The sinc-galerkin method and its applications on singular Dirichlet-type boundary value problems

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1 Secer and Kurulay Boundary Value Problems 01, 01:16 R E S E A R C H Open Access The sinc-galerkin method and its applications on singular Dirichlet-type boundary value problems Aydin Secer 1* and Muhammet Kurulay * Correspondence: asecer@yildizedutr 1 Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa, İstanbul, 3410, Turkey Full list of author information is available at the end of the article Abstract The application of the sinc-galerkin method to an approximate solution of second-order singular Dirichlet-type boundary value problems were discussed in this study The method is based on approximating functions and their derivatives by using the Whittaker cardinal function The differential equation is reduced to a system of algebraic equations via new accurate explicitapproximations of theinnerproducts without any numerical integration which is needed to solve matrix system This study shows that the sinc-galerkin method is a very effective and powerful tool in solving such problems numerically At the end of the paper, the method was tested on several examples with second-order Dirichlet-type boundary value problems Keywords: sinc-galerkin method; sinc basis functions; Dirichlet-type boundary value problems; LU decomposition method 1 Introduction Sinc methods were introduced by Frank Stenger in [1] and expanded upon by him in [] Sinc functions were first analyzed in [3] and[4] An extensive research of sinc methods for two-point boundary value problems can be found in [5, 6] In [7, 8], parabolic and hyperbolic problems were discussed in detail Some kind of singular elliptic problems were solved in [9], and the symmetric sinc-galerkin method was introduced in [10] Sinc domain decomposition was presented in [11 13] and[14] Iterative methods for symmetric sinc-galerkin systems were discussed in [15, 16] and[17] Sinc methods were discussed thoroughly in [18] Applications of sinc methods can also be found in [19, 0] and[1] The article [] summarizes the results obtained to date on sinc numerical methods of computation In [14], a numerical solution of a Volterra integro-differential equation by means of the sinc collocation method was considered The paper [] illustrates the application of a sinc-galerkin method to an approximate solution of linear and nonlinear second-order ordinary differential equations, and to an approximate solution of some linear elliptic and parabolic partial differential equations in the plane The fully sinc-galerkin method was developed for a family of complex-valued partial differential equations with time-dependent boundary conditions [19] Some novel procedures of using sinc methods to compute solutions to three types of medical problems were illustrated in [3], and sincbased algorithm was used to solve a nonlinear set of partial differential equations in [4] A new sinc-galerkin method was developed for approximating the solution of convec- 01 Secer and Kurulay; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

2 Secer and Kurulay Boundary Value Problems 01, 01:16 Page of 14 tion diffusion equations with mixed boundary conditions on half-infinite intervals in [5] The work which was presented in [6] deals with the sinc-galerkin method for solving nonlinear fourth-order differential equations with homogeneous and nonhomogeneous boundary conditions In [7],sinc methodswere used tosolve second-order ordinary differential equations with homogeneous Dirichlet-type boundary conditions Sinc functions preliminaries Let C denote the set of all complex numbers, and for all z C, define the sine cardinal or sinc function by sin(πz), y 0, πz sin c(z)= 1, y =0 (1) For h > 0, the translated sinc function with evenly spaced nodes is given by sin(π z kh h ), z kh, π sin c(k, h)(z) = z kh h 1, z = kh () For various values of k, thesincbasisfunctions(k, π/4)(x) onthewholerealline < x < is illustrated in Figure 1 For various values of h, thecentralfunctions(0, h)(x) is illustrated in Figure If a function f (x) is defined over the real line, then for h >0,theseries C(f, h)(x)= ( ) x kh f (kh) sin c h k= (3) is called the Whittaker cardinal expansion of f whenever this series converges The infinite strip D s of the complex w plane, where d >0,isgivenby { D s w = u + iv : v < d π } (4) Figure 1 The basis functions S(k, h)(x)fork = 1, 0, 1 with h = π/4

