Spectral collocation method for compact integral operators

Transcription

1 Wayne State University Wayne State University Dissertations --0 Spectral collocation method for compact integral operators Can Huang Wayne State University, Follow this and additional works at: Part of the Mathematics Commons Recommended Citation Huang, Can, "Spectral collocation method for compact integral operators" (0). Wayne State University Dissertations. Paper 35. his Open Access Dissertation is brought to you for free and open access by It has been accepted for inclusion in Wayne State University Dissertations by an authorized administrator of

2 SPECRAL COLLOCAION MEHOD FOR COMPAC INEGRAL OPERAORS by CAN HUANG DISSERAION Submitted to the Graduate School of Wayne State University, Detroit, Michigan in partial fulfillment of the requirements for the degree of DOCOR OF PHILOSOPHY 0 MAJOR: MAHEMAICS Approved by: Advisor Date

3 DEDICAION o My Parents Siyong Huang and Yueping Zheng ii

4 ACKNOWLEDGEMENS his work could not have been finished without the help of many people. Especially, I am heartily thankful to my advisor Professor Zhimin Zhang, whose endless encouragement and caring, exceptional guidance and consistent support from the initial and final level enabled me to develop a better understanding of the subject. His guidance and encouragement were indispensable during my study in Wayne State University. I also would like to express my deepest appreciation to Professor George Yin for giving me many valuable opportunities to review papers on stochastic differential equations with him. I learned a lot from his comments and discussions. I am grateful to Professor Pao-liu Chow for his service for me on stochastic partial differential equations in summer 009, also for his consistent caring and help. I am taking this opportunity to thank Doctor Fatih Celiker and Professor Weisong Shi for serving in my committee. My thanks also goes to Ms. Mary Klamo, Professor John Breckenridge, Professor Bertram Schreicer, Dr. Huiqing Zhu and Mr. James Veneri for their help and encouragement in one way or another. Finally, I wish to thank all staff in the mathematics department for their cooperation and generous assistance. I enjoyed the warm and friendly atmosphere in the department and I appreciate the support I received during my stay at Wayne State University. iii

5 ABLE OF CONENS Dedication ii Acknowledgements iii Chapter Introduction Chapter Spectral Collocation Method for Integral Equations.. 8. Preliminary knowledge Algorithm Algorithm for Volterra integral equation with weakly singular kernel Algorithm for Fredholm integral equations heoretical Analysis Volterra integral equations Fredholm integral equations Numerical Results Volterra Integral Equation Fredholm Integral Equation Chapter 3 Spectral Collocation for Integro-Differential Equations 3 3. Algorithms Algorithm for Volterra integro-differential equation Algorithm for Fredholm integro-differential equation iv

6 3. Convergence Analysis Volterra Integro differential equations Fredholm integro-differential equation Numerical Experiments Chapter 4 Eigenvalue approximation for compact integral operators Preliminary Algorithms he first algorithm for Weakly Singular Kernel he Second Algorithm for the Weakly Singular Kernel Algorithm for the Piecewise Smooth Kernel Algorithm for the Smooth Kernel Convergence Analysis Numerical Results Chapter 5 Concluding Remarks Abstract Autobiographical Statement v

7 Introduction he early history of integral equations goes back to the special integral equations studied by several mathematicians of late eighteenth century and early nineteenth century Laplace, Fourier, Poisson, Abel and Liouville while the systematic study of integral equations begins with work of Volterra [4], where he transformed an integral equation t K(t, s)y(s)ds = g(t), t [0, ] by differentiation with respect to t, into 0 an integral equation of the form y(t) = t 0 K(t, s)y(s)ds + f(t), t [0, ], (.) and Fredholm [0], where he gave necessary and sufficient conditions for solvability of integral equations of the form y(t) = 0 K(t, s)y(s)ds + f(t), t [0, ]. (.) Since then, integral equations and integro-differential equations are used as mathematical models for many and varied physical situations, for example in population dynamics [6] and financial mathematics [9, 30] and integral equations also occur as reformulations of other mathematical problems, see [, 6, 3, 44]. Among these integral equations, singular kernel problems are especially attractive, see [, 9, 8, 39]. he development of theory and application of integral equations stimulates the development of numerical treatment of integral equations. Some well-known monographs are [3, 4, 6, 5]. In the present thesis, we try to apply a spectral collocation method to solve integral or integro-differential equations and related eigenvalue prob-

8 lems. Here, compact kernels include weakly singular kernels, smooth or piecewise smooth kernels. Numerical solution to integral or integro-differential equations. Specifically, we study the numerical solution of equations (.), (.) and integrodifferential equations of the form y (t) = a(t)y(t) + t 0 K(t, s)y(s)ds + f(t), y(0) = y 0, t [0, ] (.3) and y (t) = a(t)y(t) + 0 K(t, s)y(s)ds + f(t), y(0) = y 0, t [0, ] (.4) where the kernel K(t, s) is the weakly singular kernel with 0 < µ < Volterra Equations (t s) K(t, s) = µ Fredholm Equations t s µ (.5) a(t) is a smooth function and f(t) is a given function and we assume that these equations possess a unique solution [, 44]. Numerical approximation to eigenvalues We consider eigenvalue problem of 0 K(t, s)y(s)ds = λy(t), t [0, ]. (.6) where K(t, s) is compact with aforementioned forms. Properties of eigenvectors of weakly singular kernel is studied in [38]. In this thesis, we particularly focus on the equations with weakly singular kernels because this type of equations are related to fractional differential equations. Application of fractional differential equations include Fluid Flow, Rhology, Dynamical

9 3 Processes in Self-Similar and Porous Structures, Electrical Networks, Probability and Statisics and so on. It is well-known that Cauchy problem of fractional differential equations and the Volterra integral equations are equivalent under certain conditions, see [7]. For example, ( C Da+y)(x) α = f[x, y(x)] (α > 0; a x b) y (k) (a) = b k, b k R, k = 0,,, n ; n = [ α] (.7) where ( C D α a+y) is the Caputo derivative defined as ( C D α a+y) = x y (n) (t) dt Γ(n a) a (x t) α n+ and [ ] is the integer part of a number. his equation can be reduced to the Volterra integral equation y(x) = n b j j! (x a)j + x f[t, y(t)]dt (.8) Γ(α) a (x t) α Moreover, if f(x, y) C[a, b] and satisfies the Lipschitz condition, then there exists unique solutions to the Cauchy problem (.7) in the space C α,n [a, b]: C α,n [a, b] = {y(x) C n [a, b] : C D α a+y C[a, b]} and to the Cauchy problem (.8) in the space C n [a, b], see [7, heorem 3.5]. So, if y(x) C n [a, b], then (.7) and (.8) are equivalent. Furthermore, the condition can be relaxed to y(x) L[a, b] for the equivalence of Volterra equation and Riemann- Liouville fractional differential equations, see also [7]. hus, numerical approximation of integral equations with compact kernels is of large importance and thus, a very hot topic in the past fifty years. Big effort can

