Numerical Solution of Fredholm and Volterra Integral Equations of the First Kind Using Wavelets bases

The Journal of Mathematics and Computer Science Available online at http://www.tjmcs.com The Journal of Mathematics and Computer Science Vol.5 No.4 (22) 337-345 Numerical Solution of Fredholm and Volterra Integral Equations of the First Kind Using Wavelets bases Maryam Bahmanpour Department of Mathematics, Sama Technical and Vocational Training College, Islamic Azad University, Khorasgan (Isfahan) Branch, Isfahan, Iran. ma.bahmanpour@sama-isf.ir Mohammad Ali Fariborzi Araghi Department of Mathematics, Central Tehran Branch, Islamic Azad University, P.O.Box 385.768, Tehran, Iran. m_fariborzi@iauctb.ac.ir Received: November 22, Revised: December 22 Online Publication: December 22 Abstract The Fredholm and Volterra types of integral equations are appeared in many engineering fields. In this paper, we suggest a method for solving Fredholm and Volterra integral equations of the first kind based on the wavelet bases. The Hr, continuous Legendre, CAS, Chebyshev wavelets of the first kind (CFK) and of the second kind (CSK) are used on [,] and are utilized as a basis in Galerkin or collocation method to approximate the solution of the integral equations. In this case, the integral equation converts to the system of linear equations. Then, in some examples the mentioned wavelets are compared with each other. Keywords: First kind Volterra and Fredholm integral equation; Galerkin method; Collocation method; Hr, Legendre, CAS, CFK, CSK wavelets. 2 Mathematics Subject Claification: Primary 45B5, 45D5; Secondary 65T6.. Introduction. The theory and application of integral equations are the important subjects in applied sciences. Integral equations are used as mathematical models for different physical situations. These integral equations also occur as reformulations of other mathematical problems such as ordinary and partial differential equations. For this reason, it is important to explain the appropriate methods with high accuracy to solve these kinds of integral equations numerically. 337

Many inverse problems in science and engineering lead to the solution of the following integral equations of the first kind[2,9], bb kk(, tt)ff(tt) = gg(), < bb <, () kk(, tt)ff(tt)dtt = gg(), < bb <, (2) where, gg() and kk(, tt) are known functions and ff(tt) is the unknown function to be determined. In general, these kinds of integral equations are ill-posed for a given kernel kk and driving term gg. For this reason, the special methods should be introduced to solve them. In recent years, several numerical methods for approximating the solution of the first kind of integral equation are known. Among these methods, the methods based on the wavelets are more attractive and considerable. The wavelets technique allows the creation of very fast algorithms when compared to algorithms which are ordinarily used. Various wavelet basis are applied. In [8] Maleknejad used Legendre wavelets, Xufeng Shang in [] applied the Legendre multi wavelets. In [6] and [5] Lepik and Gu proposed non-uniform Hr wavelets and Trigonometric Hermit wavelets too. Recently, wavelets basis are applied in order to solve various kinds of integral equations [,2,8]. In [7] the first kind integral equations of Volterra type are solved by using Hr wavelet. In this paper, we propose the method which was presented in [3, 4] by applying other kinds of well known wavelets such as Legendre and Chebyshev wavelets and compare the results with Hr wavelet. In this paper, we present the application of Legendre [8], CFK [], CSK and CAS [2] wavelets as a basis functions in Galerkin method for numerical solution of the Eq. () and compare them with each other. The method is tested by using of some numerical examples. Also, we apply and compare the Legendre, CFK and CAS wavelets as the basis functions in collocation method in order to solve the Eq. (2). At first, in section 2, we introduce the wavelets which are used as basis functions to approximate the unknown function. Then, in sections 3 and 4 we remind the Galerkin and the collocation methods for computing the unknown function using the mentioned wavelets. Finally, in section 5, we solve some Fredholm and Volterra integral equations of the first kind with different wavelets and compare them with each other. 2. Wavelets and their properties Wavelets constitute a family of functions constructed from dilation and translation of a single function called mother wavelet. When the dilation parameter and the translation parameter bb vary continuously we have the following family of continuous wavelets as [8], ψψ,bb (tt) = 2ψψ tt bb,, bbbbr, where, ψψ,bb (tt) forms a wavelet basis for LL 2 (R). We consider the parameters and bb as discrete kk kk values =, bb = nnbb, >, bb > where, nn and kk are positive integers. In particular, when = 2 and bb = then ψψ kk,nn (tt) form an orthonormal basis [8]. In this work, we use the following wavelets. In the sequel, the notation (.,. ) means the inner product and (.,. ) ww denotes the inner product with respect to the weight function ww(tt). 2.. Legendre wavelets Consider the well-known Legendre polynomials of order mm, LL mm (tt), which are orthogonal with respect to the weight function ww(tt) = and derived from the following recursive formula: LL (tt) =, LL (tt) = tt, LL mm + (tt) = 2mm+ ttll mm+ mm (tt) mm LL mm + mm (tt), mm =,2,3, Legandre wavelets ψψ nn,mm (tt) = ψψ(kk, nn, mm, tt) have four arguments; kk = 2,3,, nn =,2,3,, 2 kk, mm =,,2,, MM. mm is the order of Legendre polynomials and MM is a fixed positive integer. They are defined on the interval [,) as follows: 338