3 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 3 of 14 Figure Central sinc basis function S(0, h)(x) for h = π/, π/4, π/8 Figure 3 The relationship between the eye-shaped domain D E and the infinite strip D S In general, approximations can be constructed for infinite, semi-infinite and finite intervals Define the function ( ) z w = φ(z)=ln 1 z (5) which is a conformal mapping from D E, the eye-shaped domain in the z-plane, onto the infinite strip D S,where { D E = z = x + iy : ( z arg 1 z ) < d π } (6) ThisisshowninFigure3 For the sinc-galerkin method, the basis functions are derived from the composite translated sinc functions ( ) φ(z) kh S h (z)=s(k, h)(z)=sin c h (7)

4 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 4 of 14 Figure 4 Three adjacent members S(k, h) φ(x)whenk = 1, 0, 1 and h = π 8 of the mapped sinc basis on the interval (0, 1) Table 1 Conformal mappings and nodes for some subintervals of R (a, b) φ(z) z k a b ln( z a b z ) a+be kh 1+e kh 0 1 ln( z 1 z ) e kh 1+e kh 0 ln(z) e kh 0 ln(sinh(z)) ln(e kh + e kh +1) z kh sinh 1 (z) kh for z D E These are shown in Figure 4 for real values x The function z = φ 1 (w)= ew 1+e w is an inverse mapping of w = φ(z) We may define the range of φ 1 onthereallineas = { φ 1 (u) D E : < u < } (8) the evenly spaced nodes {kh} k= on the real line The image which corresponds to these nodesisdenotedby ekh x k = φ 1 (kh)= (9) 1+ekh A list of conformal mappings may be found in Table 1 [6] Definition 1 Let D E be a simply connected domain in the complex plane C,andlet D E denote the boundary of D E Leta, b be points on D E and φ be a conformal map D E onto D S such that φ(a)= and φ(b)= Iftheinversemapofφ is denoted by ϕ,define = { φ 1 (u) D E : < u < } (10) and z k = ϕ(kh), k = 1,,

5 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 5 of 14 Definition Let B(D E )betheclassoffunctionsf that are analytic in D E and satisfy where ψ(l+u) F(z) dz 0, as u =, (11) { L = iy : y < d π }, (1) and those on the boundary of D E satisfy T(F)= F(z) dz < (13) D E The proof of following theorems can be found in [] Theorem 1 Let be (0, 1), F B(D E ), then for h >0sufficiently small, where F(z) dz h j= k(φ, h) z D = e [ iπφ(z) F(z j ) φ (z j ) = i F(z)k(φ, h)(z) D sin(πφ(z)/h) dz I F, (14) h sgn(im φ(z))] z D = e πd h (15) For the sinc-galerkin method, the infinite quadrature rule must be truncated to a finite sum The following theorem indicates the conditions under which an exponential convergence results Theorem If there exist positive constants α, β and C such that F(x) φ (x) C e α φ(x), e β φ(x), x ψ((, )), x ψ((0, )), (16) then the error bound for the quadrature rule (14) is F(x) dx h N F(x j ) ( e αnh φ (x j ) C α j= N ) + e βnh + I F (17) β The infinite sum in (14) is truncated with the use of (16) to arrive at the inequality (17) Making the selections πd h = αn, (18) 4 αn N β +1, (19)

6 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 6 of 14 where is an integer part of the statement and N is the integer value which specifies the grid size, then F(x) dx = h N j= N F(x j ) φ (x j ) + O( e (παdn)1/) (0) We used Theorems 1 and to approximate the integrals that arise in the formulation of the discrete systems corresponding to a second-order boundary value problem Theorem 3 Let φ be a conformal one-to-one map of the simply connected domain D E onto D S Then δ (0) jk = [ S(j, h) φ(x) ] 1, k = j, x=xk = (1) 0, k j, δ (1) jk = h d [ ] 0, k = j, S(j, h) φ(x) = dφ x=xk ( 1) k j (k j), k j, () δ () jk = h d [ ] π S(j, h) φ(x) 3 dφ =, k = j, (3) x=xk ( 1) k j, k j (k j) 3 The sinc-galerkin method for singular Dirichlet-type boundary value problems Consider the following problem: y + P(x)y + Q(x)y = F(x) (31) with Dirichlet-type boundary condition y(a)=0, y(b)=0, (3) where P, Q and F are analytic on D We consider sinc approximation by the formula y(x) y N (x)= N c k S(k, h) φ(x), (33) k= N S(k, h)= sin[ π (x kh)] h π h (x kh) (34) The unknown coefficients c k in Eq (33) are determined by orthogonalizing the residual with respect to the sinc basis functions The Galerkin method enables us to determine the c k coefficients by solving the linear system of equations LyN F, S(k, h) φ(x) =0, k = N, N +1,,N 1,N (35) Let f 1 and f be analytic functions on D and the inner product in (35) be defined as follows: f 1, f = w(x)f 1 (x)f (x) dx, (36)