10 4 be found in [3, 4, 6, ] while numerical solutions of integro-differential equation are studied in [6, 33, 40] etc. However, in [3, 4], authors devoted to integral equations with smooth kernels and for weakly singular kernel problem, they used very basic numerical quadratures such as Simpson s Rule or Gauss quadrature to approximate the integration. he disadvantage is that the order of Simpson s Rule is low and Gauss quadrature does not yield a high accuracy for integration with weakly singular kernels. In [6], however, only Volterra equations (integro-differential) equations and related delay equations are studied. Many other numerical analysts used graded meshes to develop numerical schemes with an optimal order of convergence, see [, 5, 33]. o use this method, certain properties of solutions should be known before implementation, for example, the pole and its order. Recently, in [3, 4] a spectral Jacobi-collocation method was proposed and analyzed to solve Volterra equations. he basic idea is to collocate equations at some Jacobi points and use a highly accurate quadrature to approximate the integration in (.). In this thesis, however, instead of numerical integration, we apply exact integration of the composition of the Legendre polynomials and the kernel to solve the Volterra equation (.), which leads to less error and computation cost of our method. It will be shown that a geometric (super-geometric) rate of convergence can be achieved by using our method if the true solution y(t) satisfies Condition (R): y (k) L [0, ] ck!r k

11 5 or Condition (M): y (k) L [0, ] cm k, M > not only for Volterra equations but also for Fredholm equations as well as their corresponding integro-differential equations. Here, R is sufficiently large. If R is small, hp-version of our method is necessary and for Volterra equations, if the solution is not smooth enough, we may take some function transformations as in [4] to change the equation into a new one so that the new equation possesses better regularity. As to super-geometric convergence of spectral collocation method for differential equations, readers are referred to [4, 45]. Regarding to the numerical approximation of the eigenvalue problems with compact kernels, various methods such as Galerkin method, Petrov-Galerkin method, collocation method, Nyström method, and degenerate kernel methods have been intensively studied. he results are well-documented in the literature. Here, we mention a few related to our work. In the mid-70 s, Osborn established a general spectral approximation theory for compact operators, when a sequence of { n } approximates in a collectively compact manner. he analysis of [3] covers many methods and provides a basis for the convergence analysis of our method. In [8], Dellwo and Friedman proposed a new approach by solving a polynomial eigenvalue problem of a higher degree, base on which, Alam etc. [] obtained an accelerated spectral approximation for eigenelements. Kulkarni [4] introduced another method by involving a new approximation operator n and obtained a high-order convergence rate. In addition, a multiscale method was discussed in []. Comprehensive studies for eigenvalue

12 6 problem can be found in [5, 5, 43]. Because the regularity of eigenfunctions directly affect the convergence rate of eigenvalue approximation, we approximate eigenfunctions by some appropriate orthogonal polynomial expansions. Different from previous methods in the literature, we use the exact integration when calculating the convolution of the singular kernel with the orthogonal polynomials as we did in solving integral equations. he key ingredients here are some special identities. It is worthy to mention that in one of our numerical experiments, we obtain a thirteen-digit of accuracy of the approximation of the first eigenvalue for a weakly singular kernel even if its eigenvector is not smooth. o summarize, in this dissertation, we investigate the numerical solution of Volterra /Fredholm integral equation with weakly singular kernels, Volterra/Fredholm integrodifferential equation with weakly singular kernels and eigenvalue problem for compact integral operators including weakly singular kernels, smooth or piecewise smooth kernels and SDE with jump diffusion. Our contributions include:. By using some identities, we avoid large numerical quadrature errors accumulated with the singular kernels and thereby obtain higher accuracy for numerical approximation of integral equations and eigenvalue approximations; we avoid product integration method and therefore reduce the computational cost.. We prove geometric or super-geometric rate of convergence for numerical approximation of integral or integro-differential equations which was observed in [3, 4]. Also, we observe and prove the geometric or super-geometric rate of convergence for eigenvalue approximation given a general integraql compact operator.

13 7 3. We observe and prove a refined convergence rate of eigenvalue approximation (compared to geometric or super-geometric rate of convergence), if the kernel is positive definite. 4. For the weakly singular kernel of the form t s µ, 0 < µ <, we can obtain a very high digit of accuracy even though the eigenvector is not smooth. he rest of the dissertation is arranged as follows. In Chapter, we will provide algorithms of spectral collocation method for integral equations, including some essential identities that we use in our algorithms. Numerical Experiments and the proof of geometric or super-geometric rate of convergence for these equations are also covered in this chapter. Chapter 3 aims to provide algorithms, numerical experiments and convergence analysis for integro-differential equations. In Chapter 4, we will study the algorithms and numerical experiments for eigenvalue problems, especially for the eigenvalue approximation of weakly singular kernel problem. We use two different methods, compare their accuracy and make a convergence analysis. 5. Finally, we end this dissertation with conclusions and further remarks in Chapter

14 8 Spectral Collocation Method for Integral Equations his chapter is devoted to illustrate the algorithms for solving integral equations with weakly singular kernels. For the integral operators of the form (t s) µ or t s µ, 0 < µ <, we apply some existing identities to obtain a high digit of accuracy and also avoid product integration and therefore reduce the computational cost.. Preliminary knowledge he class of Jacobi polynomials P (α,β) k (x) are essentially used in our algorithms. It is well-known that these polynomials are polynomials solutions to the Jacobi differential equation [9], d dx [( x)+α ( + x) +β y ] + n(n + α + β + )( x) α ( + x) β y = 0. (.) Under the normalization P (α,β) k P (α,β) k (x) = k () = ( ) k+α, one has the expression, namely, k l=0 k ( k + α k l )( k + β Jacobi polynomials satisfy the three-term recursive relations: l ) (x ) l (x + ) k l. (.) P (α,β) 0 (x) =, P (α,β) (x) = [(α β) + (α + β + )x] a,k P (α,β) k+ (x) = a,kp (α,β) k (x) a 3,k P (α,β) k (x) (.3)

15 9 where a,k = (k + )(k + α + β + )(k + α + β), a,k = (k + α + β + )(α β ) + xγ(k + α + β + 3)/Γ(k + α + β), a 3,k = (k + α)(k + β)(k + α + β + ). (.4) A useful formula that relates Jacobi polynomials and their derivatives is in particular, one has d m dx m P (α,β) k (x) = m Γ(k + α + β + + m) Γ(k + α + β + ) P (α+m,β+m) k m (x); (.5) d dx P (α,β) k (x) = (α+,β+) (k + + α + β)p k (x). (.6) Let h(t) be a smooth function on [0, ] and write f(x) = h( ( + x)), x [, ]. Let p be the Chebyshev polynomial of the first kind with degree p and I p f P p [, ] interpolate f at (p + ) Chebyshev points: x i = cos i+ π, i = 0,, p. hen the p+ remainder of the interpolation is f(x) I p f(x) = f[x 0, x,, x p, x]ν(x) (.7) where ν(x) = (x x 0 )(x x ) (x x p ). Hence, f(x) I p f(x) = f[x 0, x,, x p, x] p+ p+ (x), (.8)

16 0 since the leading coefficient of p+ (x) is p+. Moreover, if f C p+ [, ], the divided difference f[x 0, x,, x p, x] = f (p+) (ξ x ) (p + )!, ξ x (, ). (.9) herefore, if h satisfies condition (R), we have ( ) p+ f I p f L [,] C (.0) 4R and if h satisfies condition (M), we obtain by the Stirling formula, f I p f L [,] Define a weighted L norm by v w α,β = ( ( ) C p+ em (.) p + 4(p + ) ( x) α ( + x) β v(x) dx From (.8), the rates of interpolation error under condition (R) and condition (M) in weighted L norm are exactly the same as those in L norm, respectively. Next, we introduce two identities, which will be essential in this paper. Lemma.. ([35]) Let a, b be positive constants and L n (x) be the Legendre polynomial with degree n on [, ], then ). b a b a (x a) α L n ( x b n! )dx = (b a) α P n (α, α) ( a ), b < a < b; α > 0, (.) (α) n+ b (b x) β L n ( x n! )dx = (b + a) β P n ( β,β) ( b ), a < b < a; β > 0, (.3) a (β) n+ a where (k) n+ = k(k + ) (k + n).

