. pp 25-41. Radiat. Transf., 79-80,495-508. (2003) (2003) 20. UloLepic and EnnTamme. Solution of Nonlinear 7. N.J.Fine On the Walsh functions Ph.d. thesis (1946) Fredholm Integral Equations via the Haar Wavelet 8. W.F.Blyth et al. Volterra Integral Equations Solved in Method. Proc. Estonian Acad. Sci. Phy. Math., 56, Fredholm form using Walsh Functions ANZIAM J. 17-27. (2007) Vol. 45(E) 269-282. (2004)

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