7 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 7 of 14 where w is the weight function For the second-order problems, it is convenient to take [] w(x)= 1 φ (x) (37) For Eq (31), we use the notations (1)-(3) together with the inner product that, given (35)[], showed to get the following approximation formulas: F(x), S(k, h) φ(x) = Q(x)y(x), S(k, h) φ(x) = P(x)y (x), S(k, h) φ(x) = w(x)f(x)s(k, h) φ(x) dx hw k F k = φ k, (38) = h N w(x)q(x)y(x)s(k, h) φ(x) dx = h ( wk F k φ k w(x)p(x)y (x)s(k, h) φ(x) dx = h ( wk P k φ k j= N c j [ (Pw) j φ δ (0) kj +(Pw) δ (0) kj j h y (x), S(k, h) φ(x) ( = w(x)y (x)s(k, h) φ(x) dx wk = h φ k = h N j= N c j [ w j φ δ(0) kj + ) c k, (39) ) c k ], (310) ) c k (w j + w jφ j ) δ (1) kj φ j h + w jφ j δ () ] kj, (311) h where w k = w(x k ) If we choose h =(πd/αn) 1/ and w(x)=1/φ (x) as given in []theaccuracy for each equation between (38)-(311)willbeO(N 1/ e (πdαn)1/ ) Using (35), (38)-(311), we obtain a linear system of equations for N +1numbersc k The N +1linearsystemgivenin(35) can be expressed by means of matrices Let m = N +1,andletS m and c m be a column vector defined by S N c N S m (x)= S N+1, c m = c N+1 (31) S N c N Let A m (y) denote a diagonal matrix whose diagonal elements are y(x N ), y(x N+1 ),,y(x N ) and non-diagonal elements are zero, and also let I m (0), I m (1) and I m () denote the matrices I m (0) = I m (1) = N N N 1 N = [ δ (0) ] jk, (313) N 1 1 N = [ δ (1) ] jk, (314)

8 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 8 of 14 I m () = π 3 π π 1 (N) (N 1) 3 (N ) π (N) (N 1) (N ) 3 = [ δ () ] jk (315) With these notations, the discrete system of equations in (35)takes the form: LyN F, S m (k, h) φ(x) [ ( w = h I m (0) A m φ [ h I m (0) A m [ + h I m (0) A m ) + 1 h I(1) m A ( m w + wφ /φ ) + 1 h I() m A m( wφ )] c m ( (Pw) φ ( Qw φ ) + 1 ] h I(1) m A m(pw) c m )] c m ha m Fw φ (316) Theorem 31 Let c be an m-vector whose jth component is c j Then the system (316) yields the following matrix system, the dimensions of which are (N +1) (N +1): c = A m Fw φ (317) Now we have a linear system of (N +1)equations of the (N +1)unknown coefficients If we solve (317) by using LU or QR decomposition methods, we can obtain c j coefficients for the approximate sinc-galerkin solution y(x) y N (x)= N c k S(k, h) φ(x) (318) k= N 4 Examples Three examples were given in order to illustrate the performance of the sinc-galerkin method to solve a singular Dirichlet-type boundary value problem in this section The discrete sinc system defined by (318) was used to compute the coefficients c j ; j = N,,N for each example All of the computations were done by an algorithm which we have developed for the sinc-galerkin method The algorithm automatically compares the sincmethod with the exact solutions It is shown in Tables -4 and Figures 5-7 that the sinc- Galerkin method is a very efficient and powerful tool to solve singular Dirichlet-type boundary value problems Example 41 Consider the following singular Dirichlet-type boundary value problem on the interval [0, 1]: d dx y(x)+ y(x) 7 = x(x 1) 1,045 x + 1 1,045 x x4 + 1/19x 5, y(0) = 0, y(1) = 0 (41)