2LL ψψ nn,mm (tt) = 2kk mm 2 kk nn tt 2nn + tt < nn 2 kk 2 kk ooooheeeeeeeeeeee, where, LL mm (tt) = (2mm + ) 2LL mm (tt). (3) 2.. CFK wavelets CFK wavelets ψψ nn,mm = ψψ(kk, nn, mm, tt) have four arguments, nn =,2,, 2 kk, kk can aume any positive integer, mm is the degree of Chebyshev polynomials of the first kind and tt denotes the time. 2TT ψψ nn,mm (tt) = 2kk mm 2 kk nn tt 2nn + tt < nn, 2 kk 2 kk ooooheeeeeeeeeeee, where, mm =, ππ TT mm (tt) = 2 (4) TT ππ mm (tt) mm >, and mm =,,, MM, nn =,2,, 2 kk. In Eq.(4), TT mm (tt) are Chebyshev polynomials of degree mm which are orthogonal with respect to the function ww(tt) = / tt 2,on the interval [,],and satisfy the following recursive formula: TT (tt) =, TT (tt) = tt, TT mm+ (tt) = 2ttTT mm (tt) TT mm (tt), mm =,2, 2.2. CSK wavelets CSK wavelets ψψ nn,mm = ψψ(kk, nn, mm, tt) also have four arguments, nn =,2,, 2 kk, kk can aume any positive integer, mm is the degree of Chebyshev polynomials of the second kind [] and tt denotes the time. 2UU ψψ nn,mm (tt) = 2kk mm 2 kk nn tt 2nn + tt < nn, 2 kk 2 kk ooooheeeeeeeeeeee, where, UU mm (tt) = 2 ππ UU mm (tt), mm (5) and mm =,,, MM, nn =,2,, 2 kk. In Eq.(5), UU mm (tt) are Chebyshev polynomials of degree mm which are orthogonal with respect to the function ww(tt) = tt 2,on the interval [,],and satisfy the following recursive formula: UU (tt) =, UU (tt) = 2tt, UU mm+ (tt) = 2ttUU mm (tt) UU mm (tt), mm =,2, The set of ψψ nn,mm (tt) are composed and be expanded by Legendre, FKC or SKC wavelet series as ff(tt) = nn= mm= cc nn,mm ψψ nn,mm (tt), (6) where, cc nn,mm = ff(tt), ψψ nn,mm (tt) ww. If the infinite series in (6) is truncated, then (6) can be written as 2 kk MM nn= mm=. ff(tt)~ cc nn,mm ψψ nn,mm (tt) 2.3. CAS wavelets 339