17 Specifically, if we choose a =, b = x, β = µ in (.3), then we obtain x L n (t) (x t) dt = n! ( + x) µ P µ n (µ, µ) (x), (.4) ( µ) n+ and a = x, b =, α = µ in (.), we achieve x L n (t) (t x) dt = n! ( x) µ P µ n ( µ,µ ) (x). (.5) ( µ) n+. Algorithm.. Algorithm for Volterra integral equation with weakly singular kernel he equation is of the form: After a change of variable y(t) t 0 y(s) ds = f(t), t [0, ]. (.6) (t s) µ t = ( + x), (.6) can be written as u(x) where x [, ] and (+x)/ 0 u(x) = y hen, we make another change of variable ( ( ( + x) s) µ y(s)ds = g(x), (.7) ) ( ) ( + x), g(x) = b ( + x). (.8) s = ( + τ), τ [, x], (.9)

18 we arrive at u(x) ( ) µ x u(τ) dτ = g(x). (.0) (x τ) µ Let u p (x) = p c j L j (x), which is the approximation of u(x). hen we require that c js satisfy the equations at the collocation points, p c j L j (x i ) By virtue of (.4), we have ( ) µ p xi c j L j (τ) (x i τ) µ dτ = g(x i), i = 0,, p. (.) ( ) µ ) p j! c j (L j (x i ) (+x i ) µ P (µ, µ) j (x i ) = g(x i ), i = 0,, p. ( µ) j+ (.).. Algorithm for Fredholm integral equations We solve a class of Fredholm integral equation of form: y(t) Make a change of variable 0 y(s) ds = f(t), t [0, ]. (.3) t s µ t = ( + x), and we obtain u(x) (+x)/ 0 ( µ ( ) µ ( + x) ( + x) s) y(s)ds s y(s)ds = g(x) (+x)/ where u(x) and g(x) are defined the same as those in (.8). Again, let (.4) s = ( + τ), τ [, ]. (.5)

19 3 We reach u(x) ( ) µ x ( ) µ u(τ) (x τ) dτ µ x u(τ) dτ = g(x), (x τ) µ x [, ] (.6) Let u p (x) = p c j L j (x) be the approximation of u(x). hen c js satisfy the equations p c j L j (x i ) ( ) µ p xi c j ( ) µ p L j (τ) (x i τ) dτ c µ j x i L j (τ) (τ x i ) µ dτ = g(x i ), i = 0,, p. (.7) Applying (.4) and (.5), we obtain p c j (L j (x i ) ( ( ) µ j! ( µ) j+ ( + x i ) µ P (µ, µ) j (x i ) ) µ j! ( x i ) µ P ( µ,µ ) j (x i ) ( µ) j+ ) = g(x i ), i = 0,, p. (.8) Remark. Note that (.) and (.), (.7) and (.8) are equivalent. However, in numerical application, we use (.) and (.8) and for the convenience of theoretical analysis, we use (.) and (.7). For the sake of analysis, we define the Lagrange interpolation at (p+) Chebyshev points, i.e I p u(x i ) = u(x i ), 0 i p. (.9) It is clear that I p u(x i ) = p u(x i )l i (x), (.30) i=0

20 4 where l i (x) is the Lagrange interpolation basis function at Chebyshev points. Hence, It is obvious that ( ) y(t) = y ( + x) = u(x), t [0, ] and x [, ]. (.3) y p (t) = y p ( ( + x) ) = u p (x), t [0, ] and x [, ]. (.3) herefore, (y y p )(t) = (u u p )(x) := e(x). (.33).3 heoretical Analysis his section is devoted to the convergence analysis of our algorithms to Volterra (.6) or Fredholm (.3) integral equations..3. Volterra integral equations We first present some useful lemmas. Lemma.3.. [4] Let {l j (x)} p be the Lagrange interpolation polynomials at the Chebyshev points, then I p := max x [,] p l j (x) = O(log p). (.34) In our analysis, we shall apply the following generalization of Gronwall s leamma, see [3].

21 5 Lemma.3.. Suppose L 0, 0 < µ < and v(t) is a non-negative, locally integrable function defined on [0, ] satisfying u(t) v(t) + L t hen there exists a constants C = C(µ) such that 0 (t s) µ u(s)ds. (.35) u(t) v(t) + CL t 0 (t s) µ v(s)ds, 0 t <. (.36) Let C r,κ ([0, ]) denote the space of function whose r-th derivatives are Hólder continuous with exponent κ, endowed with the usual norm r,κ. hen from the result of [36, 37], we have the following lemma Lemma.3.3. Let r be a non-negative integer and κ (0, ). hen for any v C r,κ ([, ]), there exists a polynomial function N v P N such that v N v CN (r+κ) v r,κ (.37) Here, N is a linear operator from C r,κ to P N. We now define a linear integral operators M by M v(x) = x v(τ) dτ. (.38) (x τ) µ Lemma.3.4. [3] Let κ (0, µ) and M be defined by (.38) under the condition 0 < κ < µ. hen for any function v C([0, ]), there exists a positive constant C such that M v 0,κ C v. (.39)

22 6 heorem.3.5. Let y(t) and u p (x) be the exact solution and spectral approximation of (.6), respectively. ) If y satisfies the condition (R), then, for sufficiently large p, ( ) p+ u u p C, (.40) 4R ) If y satisfies the condition (M), then, for sufficiently large p, u u p Proof: Using (.), we have u i = ( ) µ xi ( ) C p+ em. (.4) p + 4(p + ) u p (τ) (x i τ) µ dτ + g(x i), (.4) where u i = p c j L j (x i ). Note that the true solution at Chebyshev points satisfies u(x i ) = hus, let e(x) = u(x) u p (x), u(x i ) u i = ( ) µ xi ( ) µ xi u(τ) (x i τ) µ dτ + g(x i). (.43) e(τ) dτ. (.44) (x i τ) µ Multiplying both sides by l i (x) and summing up from 0 to p and applying the fact that u p (x) = p c j L j (x) = p u j l j (x) give I p u u p = ( ) µ [ ] x e(τ) I p (x τ) dτ, (.45) µ