9 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 9 of 14 Table The numerical results for the approximate solutions obtained by sinc-galerkin in comparison with the exact solutions of Eq (41)forN = 100 x Exact solution Sinc-Galerkin Absolute error E E E E 11 Table 3 The numerical results for the approximate solutions obtained by sinc-galerkin in comparison with the exact solutions of Eq (4)forN = 100 x Exact solution Sinc-Galerkin Absolute error E E E E 10 Table 4 The numerical results for the approximate solutions obtained by sinc-galerkin in comparison with the exact solutions of Eq (43)forN = 100 x Exact solution Sinc-Galerkin Absolute error E E E E E E E E E E E E E E E 8 The exact solution of (41)is y(x)= 1,834,59 887,331,445 x + 917,96 887,331,445 x + 458, ,331,445 x3 188,07 34,571,355 x4 + 1, ,5,685 x5 + 78,597 x x7 We choose the weight function according to [], φ(x)=ln( 1 1 ), w(x)= 1 x φ, and by taking (x) d = π/, h = N, x k = ekh for N =8,16,3,100,thesolutionsinFigure5 and Table are 1+e kh achieved Example 4 Let us have the following form of a singular Dirichlet-type boundary value problem on the interval [0, 1]: dx y(x) 1 d x dx y(x)+ y(x) x(x +1) = x3, y(0) = 0, y(1) = 0 d (4)

10 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 10 of 14 Figure 5 Approximation to the exact solution: the red colored curve displays the exact solution and thegreenoneistheapproximatesolutionofeq(41) Figure 6 Approximation to the exact solution: the red colored curve displays the exact solution and thegreenoneistheapproximatesolutionofeq(4) Figure 7 Approximation to the exact solution: the red colored curve displays the exact solution and thegreenoneistheapproximatesolutionofeq(43) The problem has an exact solution like y(x)= (14 ln(x +1)x +14ln(x +1) 14x +6x 1x ln() x 3 +4x 3 ln() + x 4 x 4 ln() + 9x 5 18x 5 ln() ) / ( 1 + ln() ), where φ(x)=ln( 1 1 ), w(x)= 1 x φ Bytakingd = π/, h = (x) N, x k = ekh for N =8,16,3,100, 1+e kh we get the solutions in Figure 6 and Table 3 Example 43 The following problem is given on the interval [ 1, 4]: d dx y(x) d y(x)= (x 4), dx y( 1) = 0, y(4) = 0, (43) where the exact solution of (43)isy(x)= x e 4 x e 1 +15e x 6xe 4 +6xe 1 7e 4 8e 1 (e 4 e 1 )

11 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 11 of 14 In this case, φ(x)=ln( x+1 1 ), w(x)= 4 x φ, and by taking d = π/, h = (x) N, x k = 1+4ekh for 1+e kh N =8,16,3,100,wegetresultsinFigure7 and Table 4 5 Conclusion The sinc-galerkin method was employed to find the solutions of second-order Dirichlettype boundary value problems on some closed real interval The main purpose was to find the solution of boundary value problems which arise from the singular problems The examples show that the accuracy improves with increasing number of sinc grid points NWe have also developed a very efficient and rapid algorithm to solve second-order Dirichlettype BVPs with the sinc-galerkin method on the Maple computer algebra system All of the above computations and graphical representations were prepared by using Maple We give the Maple code in the Appendix section Appendix: Maple code which we developed for the sinc-galerkin approximation >restart: > with(linalg): > with(linearalgebra): >N:=100: > P(x):=-(1); P(x):= 1 >Q(x):=0; Q(x):=0 > F(x):=-(x-4); F(x):= x +4 > Boundaries:=z(-1)=0,z(4)=0; Boundaries := z( 1) = 0, z(4) = 0 > Example3:=diff(z(x),x$)+P(x)*diff(z(x),x$1)+Q(x)*z(x)=F(x); ( d ) ( ) d Example 1 := dx z(x) dx z(x) = x +4 > Exact_sol:=unapply(simplify(rhs(dsolve({Example3,Boundaries},z(x)))),x); Exact_sol := x 1 x e 4 x e ( 1) +15e x 6xe 4 +6xe ( 1) 7e 4 8e ( 1) e 4 e ( 1) > delta[0]:=unapply(piecewise(j=k,1,j<>k,0),j,k):