CAS wavelets ψψ nn,mm (tt) = ψψ(kk, nn, mm, tt) have four arguments, nn =,,, 2 kk, kk can aume any nonnegative integer, mm is any integer and tt is the normalized time and weight function is ww(tt) =. They are defined on the interval [,) as ψψ nn,mm (tt) = 2kk 2CCCCCC mm 2 kk nn tt nn tt < nn+ 2 kk 2 kk (7) ooooheeeeeeeeeeee, where, CCCCCC mm (tt) = cos(2mmmmmm) + sin(2mmmmmm). (8) The set of ψψ nn,mm (tt) are composed and be expanded by CAS wavelet series as ff(tt) = nn= mmmm Z cc nn,mm ψψ nn,mm (tt), (9) where cc nn,mm = ff(tt), ψψ nn,mm (tt). If the infinite series in (9) is truncated, then (9) can be written as 2 kk MM ff(tt)~ nn= mm= MM cc nn,mm ψψ nn,mm (tt).[2] () 3. Galerkin Method In Galerkin method, we choose a sequence of finite dimensional subspaces XX nn XX = LL 2 [,] with dim(xx nn ) = nn. Suppose that XX nn = {φφ, φφ 2,, φφ nn } where {φφ ii } nn ii= are the orthonormal Legendre or Chebyshev wavelet basis and yy nn is a function belongs to XX nn as an approximate solution of Eq. (). Hence, we can write, yy nn (tt) = nn jj = cc jj φφ jj (tt). By substituting into Eq. (), we have bb bb rr nn () = kk(, tt)yy nn (tt) gg() = kk(, tt) nn cc jj = jj φφ jj (tt) gg(), bb, where, rr nn is called the residual in the approximation of Eq.() [2]. Now, to determine unknown coefficients, we impose the following requirements: rr nn (), φφ ii () ww = ii =,2,, nn, nn, bb. Thus, we obtain the system of algebraic equation: AAAA = GG. () In this system, the unknowns vector is CC = [cc, cc 2,, cc nn ] TT and the elements of AA and GG are: bb iiii = ( kk(, tt)φφ ii (tt)dtt, φφ jj ()) ww, gg ii = gg(), φφ ii () ww, ii, jj =,2,, nn. In this section, we apply the Galerkin method based on the mentioned wavelets as basis for solving Eq.(). In the tables, the absolute errors are computed for each one of the wavelets. [a,b]=[,] Algorithm3.. nn Step : Consider the basic functions of Legendre, CAS, FKC or SKC wavelets {φφ ii ()} ii= Step 2: Calculate matrix AA, nn nn, with entry iiii = ww()kk(, tt)φφ ii (tt)φφ jj () and matrix FF, nn with entry ff ii = ww()φφ ii ()gg() and then solve the system of AAAA = FF to obtain CC. Step 3: Approximate yy() in Eq.() with yy nn (tt) = cc ii φφ ii (tt), where cc ii is i-th entry of CC. Step 4: Compute the error of the method using EE = ff() ff nn () = where f and ff denote the exact solution and the numerical solution, respectively. 4. Collocation Method nn ii= 34

In this section, as we mentioned in the previous section to approximate the unknown function in the integral Eq. (2). Let ff nn (tt) = nn jj = cc jj φφ jj (tt) be the approximate solution of Eq. (2). Now, by substituting in integral Eq. (2) we get the following residual function: rr nn () = kk(, tt) nn cc jj φφ jj (tt) dtt gg(). jj = Now, for determining unknown coefficients cc jj we can choose an expansion method [6]. We choose the collocation method to find cc jj. In the collocation method, we select some collocation points then put the residual equation in these points equal to zero. So, we have rr nn ( ii ) = ii kk( ii, tt) nn jj = cc jj φφ jj (tt) dtt gg( ii ) =, where, the collocation points are: ii = + ii(bb ), ii =,2,, MM2 kk nn. Therefore, the following system of linear equations is formed [6,], AAAA =G, (2) where, the elements of AA and GG are: iiii = [ ii nn kk( ii, tt)φφ jj (tt)] ii,jj = nn. and bb ii = [gg( ii )] ii= In the Legendre, CFK or CSK wavelet nn = MM2 kk. Algorithm 4.. Step : Consider the basic functions of Hr [5], Legendre [8] and Chebyshev [] wavelets as {φφ ii ()} nn ii=. Step 2: Calculate matrix AA with entry iiii = ii KK( ii, tt) φφ jj (tt) and vector GG with entry GG ii = gg( ii ) and then solve the system AAAA = GG to obtain CC. Step 3: Approximate ff() in Eq.() with ff nn (tt) = nn ii= cc ii φφ ii (tt), where cc ii is i-th entry of CC. Step 4: Compute the error of the method by using EE = ff() ff nn () =, where ff and ff denote the exact solution and the numerical solution respectively. 5. Illustrative examples In this section, we apply the collocation method based on the mentioned wavelets as basis for solving Eq. (). In the tables, the absolute errors are computed for each one of the wavelets. The programs have been provided with Mathematica 6 according to the algorithms and 2 in the interval [, bb] = [,]. We use the points mentioned in (6) and construct the system (7) in the algorithm. Example 5.. Consider the following Fredholm integral equation of the first kind: kk(, tt)yy(tt) = ee + ( ee), tt( ) > tt, kk(, tt) = (tt ) tt. The exact solution is yy() = ee. Table illustrates the results of the example. Table. Comparison of Legendre,FKC, CKC and CAS wavelets methods with k = andm = 2 CFK wavelet CSK wavelet CAS wavelet Legendre wavelet.6359.3732.28773.46845..97782.779.557322.28883 34