23 7 We write e(x) = u(x) I p u(x) + I p u(x) u p (x). (.46) Obviously, e(x) = ( ) µ x e(τ) (x τ) dτ + u(x) I pu(x) µ }{{} I ( ) µ { [ ]} x e(τ) + (I p I) (x τ) dτ, (.47) µ }{{} I where I is the identity operator. By the Gronwall s inequality, namely, Lemma.3., we have e(x) I + I + C herefore, x (x τ) µ I + I dτ. (.48) e C( I + I ). (.49) It follows from (.0) and (.) that if y(t) satisfies condition (R), then and if y(t) satisfies condition (M), then I ( ) p+ I C, (.50) 4R ( ) C p+ em. (.5) p + 4(p + ) Now let s estimate I. We derive from Lemma.3.3 and Lemma.3.4 that M e p M e Cp κ M e 0,κ, κ (0, µ). (.5)

24 8 Hence, I = = ( ) µ (I p I)M e (.53) ( ) µ (I p I)(M e p M e) (.54) ( ) µ ( + I p ) M e p M e (.55) Cp κ log p e. (.56) herefore, for sufficiently large p, we obtain which implies I C e, (.57) ( ) p+ e C (y satisfies condition (R)); (.58) 4R e ( ) C p+ em (y satisfies condition (M)). (.59) p + 4(p + ).3. Fredholm integral equations For our method, the error analysis of the L norm for Fredholm equations is similar to that of Volterra equations. However, the analysis for Fredholm type equations is not included in [3, 4]. Let us start with a property of weakly singular integral operators. Lemma.3.6 ([39] Corollary..) Weakly singular integral operators are compact from L to C[0, ]( and hence from L to L ) and from C[0, ] to C[0, ].

25 9 We now define a linear integral operators M by M v(x) = x v(τ) dτ. (.60) (τ x) µ Lemma.3.7 Let κ (0, µ) and M be defined by (.60) under the condition 0 < κ < µ. hen for any function v C[0, ], there exists a positive constant C such that M v 0,κ C v. (.6) Proof: We need to prove M v(t ) M v(t ) t t κ C v. (.6) Without loss of generality, we assume t < t. hen, we have M v(t ) M v(t ) t t κ = (t t ) κ v(s) t (s t ) ds v(s) µ t (s t ) ds µ [ ] = (t t ) κ (s t ) v(s)ds µ (s t ) µ t t t v(s) (s t ) ds µ I + I, (.63) where [ ] I = (t t ) κ t (s t ) v(s)ds µ (s t ) µ t I = (t t ) κ v(s) (s t ) ds µ. t We now estimate these two terms one by one. By simple calculation and the condition κ (0, µ), we derive

26 0 ) If t t t, I (t t ) κ µ ( t ) µ ( t ) µ + (t t ) µ v (t t ) κ µ ( µ)( ξ) µ (t t ) + (t t ) µ v, ξ (t, t ) (t t ) κ µ ( µ)( t ) µ (t t ) + (t t ) µ v C(t t ) µ κ v C v. (.64) due to t t <. ) If t t > t, then I (t t ) κ µ = (t t ) µ κ µ Noting that the function ( t ) µ ( t ) µ + (t t ) µ v ( ) µ ( ) + t + t µ t t t t v (.65) k(x) = + x µ ( + x) µ, x [0, ] is non- negative and achieves its maximum at x =, we have I C(t t ) µ κ v C v. (.66)

27 As for I, t I = (t t ) κ t (s t ) v(s)ds µ t (t t ) κ v (s t ) ds µ t (t t ) κ v (t t ) µ µ C v. (.67) hen, the desired result is followed by combining (.64), (.66) and (.67). heorem.3.8 Let y(t) and u p (x) be the exact solution and spectral approximation of (.3), respectively. ) If y satisfies the condition (R), then, for sufficiently large p, ( ) p+ u u p C, (.68) 4R ) If y satisfies the condition (M), then for sufficiently large p, ( ) u u p C p+ em. (.69) p + 4(p + ) Proof: Using (.7), we have u i = where ( ) µ ( ) µ xi (x i τ) µ u p (τ)dτ + (τ x i ) µ u p (τ)dτ +g(x i ), (.70) x i u i = p c j L j (x i ). he true solution at Chebyshev points satisfies ( ) µ ( ) µ xi u(x i ) = (x i τ) µ u(τ)dτ + (τ x i ) µ u(τ)dτ +g(x i ), (.7) x i

28 hus, let e(x) = u(x) u p (x), ( ) µ xi u(x i ) u i = (x i τ) µ e(τ)dτ + ( ) µ (τ x i ) µ e(τ)dτ. (.7) x i Multiplying both sides by l i (x) and summing up from 0 to p, and applying the fact that u p (x) = p c j L j (x) = p u j l j (x) give I p u u p = We write ( ) µ I p [ x (x τ) µ e(τ)dτ ] + ( ) µ [ ] I p (τ x) µ e(τ)dτ. x (.73) e(x) = u(x) I p u(x) + I p u(x) u p (x). (.74) Obviously, e(x) = ( ) µ x τ µ e(τ)dτ + u(x) I p u(x) }{{} + ( where I is the identity operator. ) µ { (I p I) [ I x τ µ e(τ)dτ ]} } {{ } I By Lemma.3.6 and the Fredholm Alternative, we have, (.75) e C( I + I ). (.76) It follows from (.0) and (.) that if y(t) satisfies condition (R), then ( ) p+ I C, (.77) 4R and if y(t) satisfies condition (M), then ( ) I C p+ em. (.78) p + 4(p + )

29 3 We derive from Lemma.3.3,.3.4 and.3.7 that M i e p M i e Cp κ M i e 0,κ, κ (0, µ), i =,. (.79) Hence, I = = ( ) µ (I p I)M e + (I p I)M e ( ) µ ) ( (I p I)(M e p M e) + (I p I)(M e p M e) ( ) µ ( + I p )( M e p M e + M e p M e ) Cp κ log p e. (.80) he desired results (.68) and (.69) follow from (.77), (.78) and (.80) for sufficiently large p..4 Numerical Results.4. Volterra Integral Equation In this subsection, we will find numerical approximation to solutions of two examples to demonstrate our heorem.3.5. Unlike the numerical scheme in [3, 4] which they called spectral Jacobi method, our scheme is of the form LC p AC p = G p, (.8)

30 4 where C p = [c 0,, c p ] and G p = [g(x 0 ),, g(x p )] and the elements of the matrix A = (a ij ) and L = (l ij ) are given by a ij = l ij = L j (x i ), ( ) µ j! ( + x i ) µ P (µ, µ) j (x i ), ( µ) j+ which is obtained from (.). In this subsection, reference curve is the graph of function f(p) = p+ associating with condition (R) and reference curve is the graph of f(p) = p+ ( e.5 4(p+) )p+ corresponding to condition (M). Example.4. Consider a Volterra integral equation of the form (.) on [0, 6] with α = and b(t) = (t + ) /3 3π 8 (t + ) (t + ) arctan ( ) t (3t + 0). t 4 he exact solution for this example is y(t) = ( + t) 3/. Obviously, this solution satisfies condition (R). Hence, we expect a geometric rate of convergence. Numerical errors for our method and the spectral Jacobi method are presented in able and Figure. We see that our method outperforms the spectral Jacobi method for larger p > 0. Example.4. Consider a Volterra equation of the form (.) on [0, 4] with µ = and b(t) = 3 et ( γ(, t)), where γ(a, x) is the lower incomplete gamma function 3 defined by: γ(a, x) = x 0 t a e t dt.