12 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 1 of 14 > delta[1]:=unapply(piecewise(j=k,0,j<>k,((-1)^(k-j))/(kj)),j,k): > delta[]:=unapply(piecewise(j=k,(-pi^)/3,j<>k,-*(-1)^(kj)/(k-j)^),j,k): > d:=pi/: >h:=/sqrt(n): > xk:=unapply((-1+4*exp(k*h))/(1+exp(k*h)),k); xk := k 1 + 4e(1/5k) 1+e (1/5k) > phi:=unapply(log((x+1)/(4-x)),x); ( ) x +1 φ := x ln x +4 > Dphi:=unapply(simplify(diff(phi(x),x)),x): > Dphi:=unapply(simplify(diff(phi(x),x$)),x): > g:=unapply(1/dphi(x),x): > Dg:=unapply(simplify(diff(g(x),x$1)),x): > Dg:=unapply(simplify(diff(g(x),x$)),x): > sys:=[]: >forpfrom-ntondo sys:=[op(sys), h*(sum(y[j]*((1/h^)*delta[](p,j)* (Dphi(xk(j))*g(xk(j)))+ (1/h^1)*delta[1](p,j)* ((Dphi(xk(j))/Dphi(xk(j)))* g(xk(j))+*dg(xk(j)))+ (1/h^0)*delta[0](p,j)* ((Dg(xk(j))/Dphi(xk(j))))), j=-nn) -sum(y[j]*((1/h^1)*delta[1](p,j)* (subs(x=xk(j),p(x)*g(x)))+ (1/h^0)*delta[0](p,j)* (subs(x=xk(j),diff(p(x)*g(x),x)))/dphi(xk(j)) ),j=-nn) +y[p]*subs(x=xk(p),g(x)*q(x))/dphi(xk(p)) -subs(x=xk(p),g(x)*f(x))/dphi(xk(p)))=0]: od: >evalf(sys): > vars:=seq(y[i],i=-nn):

13 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 13 of 14 > A,b:=LinearAlgebra[GenerateMatrix](evalf(sys),[vars]): > c:=linsolve(a,b): > ApproximateSol:=unapply(sum(c[j+N+1]*sin(Pi*(phi(x)- j*h)/h)/(pi*(phi(x)-j*h)/h),j=-nn),x): > plot([exact_sol(x),approximatesol(x)],x=-14, title ="Sinc- Galerkin Approximation",labels=["x","y"], legend = ["Exact Solution","Sinc-Galerkin"],); > Digits := 15; Digits := 15 > Exact:=[]:Apprx:=[]:Err:=[]: for s from -08 by 0 to do Exact:=[op(Exact),evalf(Exact_sol(s))]: Apprx:=[op(Apprx),evalf(ApproximateSol(s))]: Err:=[op(Err),evalf(abs(evalf(ApproximateSol(s))- evalf(exact_sol(s))))]: od: > latex(exact);latex(apprx);latex(err); [ , , , , , , , , , , , , , , ] [ , , , , , , , , , , , , , , ] [ , , , , , , , , , , , , , , ]