.2.4375.279823.37895.98.3.6763.89238.3878.37496.4.28255.32386.5777.8663.5.422557.47835.492577.389.6.397554.327474.78.3553.7.987.64749.72787.563.8.28722.39959.726.26799.9.85372.865262.28.5982 Table shows that the results of the example for FKC, SKC and CAS wavelets are better than the result of Legendre wavelets. Example 5.2. Consider the following Fredholm integral equation: ( 2 + tt 2 ) 3 yy(tt) = +2 3 3, with exact solution yy() =. The numerical results are represented in tables 2 and 3. With comparing the results of tables 2 and 3, we can observe that the growth of the error in FKC and SKC wavelets, as k and M increase, is le than the growth in the Legendre wavelets. Table 2. Comparison of Legendre, FKC and SKC wavelets method with kk = 2 and MM = 2 CFK wavelet CSK wavelet Legendre wavelet 4.79773e 9 2.279e 9 4.85723e 5..839e 9 6.6598e 7.2645e 6.2.955e 9 7.82395e 6.352e 5.3 4.862e 9 2.22999e 9.8794e 4.4 7.884e 9 3.67758e 9.7486e 4.5.243e 8 7.22839e 9 4.33e 4.6 8.37393e 9 4.95e 9 2.8886e 4.7 4.6473e 9 2.6783e 9.4988e 4.8 8.3553e 3.93559e.88738e 5.9 2.93367e 9.88472e 9.233e 4 Table 3. Comparison of Legendre, CFK and CSK wavelets method with kk = 3 and MM = 3 CFK wavelet CSK wavelet Legendre wavelet.584e 6 3.6294e 6.997e 8. 5.7364e 5 8.9372e 7 6.94946e 8.2.7645. 38299 3.793e 7.3.5328. 84774.59.4.88. 25236.22.5.3978. 568369.4944.6.53297. 2785.26823.7.886. 238855.286.8.84679. 233272.2768.9.7588. 85788.3444 342