31 5 able : Example.4.: L error and weighted L error for t [0, 6] N L Our Method e e e e-04 Spectral Jacobi Mehtod 8.495e e e e-04 w /, / Our method 7.057e e e e-04 w µ,0 Spectral Jacobi Method.349e e e e-04 N L Our method 8.709e e e e-09 Spectral Jacobi Method 7.087e e e e-07 w /, / Our method 8.395e e e e-09 w µ,0 Spectral Jacobi Method 8.093e e e e Our Method Spectral Jacobi Method Reference Curve 0 0 Our Method: w /, / Spectral Jacobi Method: w µ,0 Reference Curve e L Weighted L norms p p Figure. L error for both methods Weighted L error for both methods

32 6 able : Example.4.: L error and weighted L error for t [0, 4] N L Our Method e-0.40e e e-06 Spectral Jacobi Mehtod 4.038e e e e-06 w /, / Our method 5.835e-0.0e e e-06 w µ,0 Spectral Jacobi Method e-0.739e e e-05 N L Our method.35e e e e-3 Spectral Jacobi Method e e-09.e-.38e- w /, / Our method 9.434e e e-.3305e-3 w µ,0 Spectral Jacobi Method.359e e e-.74e Our Method Spectral Jacobi Method Refence Curve Our Method: w /, / Spectral Jacobi Method: w µ,0 Refence Curve e L Weighted L norms p p Figure. L error for both methods Weighted L error for both methods

33 7 he exact solution of this equation is y(t) = e t, which satisfies condition (M). We expect a supergeometric rate of convergence for numerical approximations, see able and Figure..4. Fredholm Integral Equation From algorithm for (.3), we can obtain the scheme LC p AC p = G p, where C p = [c 0,, c p ], G p = [g(x 0 ),, g(x p )], and the elements of the matrix A = (a ij ) and L = (l ij ) are given by ( ) µ j! a ij = ( + l ij = L j (x i ). ( µ) j+ ( + x i ) µ P (µ, µ) j (x i ) ) µ j! ( x i ) µ P ( µ,µ ) j (x i ), ( µ) j+ In this subsection, reference curve is the graph of function f(p) = 4 p+ associating with condition (R) and reference curve is the graph of f(p) = corresponding to condition (M). e p+ ( 4(p+) )p+ Example.4.3 Consider a Fredholm equation of the form y(t) = 6 Specifically, we choose µ = / and b(t) = t + π (t + ) + (t + ) arctan ( t 0 y(s) ds + b(t). t [0, 6] (.8) t s µ (t + ) log( 6 t + 4 ) + (t + ) log( t + ) ) t t

34 8 able 3: Example.4.3: errors for t [0, 6] N L 4.805e e e e-07 w /, / 6.563e e e e-07 N 4 8 L.4987e e e e-4 w /, / 3.484e e-0.349e e-3 so that the true solution is y(t) = t +. Clearly, the solution satisfies condition (R). Numerical errors are reported in able 3 and Figure 3. Example.4.4 Consider a Fredholm equation of the form We choose µ = / and y(t) = 0 0 y(s) ds + b(t). t [0, 0] (.83) t s µ b(t) = sin(t) + [ ( ) ( )] t t π cos(t)s sin(t)c π π [ ( ) ( )] (0 t) (0 t) π sin(t)c + cos(t)s π π Here, C(u) and S(u) are Fresnel integrals defined by C(u) = u 0 cos( πx )dx, S(u) = u 0 sin( πx )dx hen, the true solution is y(t) = sin(t). It is obvious that the solution satisfy condition (M). Numerical results are given in able 4 and Figure 4.

35 9 0 Our Method Refence Curve 0 Our Method Refence Curve e L 0 8 Weighted L norm p p Figure 3. L error for our method w /, / error for our method 0 Our Method Refence Curve 0 Our Method Refence Curve e L Weighted L norm p p Figure 4. L error for our method w /, / error for our method

36 30 able 4: Example.4.4: errors for t [0, 0] N L.55e-0.44e e e-05 w /, / e e-0.953e-03.85e-05 N L.833e e-.890e-.076e-4 w /, /.8859e-08.75e e e-4

37 3 3 Spectral Collocation Method for Integro-Differential Equation In this chapter, we focus on the spectral collocation method for integro-differential equations of first order whereas our algorithms can be generalized to arbitrary order of integro-differential equations with weakly singular kernels. 3. Algorithms 3.. Algorithm for Volterra integro-differential equation We consider the following equation y (t) = a(t)y(t) + t 0 y(s) (t s) µ ds + f(t), y(0) = y 0, t [0, ] (3.) We take the same notations and variable transformation as in algorithm for (.6), then we obtain ( ) µ x u(s) u (x) = f(x)u(x) + ds + g(x), (3.) (t s) µ where f(x) = a( ( + x)). Let and note that from (.6) u p (x) = y 0 + p c j (L j (x) + L j (x)) j= d dx L n(x) = n + P (,) n (x). (3.3)

38 3 We have [ ] p c j (j + )P (,) j (x i) + jp (,) j (x i) j= + ( = f(x i ) ( y 0 + ) p c j (L j (x i ) + L j (x i )) ) µ xi y 0 + p c j (L j (τ) + L j (τ)) j= dτ + g(x (x i τ) µ i ), i =,, p. (3.4) By virtue of (.4) again, we obtain p j= ( c j [ j + P (,) j (x i) + j P (,) j (x i) (L j (x i ) + L j (x i ))f(x i ) (3.5) ) µ j! ] P (µ, µ) j (x i ) ( + x i ) µ P (µ, µ) j (x i ) ( µ) j+ = g(x i ) + f(x i )y 0 + j= ( ) µ (j )! ( + x i ) µ ( µ) j ( ) µ y 0 µ ( + x i) µ, i =,, p. 3.. Algorithm for Fredholm integro-differential equation We investigate algorithm for the equation y (t) = a(t)y(t) + 0 y(s) t s µ ds + f(t), y(0) = y 0, t [0, ]. (3.6) We take the same notation and variable transformation as in algorithm for (.3) and we obtain ( ) µ x u (x) = f(x)u(x) + ( ) µ u(τ) (x τ) dτ + µ x u(τ) dτ + g(x), (τ x) µ x [, ]. (3.7) Again, f(x) has the same definition as in the previous algorithm. Let u p (x) = y 0 + p c j (L j (x) + L j (x)) be the approximation of u(x). hen c j j=