14 Secer and Kurulay Boundary Value Problems 01, 01:16 Page 14 of 14 Competing interests The authors declare that they have no competing interests Authors contributions AS proposed main idea of the solution schema by using Sinc Method for linear BVPs He developed computer algorithm and worked on theoretical aspect of problem MK searched the materials about study and compared with other techniques, contributed with his experience on Nonlinear Approximation methods Author details 1 Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa, İstanbul, 3410, Turkey Department of Mathematics, Faculty of Art and Sciences, Yildiz Technical University, Davutpasa, İstanbul, 3410,Turkey Received: September 01 Accepted: 15 October 01 Published: 9 October 01 References 1 Stenger, F: Approximations via Whittaker s cardinal function J Approx Theory 17,-40 (1976) Stenger, F: A sinc-galerkin method of solution of boundary value problems Math Comput 33, (1979) 3 Whittaker, ET: On the functions which are represented by the expansions of the interpolation theory Proc R Soc Edinb 35, (1915) 4 Whittaker, JM: Interpolation Function Theory Cambridge Tracts in Mathematics and Mathematical Physics, vol 33 Cambridge University Press, London (1935) 5 Lund, J: Symmetrization of the sinc-galerkin method for boundary value problems Math Comput 47, (1986) 6 Lund, J, Bowers, KL: Sinc Methods for Quadrature and Differential Equations SIAM, Philadelphia (199) 7 Lewis, DL, Lund, J, Bowers, KL: The space-time sinc-galerkin method for parabolic problems Int J Numer Methods Eng 4, (1987) 8 McArthur, KM, Bowers, KL, Lund, J: Numerical implementation of the sinc-galerkin method for second-order hyperbolic equations Numer Methods Partial Differ Equ 3, (1987) 9 Bowers, KL, Lund, J: Numerical solution of singular Poisson problems via the sinc-galerkin method SIAM J Numer Anal 4(1), (1987) 10 Lund, J, Bowers, KL, McArthur, KM: Symmetrization of the sinc-galerkin method with block techniques for elliptic equations IMA J Numer Anal 9, 9-46 (1989) 11 Lybeck, NJ: Sinc domain decomposition methods for elliptic problems PhD thesis, Montana State University, Bozeman, Montana (1994) 1 Lybeck, NJ, Bowers, KL: Domain decomposition in conjunction with sinc methods for Poisson s equation Numer Methods Partial Differ Equ 1, (1996) 13 Morlet, AC, Lybeck, NJ, Bowers, KL: The Schwarz alternating sinc domain decomposition method Appl Numer Math 5, (1997) 14 Morlet, AC, Lybeck, NJ, Bowers, KL: Convergence of the sinc overlapping domain decomposition method Appl Math Comput 98, 09-7 (1999) 15 Alonso, N, Bowers, KL: An alternating-direction sinc-galerkin method for elliptic problems J Complex 5, 37-5 (009) 16 Ng, M: Fast iterative methods for symmetric sinc-galerkin systems IMA J Numer Anal 19, (1999) 17 Ng, M, Bai, Z: A hybrid preconditioner of banded matrix approximation and alternating-direction implicit iteration for symmetric sinc-galerkin linear systems Linear Algebra Appl 366, (003) 18 Stenger, F: Numerical Methods Based on Sinc and Analytic Functions Springer, New York (1993) 19 Koonprasert, S: The sinc-galerkin method for problems in oceanography PhD thesis, Montana State University, Bozeman, Montana (003) 0 McArthur, KM, Bowers, KL, Lund, J: The sinc method in multiple space dimensions: model problems Numer Math 56, (1990) 1 Stenger, F: Numerical methods based on Whittaker cardinal, or sinc functions SIAM Rev 3, (1981) Stenger, F: Summary of sinc numerical methods J Comput Appl Math 11, (000) 3 Stenger, F, O Reilly, MJ: Computing solutions to medical problems via sinc convolution IEEE Trans Autom Control 43, 843 (1998) 4 Narasimhan, S, Majdalani, J, Stenger, F: A first step in applying the sinc collocation method to the nonlinear Navier Stokes equations Numer Heat Transf, Part B 41, (00) 5 Mueller, JL, Shores, TS: A new sinc-galerkin method for convection-diffusion equations with mixed boundary conditions Comput Math Appl 47, (004) 6 El-Gamel, M, Behiry, SH, Hashish, H: Numerical method for the solution of special nonlinear fourth-order boundary value problems Appl Math Comput 145, (003) 7 Lybeck, NJ, Bowers, KL: Sinc methods for domain decomposition Appl Math Comput 75, 4-13 (1996) doi:101186/ Cite this article as: Secer and Kurulay: The sinc-galerkin method and its applications on singular Dirichlet-type boundary value problems Boundary Value Problems 01 01:16