Example 5.3. In this example, we consider the following Volterra integral equation: ff(tt)dtt = (ππ 2 +tt 2 4 + 4 ln(4)). The exact solution is ff(tt) = tt 2 tt +. Table 4 illustrates the results of this example. Table 4. Comparison of Hr, Legendre and CFK wavelets with k = 3, M = 4 Hr wavelet Legendre wavelet CFK wavelet /6.27424.22e 6 2/6.223668 2.2245e 6 2.2245e 6 3/6.84296 3.3367e 6 4/6.45393.88738e 5 2.9976e 5 5/6.6433 9.992e 6.33227e 5 6/6.674223 7.7756e 6 2.2245e 5 7/6.283853.5543e 5.9984e 5 8/6.6646 8.54872e 5 9.2485e 5 9/6.49729.5543e 5 2.9976e 5 /6.8783 2.2245e 6 5.552e 5 /6.2784.66533e 5 5.8848e 5 2/6.6693.6982e 4 5.73e 4 3/6.25966 5.8848e 5 3.88578e 5 4/6.24528.66533e 4.8794e 4 5/6.2849.25455e 4 8.32667e 5 Example 5.4. In this example, we consider the following Volterra integral equation: ee tt + 4ππ cos (4ππππ)+sin (4ππππ) 4ππee 2 ff(tt)dtt =. + 2 (+6ππ 2 ) The exact solution is ff(tt) = sin(4ππππ). Table 5 shows the result for example 5.4. Table 5. Comparison of Hr, Legendre and CFK wavelets with k = 3, M = 4 Hr wavelet Legendre wavelet CFK wavelet /6.54976.962564.962564 2/6.2496.8575.8575 3/6.495292.3583.3583 4/6.483232.94333.94333 5/6.6889.65563.65563 6/6.238234.6735.6735 7/6.388895.2698.2698 8/6.23382845.52.52 9/6.482326.432794.432794 /6.8647874.749734.749734 /6.8767629.72492.72492 2/6.388923.3242299834.3242299834 3/6.3444788.2965663753.29656637533 343

4/6.72956486.63572632474.635726324742 5/6.8464694.53945627967.539456279676 As we observe in the tables 4 and 5, the result of the examples for Legendre and Chebyshev wavelets is more accurate than the result of Hr wavelet. 6. Conclusion The aim of this work is to propose a method for solving Volterra integral equation of the first kind using Legendre and Chebyshev wavelets. The presented method converts the integral equation into a system of linear algebraic equations. We applied the Legendre and Chebyshev wavelets as basis function and observed that these wavelets can behave better than the Hr wavelet on [,]. Acknowledgement. The authors are grateful to research affairs of Sama Technical and Vocational Training College, Islamic Azad University Khorasgan (Isfahan) Branch, for supporting this research. References. [] E. Babolian, F. Fattahzadeh, Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Compt., (27), pp. 6-22. [2] L. M. Delves, J. L. Mohamed, Computational Methods for integral equations, Cambridge University Pre, (985). [3] M. A. Fariborzi Araghi, M. Bahmanpour, Numerical Solution of Fredholm Integral Equation of the First kind using Legendre, Chebyshev and CAS wavelets, International J. of Math. Sci. & Engg. Appls. (IJMSEA) Vol. 2 No. IV (28), pp. -9. [4] M. A. Fariborzi Araghi, M. Bahmanpour, Numerical Solution of Fredholm Integral Equation of the First Kind by Using Chebyshev Wavelets, Proceeding of the International Conference of Approximation on Scientific Computing (ICASC'8), 26-3 Oct. (28), ISCAS, Beijing, China. [5] J. S. Gu, W. S. Jiang, The Hr wavelets operational matrix of integration, Int. J. Sys. Sci. 27, (996), pp. 623-628. [6] U. Lepik, Solving integral and differential equations by the aid of non-uniform Hr wavelets, Appl. Math. Compt. 98 (28), pp. 326-332. [7] K. Maleknejad., R. Mollapourasl and M. Alizadeh, Numerical solution of Volterra type integral equation of the first kind with wavelet basis, Appl. Math. Compt. 94 (27), pp. 4-45. [8] M. Maleknejad, S. Sohrabi, Numerical solution of Fredholm integral equation of the first kind by using Legendre wavelets, Appl. Math. Compt., (27), pp. 836-843. [9] M. Rabbani, R. Jamali, Solving Nonlinear System of Mixed Volterra-Fredholm Integral Equations by Using Variational Iteration Method, The Journal of Mathematics and Computer Science, Vol.5 No.4 (22), pp. 28-287. [] T. J. Rivlin, Chebyshev Polynomials, John Wiley and Sons, Second Edition, (99). 344

[] X. Shang and D. Han, Numerical solution of Fredholm integral equation of the first kind by using linear Legendre multi-wavelets, Appl. Math. Compt. 9 (27), pp. 44-444. [2] S. Yousefi, A. Banifatemi., Numerical solution of Fredholm integral equations by using CAS wavelets, Appl. Math. Compt.,( 26), pp. 458-463. 345