39 33 must satisfy the equation g(x i ) = ( ) p c j (L j(x i ) + L j (x i )) f(x i )(y 0 + j= ( ( ) µ xi y 0 + p c j (L j (τ) + L j (τ)) dτ (x i τ) µ p c j (L j (x i ) + L j (x i ))) ) µ y 0 + p c j (L j (τ) + L j (τ)) dτ. (3.8) x i (τ x i ) µ Apply (.5) and (3.3), we obtain p j= ( [ j + c j P (,) j (x i) + j P (,) j (x i) (L j (x i ) + L j (x i ))f(x i ) (3.9) ) µ ( j! ( + x i ) µ P (µ, µ) (j )! j (x i ) + ( + x i ) µ P (µ, µ) j (x i ) ( µ) j+ ( µ) j j! + ( x i ) µ P ( µ,µ ) (j )! j (x i ) + ( x i ) µ P (µ, µ) j (x i ) ( µ) j+ ( µ) j ( ) µ ( ) y 0 = g(x i ) + f(x i )y 0 + µ ( + x i) µ + y 0 µ ( x i) µ, j= )] i =,, p. (3.0) 3. Convergence Analysis his section concerns the L norm of errors for both type of equations. We will apply the Fredholm Alternative in the proof of this subsection. Lemma 3.. (he Fredholm Alternative) Let X be a Banach space, and let A L(X, X) be a compact operator. hen the equation x = Ax + g, g X has a unique solution x X if and only if the homogeneous equation z = Az has only the trivial solution z = 0. In such a case, the operator I A has a bounded inverse

40 34 (I A) L(X, X). 3.. Volterra Integro differential equations heorem 3.. Let y and y p be the exact solution and spectral approximation of (3.), respectively. ) If y(t) satisfies the condition (R), then, for sufficiently large p, y y p ( ) p+ C(p + ), (3.) R 4R ) If y satisfies the condition (M), then, for sufficiently large p, ( ) y y p C log p p+ em. (3.) p + 4(p + ) Proof: Using (3.6), we have du p dx = f(x i )u i + xi ( ) µ xi (x i τ) µ u p (τ)dτ + g(x i ), (3.3) where p u i = y 0 + c j (L j (x i ) + L j (x i )) j= Note that the true solution at Chebyshev points satisfies du = f(x i )u(x i ) + dx xi hus, let e(x) = u(x) u p (x), we have ( ) µ xi (x i τ) µ u(τ)dτ + g(x i ), (3.4) ( ) ( ) du dx du µ p xi = f(x dx i )e(x i ) + (x i τ) µ e(τ)dτ. (3.5) xi

41 35 Multiplying both sides by l i (x) and summing up from to p and applying the fact that u p (x) = y 0 + p c j (L j (x) + L j (x)) = y 0 l 0 (x) + p u j l j (x), we obtain We write = j= j= I p u (x) I p u p(x) I p [f(x)(i p u(x) u p (x)) ( ) µ [ ] [ ] x I p (x τ) µ e(τ)dτ + I p f(x)(u(x) I p u(x)). (3.6) ] e(x) = u(x) I p u(x)+i p u(x) u p (x), e (x) = u (x) I p u (x)+i p u (x) u p(x). (3.7) hus, by (3.6) and simple calculation, e (x) f(x)e(x) = u (x) I p u (x) f(x)(u(x) I p u(x)) (I I p )[f(x)(i p u(x) u p (x))] +I p u (x) u p(x) I p [f(x)(i p u(x) u p (x))] = u (x) I p u (x) f(x)(u(x) I }{{} p u(x)) (I I }{{} p )[f(x)(i p u(x) u p (x))] }{{} I I 4 I ( ) µ { [ ]} x + (I p I) (x τ) µ e(τ)dτ + I p [f(x)(u(x) I p u(x))] }{{}}{{} I 5 + ( ) µ x (x τ) µ e(τ)dτ. (3.8) Denoting I(x) = 5 I k (x) and integrating both sides from to x, we derive e(x) = = k= x x I(s)ds + I(s)ds + x x ( f(s) + ( ) x ) µ (v s) µ dv e(s)ds ( f(s) + ( s ) µ (x s) µ µ ) e(s)ds x Ĩ(x) + L e(s)ds, (3.9) I 3

42 36 where Ĩ(x) = x µ I(s)ds, L = + max f(x). µ µ x [,] By the Gronwall inequality, we have herefore, x e(x) Ĩ(x) + C Ĩ(τ) dτ. (3.0) e C Ĩ C I. (3.) Now, let us estimate terms from I to I 5. ) Estimates of I : If y satisfies condition (R), then I and if y satisfies condition (M), then ) Estimates of I : on [0, ]. I ( ) p+ C(p + ), (3.) R 4R ( ) C p+ em. (3.3) p + 4(p + ) he result follows directly from (.0) and (.) and the fact that f(x) M Hence, If y satisfies condition (R), then and if y satisfies condition (M), then I ( ) p+ I C, (3.4) 4R ( ) C p+ em p + 4(p + ).. (3.5)

43 37 3) Estimates of I 3 : Since f(x) is analytic and I p u(x) u p (x) are polynomials, the product of them is analytic also. I 3 ( ) C p+ e. (3.6) p + 4(p + ) 4) Estimates of I 4 It is exactly the same as I in the proof of heorem.3.5. herefore, if p is sufficiently large, from the estimate of I in heorem.3.5, we obtain I 4 4C e. (3.7) 5) Estimates of I 5 I 5 C log p u(x) I p u(x). (3.8) herefore, ( ) p+ I 5 C log p, (y satisfies condition (R)); (3.9) 4R ( ) I 5 C log p p+ em (y satisfies condition (M)). (3.30) p + 4(p + ) Combine all estimates above, we derive ( ) p+ C(p + ) u u p, (y satisfies condition (R)); (3.3) R 4R ( ) u u p C log p p+ em (y satisfies condition (M)). (3.3) p + 4(p + )

44 Fredholm integro-differential equation heorem 3..3 Let y and y p be the exact solution and spectral approximation of (3.6), respectively. hen ) If y satisfies the condition (R), then, for sufficiently large p, ( ) p+ C(p + ) y y p, (3.33) R 4R )If y satisfies the condition (M), then, for sufficiently large p, ( ) y y p C log p p+ em. (3.34) p + 4(p + ) Proof: Follow exactly the same technique as in the proof of heorem 3.., we obtain = I p u (x) I p u p(x) I p [f(x)(i p u(x) u p (x)) ( ) µ [ ] [ ] I p x τ µ e(τ)dτ + I p f(x)(u(x) I p u(x)). (3.35) ] Again, we write e(x) = u(x) I p u(x)+i p u(x) u p (x), e (x) = u (x) I p u (x)+i p u (x) u p(x). (3.36) hus, by (3.0) and simple calculation, e (x) f(x)e(x) = u (x) I p u (x) f(x)(u(x) I p u(x)) (I I p )[f(x)(i p u(x) u p (x))] +I p u (x) u p(x) I p [f(x)(i p u(x) u p (x))] = u (x) I p u (x) f(x)(u(x) I }{{} p u(x)) (I I }{{} p )[f(x)(i p u(x) u p (x))] }{{} + ( I ) µ { (I p I) [ I x τ µ e(τ)dτ ]} } {{ } I 4 I 3 + I p [f(x)(u(x) I p u(x))] }{{} I 5

45 Denoting Ĩ(x) = 5 k= + ( 39 ) µ x τ µ e(τ)dτ. (3.37) I k (x) and adopting the idea from heorem 4.. in [33], we let z(x) = e (x). From our algorithm, it is clear that z( ) = 0. hen, by change the order of integration, x z(x) = Ĩ(x) + f(x) z(s)ds + x = Ĩ(x) + f(x) z(s)ds + + ( ) µ µ x ( ( ) µ s ) µ µ ( x) µ z(u)du. x x s µ dsz(u)du (x u) µ z(u)du (3.38) Define ( ) µ x Az = f(x) z(s)ds+ µ x Clearly, A is compact by the Arzelá-Ascoli theory. Hence, (3.38) can be written as (x u) µ z(u)du+ ( ) µ µ x ( x) µ z(u)du. z = Az + Ĩ. From our assumption and the Fredholm Alternative, (I A) has a bounded inverse. hus, But e(x) = x z(s)ds, x [, ], then we have z C Ĩ. (3.39) e z C Ĩ. (3.40)

46 40 Clearly, I, I, I 3 and I 5 are the same as those of heorem 3.. and I 4 is the same as I in heorem.3.8, we obtain the same estimates of these terms. Combine all estimates above, we derive I he result follows. ( ) p+ C(p + ), (y satisfies condition (R)); (3.4) R 4R ( ) I C log p p+ em (y satisfies condition (M)). (3.4) p + 4(p + ) 3.3 Numerical Experiments Our scheme for integro differential equations has the form DC p + LC p AC p = G p, where C p = [c 0,, c p ] and G p = [g(x 0 ),, g(x p )] and the elements of the matrix A = (a ij ), L = (l ij ) and D = (d ij ) are given by Scheme for (3.): a ij = ( + ( ) µ j! ( µ) j+ ( + x i ) µ P (µ, µ) j (x i ) ) µ (j )! ( + x i ) µ P (µ, µ) j (x i ), ( µ) j l ij = (L j (x i ) + L j (x i ))f(x i ), d ij = j + P (,) j (x i) + j P (,) j (x i)

47 4 Algorithm for (3.6): a ij = ( ( ( ( ) µ j! ( µ) j+ ( + x i ) µ P (µ, µ) j (x i ) ) µ (j )! ( + x i ) µ P (µ, µ) j (x i ) ( µ) j ) µ j! ( x i ) µ P ( µ,µ ) j (x i ) ( µ) j+ ) µ (j )! ( x i ) µ P ( µ,µ ) j (x i ) ( µ) j l ij = (L j (x i ) + L j (x i ))f(x i ), d ij = j + P (,) j (x i) + j P (,) j (x i) In this whole section, reference curve associate with Example 3.3. and Example 3.3. is the graph of f(p) = p+ 3 p+ and reference curve associate with Example and Example is the graph of f(p) = p+ ( 7 5 )p+. Reference curve is the graph of f(p) = log p p+ ( em 4(p+) )p+ for all four examples. Example 3.3. Now let us consider integro-differential equations. First, we consider an equation of the form (3.) on [0, 6] with µ =, a(t) = et and b(t) = 3 t + e t (t + ) /3 3π 8 (t + ) (t + ) arctan ( ) t (3t + 0). t 4 he exact solution for this example is y(t) = ( + t) 3/. Numerical errors of are reported in able 5 and left part of Figure 5. Example 3.3. Consider an integro-differential equation of the form (3.) on [0, 6] with µ =, a(t) = sin(t) and b(t) = 4 et e t (sin(t) γ(, t)), where γ(a, x) is the 3 lower incomplete gamma function defined above. he exact solution of this equation

48 4 able 5: Example 3.3.: errors for t [0, 6] N L.5788e e e e-09 N L e-.043e-.453e- 5.54e-3 able 6: Example 3.3.: errors for t [0, 6] N L 4.935e e-0.953e e-07 N L.7735e-.648e e e-3 is y(t) = e t. Numerical results are listed in able 6 and right part of Figure 5. Example Consider a Fredholm integro-differential equation of the form (3.6) on [0,6]. We choose µ = /, a(t) = e t and b(t) = ( ) t + et t + π (t + ) + (t + ) arctan t t t (t + ) log( 6 t + 4 ) + (t + ) log( t + ) so that the true solution is y(t) = t +. Readers are referred to able 7 and left part of Figure 6 for numerical report. Example Consider an integro-differential equation of the form (3.6) on [0,

49 Our Method Reference Curve 0 5 Our Method Reference Curve e L e L p p Figure 5. Left: L error for Example 3.3. Right: L error for Example Our Method Reference Curve Our Method Reference Curve e L 0 8 e L p p Figure 6. Left: L error for Example Right: L error for Example 3.3.4

50 44 able 7: Example 3.3.3: errors for t [0, 6] N L 4.506e e e e-09 N L.456e-.304e-.75e e-4 able 8: Example 3.3.4: errors for t [0, 0] N L 3.07e e e e-06 N L.763e e-.344e-3.087e-4 0]. We choose µ = / and b(t) = cos(t) a(t) sin(t) + [ ( ) ( )] t t π cos(t)s sin(t)c π π [ ( ) ( )] (0 t) (0 t) π sin(t)c + cos(t)s π π so that the true solution is y(t) = sin(t). Here, C(u) and S(u) are Fresnel integrals defined as in Example.4.4 and 0, t 5; a(t) =, t > 5. (3.43) Numerical results are given in able 8 and right part of Figure 6.

51 45 From our numerical experiments, we see the condition that we put on a(t) can be violated as long as the solution y(t) is sufficiently smooth.

52 46 4 Eigenvalue approximation for compact integral operators In this chapter, we focus on problems of the form 0 k(t, s)u(s)ds = λu(t), t [0, ], (4.) where k(t, s) = t s µ for 0 < µ < and k(t, s) is piecewisely smooth or smooth. We will develop algorithms for all these types of kernels and prove a convergence rate for eigenvalue approximation respectively. 4. Preliminary Hypergeometric function, as a generalized function of standard function will be used in our algorithms. It is defined as follows: pf q (z) = r=0 (a ) r (a p ) r z r (b ) r (b q ) r r! (4.) where (a j ) r and (b j ) r are the Pochhammer symbols defined as (k) r = k(k + ) (k + r ), (k) 0 =. (4.3) he above series is defined when none of b js, j =,, q is a negative integer or zero. A b j can be zero provided there is a numerator parameter a k such that (a k ) r becomes zero first before (b j ) r becomes zero. If any numerator parameter a j is a negative integer or zero then the series terminates and becomes a polynomial in z. From the ratio test, it is evident that the series is convergent for all z if q p, it is

53 47 convergent for z if p = q + and diverge if p > q +. Note that the library function hypergeom in Matlab is very slow in computation, we call hypergeom in Maple from Matlab to save time and preserve a high digit of accuracy. Let : X X be a compact linear operator on a Banach space X and σ( ) and ρ( ) be the spectrum and resolvent of respectively. Let λ be a nonzero eigenvalue of with multiplicity m and let Γ be a circle centered at λ which lies in ρ( ) and which encloses no other points in σ( ). hen, the spectral projection associated with and λ is defined by E = ( zi) dz πi Γ and max z Γ ( zi) C Let n be a sequence of operator in B(X) that converges to in a collectively way, i.e., the set { n x : x, n =,, } is sequentially compact. For n large enough, Γ ρ( n ) and the associated projection, E n = ( n zi) dz πi Γ exists and max z Γ ( n zi) C. Clearly, dim(e) = dim(e n ) = m and n E n = E n n. Furthermore, the spectrum of n inside Γ, contains m approximations of λ, i.e. λ n,, λ n,,, λ n,m, counted according to their algebraic multiplicities [5, 3].Let hen we have the following theorem. ˆλ n = λ n, + λ n, + + λ n,m m

54 48 heorem 4..[3] For all n sufficiently large, λ ˆλ n C ( n ) R(E), where R(E) is the range of the projection E. his is a rather general result. We may refine the result if the kernel is positive definite. Let a(u, v) = k(t, s)u(s)v(t)dsdt, b(u, v) = u(t)v(t)dt, (4.4) where v is a test function in L space V. If the bilinear operator a(u, v) is coercive, then we can list eigenvalues of by λ λ λ 3 0 with zero the only possible cluster point. Let us consider a numerical approximation of the first eigen-pair (λ, u). Let (λ p, v p ) be their Galerkin approximation, and let u p be the Legendre expansion of u. We have λ = a(u, u) b(u, u) = sup a(v, v) v V b(v, v), λ p = a(v p, v p ) b(v p, v p ) = max a(v, v) v P p b(v, v). (4.5) Here P p is the polynomial space with degree no more than p. Denote λ p = a(up,up) b(u p,u p), then we have the following lemma. Lemma 4.. Let λ, λ p and λ p be defined as above and a(u, v) be coercive, then 0 λ λ p λ λ p = λ u u p b u b u u p a. (4.6) u b Proof: From Lemma 9. of [5] on page 70, we have 0 ν p ν ν p ν u u p b u a ν u u p a, (4.7) u a

55 49 where ν = λ, ν p = λ p and ν p = λ p. Hence, 0 λ λ p λ λ p u u p b u a λ p λ u u p a. (4.8) u a Using the fact that a(u p, u p ) = λ p b(u p, u p ), we derive (4.6) from (4.8). where Next, we introduce some identities, which will be essential in this paper. Lemma 4..3[34] Let α >, β > and 0 < ν <, then for < x < holds ( t) α ( + t) β P m (α,β) (t)dt = x t ν πν cos Φ (x) + cos π( ν β)φ (x) Γ(ν) cos πν, m = 0,,, (4.9) Φ (x) = Φ (x) = Γ(m + α + )Γ(m + ν)γ(β ν + )( )m α β+ν Γ(m + α + β ν + )m! F (m + ν, ν m α β ; β + ν; + x ), (4.0) Γ(m + β + )Γ(ν β )( )m+ α ( + x) ν β m! F (m + β +, m α; β ν + ; + x ). (4.) Here, F (a, b; c; z) is known as Gauss hypergeometric functions. For the sake of convergence analysis, we need to introduce the error estimate of Gauss quadrature and some knowledge of matrix analysis. Lemma 4..4 [7] Let f C (n) and x i and w i are Gauss points and their corresponding weights on the interval [a, b]. hen b a f(x)dx n i=0 w i f(x i ) = (b a)n+ (n!) 4 (n + )[(n)!] 3 f (n) (ξ), ξ (a, b) (4.)

56 50 4. Algorithms In this section, we develop algorithms for eigen-problem with all three kinds of kernels. Models that we consider in this section are () Weakly singular kernels: λy(t) = () Piecewise smooth kernel: 0 y(s) ds, 0 < µ <, t [0, ], (4.3) t s µ λy(t) = (3) Smooth kernel 0 k(t, s)y(s)ds, t [0, ], (4.4) t s/, if 0 t s where k(t, s) = s/, if 0 s < t, λy(t) = 0 e st y(s)ds, t [0, ]. (4.5) 4.. he first algorithm for Weakly Singular Kernel It is clear that (4.3) is equivalent to λy(t) = t We make a change of variable t = +x and obtain (+x)/ 0 0 y(s) (t s) ds + y(s) ds. (4.6) µ t (s t) µ ( µ ( ) + x s) y(s)ds + s + x µ y(s)ds = λu(x), (4.7) (+x)/

57 5 where x [, ] and u(x) = y ( ) +x. Next, we make another change of variable, s = +τ. and reach ( ) µ ( ) µ x (x τ) µ u(τ)dτ + (τ x) µ u(τ)dτ = λu(x), x Let u p (x) = equation p ( ) µ p xi c j c j L j (x) be the approximation of u(x). ( L j (τ) (x i τ) dτ + µ Substituting (.4) and (.5), we obtain [( p c j ( + ) µ p c j ) µ j! ( µ) j+ ( + x i ) µ P (µ, µ) j (x i ) ) µ j! ( µ) j+ ( x i ) µ P ( µ,µ ) j (x i ) x i ] x [, ] (4.8) Obviously, c js satisfy the L j (τ) (τ x i ) µ dτ = λ p p c j L j (x i ). (4.9) p = λ p c j L j (x i ), i = 0,, p. (4.0) If we write a ij = ( ) µ j! ( + ( µ) j+ ( + x i ) µ P (µ, µ) j (x i ) ) µ j! ( µ) j+ ( x i ) µ P ( µ,µ ) j (x i ) b ij = L j (x i ). then we have AC p = λ p BC p, where A = (a ij ), B = b ij, C p = (c 0, c,, c p ).

58 5 4.. he Second Algorithm for the Weakly Singular Kernel From heorem 3 of [38], we derive that the first true eigenvector is of the form y(t) = ˆd t µ + ˆd ( t) µ + a smoother function φ(t). (4.) Hence, we approximate the eigenvector by u p (t) = d t µ + d ( t) µ + p c j P j (t), where P j (t) is the shifted Legendre polynomials on [0, ], j = 0,,, p. Substituting it into (4.3) and taking the same change of variable as the previous algorithm, we obtain ( ) µ ( ( + τ) µ ) d x τ dτ + d ( τ) µ µ x τ dτ µ ( ) µ p ( x ) + c j (x τ) µ L j (τ)dτ + (τ x) µ L j (τ)dτ x ( ) µ ( ) µ + x x p = d + d + c j L j (x), x [, ]. (4.) From lemma.. and lemma 4..3, and (4.), we obtain ( ) µ Γ( µ) µ Γ(3 µ) F (µ, µ ; µ ; + x ( ) µ i )d + Γ(µ) ( Γ( µ)γ(µ)γ( µ) F µ (µ, µ ; µ; + x i Γ(µ ) ) Γ(3 µ) µ ( + x i ) µ F (, µ ; µ; + x ) [( ) µ p i j! ) d + c j ( + x i ) µ P (µ, µ) j (x i ) ( µ) j+ ( ) µ ] ( p j! + ( x i ) µ P ( µ,µ ) j (x i ) = λ p c j L j (x i ) + d ( + x i ) µ ( µ) j+ +d ( x i ) ), µ i = 0,, p +. (4.3) Note that the first hypergeometric function is not well-defined when µ = /. However, the integration of the two singular terms with the kernel are simpler, in