Cubic spline collocation for a class of weakly singular Fredholm integral equations and corresponding eigenvalue problem

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1 Cubic spline collocation for a class of weakly singular Fredholm integral equations and corresponding eigenvalue problem ARVET PEDAS Institute of Mathematics University of Tartu J. Liivi 2, 549 Tartu ESTOIA arvet.pedas@ut.ee MIKK VIKERPUUR Institute of Mathematics University of Tartu J. Liivi 2, 549 Tartu ESTOIA azzo@ut.ee Abstract: The numerical solution of a class of linear Fredholm integral equations of the second kind and corresponding eigenvalue problem by the collocation method is discussed. The kernel of the integral equation may have weak diagonal and boundary singularities. Using C 2 -smooth cubic splines and special non-uniform grids, the convergence rate of proposed numerical schemes is studied. Key Words: Weakly singular integral equation, boundary singularity, cubic spline collocation method, graded grid, eigenvalue problem Introduction Let R = (, ), = {, 2,...}, = {}. By C k (Ω) (k ) we denote the set of all k times continuously differentiable functions on Ω, C (Ω) = C(Ω). In particular, by C[, ] we denote the Banach space of continuous functions z on [, ] with the norm z = sup x z(x). Let D = {(x, y) R 2 : x, < y <, x y}. We consider a linear Fredholm type integral equation of the second kind λu = T K u + f, () λ is a parameter, f C[, ] and T K is an integral operator defined by the formula (T K u)(x) = K(x, y)u(y)dy, x. (2) The numerical solution of () with kernels of the form K(x, y) = g (x, y) x y ν + g 2 (x, y), < ν <, (3) g and g 2 are sufficiently smooth functions on [, ] [, ], has been considered by many authors (see, e.g. [,, 2]). If g, then the main difficulty with this type of equations is that their solutions (if they exist) are typically singular near the boundary of the interval [, ] their derivatives become unbounded (see, e.g. [8, 2] and Lemma 3 below). This complicates the construction of high order methods for the numerical solution of equation (). In collocation methods the singular behavior of the exact solution can be taken into account by using polynomial splines on special non-uniform grids which are properly graded in order to compensate the generic boundary singularities of the derivatives of the exact solution [,, 2]. In this paper we will consider a more complicated situation for ()-(2) the kernel K(x, y) of the operator T K may have a diagonal singularity at x = y and additionally a boundary singularity at y = or/and at y =. More precisely, we assume that K(x, y) = g (x, y) x y ν y ρ ( y) ρ +g 2 (x, y) (4) < ν <, ρ < ν, ρ < ν, (5) (x, y) D, g, g 2 C m ([, ] [, ]), m. (6) The set of kernels K(x, y) satisfying (4)-(6) will be denoted by W m,ν,ρ,ρ. Clearly, W m,ν,ρ,ρ W,ν,ρ,ρ for any m. We construct, for equation () and for corresponding eigenvalue problem λu = T K u, a collocation method which is based on C 2 -smooth cubic splines and graded grids reflecting the singular behavior of the exact solution. Our purpose is to study the convergence behavior of proposed algorithms for all values of the non-uniformity parameter of the grid. ote that the numerical solution of () with kernels of ISB:

2 the form (3) by cubic and quadratic collocation methods has been considered in [6, 7, 2] and [4, 5], respectively. Throughout the paper c is a positive constant which may have different values by different occurrences. For a Banach space E, by L(E) we will denote the Banach space of linear bounded operators A : E E with the norm A L(E) = sup{ Ax E : x E, x E = }. 2 Regularity of the solution For given k and α, β (, ), let C k,α,β (, ) be the set of functions z C[, ] C k (, ) such that z (j) (x) cx α j ( x) β j, < x <, (7) j =,..., k. Clearly, C k [, ] C k,α,β (, ), k, α, β (, ). Due to [8] it is easy to see that the following lemma holds. Lemma Let K W,ν,ρ,ρ, < ν <, ρ < ν, ρ < ν. Then T K, defined by the formula (2), is compact as an operator from C[, ] into C[, ]. Denote (λi T K ) = {u C[, ] : λu = T K u}, I is the identity mapping in the space C[, ]. A consequence of Lemma is the following result. Lemma 2 Let f C[, ], K W,ν,ρ,ρ, < ν <, ρ < ν, ρ < ν, and let (λi T K ) = {}. Then equation () is uniquely solvable and its solution belongs to C[, ]. It follows from [8] that the regularity of a solution to () can be characterized by the following lemma. Lemma 3 Let K W m,ν,ρ,ρ, f C m,ν+ρ,ν+ρ (, ), m, < ν <, ρ < ν, ρ < ν, and let equation () have a solution u C[, ]. Then u belongs to C m,ν+ρ,ν+ρ (, ), and hence u C m (, ) and (see (7)) u (j) (x) cx ν ρ j ( x) ν ρ j, (8) < x < and j =,..., m. ote that, in Lemma 3 have not supposed that equation () is uniquely solvable. If f =, then the statement of Lemma 3 yields the estimate (8) for the derivatives of the eigenfunctions u of the integral operator T K. 3 Cubic spline collocation method For given ( 2) and r [, ) let (r) = {t,..., t 2 : = t <... < t 2 = } be a grid on [, ] such that t j = ( ) j r, j =,...,, (9) 2 t +j = t j, j =,...,. () If r =, then (r) is a uniform grid with the knots t j = jh, h = /(2), j =,..., 2. If r >, then the grid points (9)-() are more densely clustered near the boundary of the interval [, ]. Denote by P 3 the set of polynomials of degree 3 and let S (2) 3 ( (r) ) be the space of cubic splines on (r) : S (2) 3 ( (r) ) = {z C 2 [, ] : z [tj,t j ] P 3, j =,..., 2}, z [tj,t j ] is the restriction of z to [t j, t j ]. We find an approximation u for u, the solution to (), determining u from the following conditions: lim u t t i,t<t i (t) = u S (2) 3 ( (r) ), 2, () λu (t i ) K(t i, s)u (s)ds = f(t i ), i =,..., 2, (2) lim u t t i,t>t i (t), i =, i = 2. (3) The settings ()-(3) form a system of equations whose exact form is determined by the choice of a basis in the space S (2) 3 ( (r) ) of cubic splines on (r) For example, we can seek u in the form u (t) = 2 j= 3. B j (t), t [, ], (4) { } are unknown coefficients and {B j (t)} are the cubic B-splines with the minimal support [t j, t j+4 ]. About the construction of B-splines see, for example, [9]. We note that, in addition to the points = t < t <... < t 2 = of the grid (r), for the construction of B-splines B j(t) (j = 3, 2,..., 2 ), six additional points t 3 < t 2 < t < and < t 2+ < t 2+2 < t 2+3 from outside of the interval [, ] are necessary. We choose them as follows: t i = i(t t ), t 2+i = + i(t 2 t 2 ), i =, 2, 3. Actually, B j (t) can be presented in the form (see [9]) j+4 B j (t) = 4 k=j (t k t) 3 + ω j (t, t R, (5) k) ISB:

3 j = 3,..., 2, { (t k t) 3 (tk t) + = 3 for t k t, for t k t <, ω j (t) = (t t j )(t t j+ )(t t j+2 )(t t j+3 )(t t j+4 ), with t R, j = 3,..., 2. Substituting (4) into (2)-(3), we obtain equations with respect to coefficients c 3, c 2,..., c 2 : j= 3 λ i j=i 3 t t,t<t j (t) = = j= 2 2 B j (t i ) 2 2 j= 3 t t,t>t j (t), K(t i, s)b j (s)ds (6) = f(t i ), i =,..., 2, (7) j=2 5 2 j=2 4 t t 2,t<t j (t) 2 t t 2,t>t j (t). (8) 2 Theorem Let f C[, ], K W,ν,ρ,ρ, < ν <, ρ < ν, ρ < ν. Furthermore, assume that (λi T K ) = {} and the grid points (9)-() of the grid (r) are used. Then equation () has a unique solution u C[, ], the settings () - (3) determine for sufficiently large, say ( 2), a unique approximation u for u, and u (x) u(x) as. (9) x [,] Proof: It follows from Lemma 2 that the equation () has a unique solution u C[, ]. The conditions ()-(3) have an operator equation representation λu = P T K u + P f. (2) Here P : C[, ] C[, ] is an interpolation operator which assigns to each function u C[, ] its piecewise cubic interpolation function P u S (2) 3 ( (r) ) C[, ] satisfying the following conditions: and (P u)(t i ) = u(t i ), i =,..., 2 lim (P u) (t) = lim (P u) (t), t t j,t<t j t t j,t>t j j = and j = 2. We find (see [2]) that P L(C[, ]), P L(C[,]) c,, with a constant c > which is independent of, and u P u as for any u C[, ]. (2) This, together with Lemma, yields that T K P T K L(C[,]) as. (22) Further, it follows from Lemma 2 that λi T K is invertible in C[, ] and (λi T K ) L(C[, ]). This together with (2) yields that λi P T K is invertible in C[, ] for all sufficiently large, say, and (λi P T K ) L(C[,]) c,, (23) with a constant c > which is independent of. Thus, for equation (2) (method ()- (3)) provides a unique solution u S (2) 3 ( (r) ) C[, ]. We have, for it and u, the solution of equation (), that u u = (λi P T K ) (P u u) and, due to (23) u u c P u u,, () with a constant c > which is independent of. This, together with u C[, ] and (2) yields (9). Lemma 4 Let u C 4,ν+ρ,ν+ρ (, ), < ν <, ρ < ν, ρ < ν. Then, for sufficiently large, say ( 2), P u u cɛ (r,ν,ρ), (25) with ɛ (r,ν,ρ) r( ν ρ) 4 for r < ( ν ρ) =, 4 4 for r ( ν ρ). (26) Here ρ = {ρ, ρ } and c is a positive constant not depending on. Proof: We outline only the basic idea on which the proof of (25) is based. Let We have h i = t i+ t i, i =,..., 2, (P u)(t) = ( τ) 2 ( + 2τ)u i + τ 2 (3 2τ)u i+ +h i τ( τ) [ ( τ)(p u) (t i ) τ(p u) (t i+ ) ], ISB:

4 t [t i, t i+ ], τ = t t i h i, i =,..., 2. Moreover, we have δ i v i, i 2 i 2 Denote δ i = (P u) (t i ) u (t i ), i =,..., 2. v = p t s 3 u (4) (s) ds 6h Using these and suitable Taylor expansions,we obtain (P u)(t) u(t) = h3 h 2 τ( τ) 2 (δ + δ 2 ) t ( τ)2 h τ( τ)δ s 3 u (4) (s)ds 6 ( 2u +h 3 h τ( τ) 2 (4) (η ) u (4) ) (η 2 ) t t (t s) 3 u (4) (s)ds, (27) t [, t ], τ = t h, η, η 2 (t, t 2 ); (P u)(t) u(t) = h i τ( τ) [δ i ( τ) δ i+ τ] + h4 i τ 2 [ ] (3 2τ)u (4) (η ) τ 2 u (4) (η 3 ) h4 i τ 2 ( τ) u (4) (η 2 ), (28) 6 t [t i, t i+ ], τ = t t i, η 3 (t i, t), h i η, η 2 (t i, t i+ ), i =,..., 2 2; (P u)(t) u(t) = h 2 τ( τ)δ 2 h3 2 h 2 τ 2 ( τ)(δ δ 2 ) 2 2 ( u +h 3 2 h 2 2 τ 2 (4) (η 2 ) 2u (4) ) (η ) ( τ) 2 t (t s) 3 u (4) (s)ds 6 t 2 + τ 2 ( s) 3 u (4) (s)ds, (29) 6 t 2 t [t 2, ], τ = t t 2 h 2, η, η 2 (t 2 2, t 2 ). p i = + 7p ih 3 i + q h 2 (6h + 7h ) s [t,t 2 ] u(4) (s) ; v i = 7q ih 3 i s [t i,t i+ ] u(4) (s) s [t i,t i ] u(4) (s), i = 2,..., 2 2; v 2 = q 2 ( s) 3 u (4) (s) ds 6h 2 t 2 + p 2 h h p 2 h h i h i + h i, q i = s [t 2 2,t 2 ] u(4) (s) s [t 2 2,t 2 ] u(4) (s), h i h i + h i, i =,..., 2. ow, in analogy to the proof of Lemma 3 in [6] (see also [2]), we find that the upper bound for all terms in the right-hand side of (27)-(29) is c ɛ (r,ν,ρ) c is a positive constant which does not depend on, ɛ (r,ν,ρ) is defined by (26) and ρ = {ρ, ρ }. Theorem 2 Let K W 4,ν,ρ,ρ, f C 4,ν+ρ,ν+ρ (, ), < ν <, ρ < ν, ρ < ν. Assume that (λi T K ) = {} and the grid points (9)-() of the grid (r) are used. Then method ()-(3) determines for ( 2) a unique approximation u for u, the solution to (), and u (x) u(x) cɛ (r,ν,ρ), (3) x [,] ɛ (r,ν,ρ) is defined by the formula (26), ρ = {ρ, ρ } and c is a positive constant not depending on. Proof: It follows from Theorem that method ()- (3) determines for a unique approximation u S (2) 3 ( (r) ) to u C[, ], the solution of equation (). With the help of Lemma 3 we obtain that u C 4,ν+ρ,ν+ρ (, ). This together with the estimate () and Lemma 4 yields the estimate (3). ISB:

5 4 Eigenvalue problem Let us consider an eigenvalue problem: λu(t) = K(t, s)y(s)ds, t. (3) Using the operator T K (see (2)), equation (3) can be written in the form λu = T K u. For given ( 2) we construct an approximating eigenvalue problem to (3) in the form λu (t i ) = u S (2) 3 ( (r) ), (32) K(t i, s)u (s)ds, i =,..., 2, (33) lim u t t i,t<t i (t) = lim u t t i,t>t i (t), i =, i = 2, (34) the interpolation points {t i } (the knots of the grid (r) ) are defined by (9)-(). Choosing a basis in the space S (2) 3 ( (r) ) and presenting u as a linear combination of basis functions, (32)-(34) will take the form of a finite dimensional eigenvalue problem of linear algebra. In particular, we may use the presentation (4) with the basis functions (5). Using P, problem (32)-(34) can be rewritten in the form λu = P T K u. Due to Lemma, T K is compact as an operator from C[, ] into C[, ]. Therefore, all eigenvalues of T K form a discrete set in the complex plane C with zero as the only possible limit and each point λ of the spectrum σ(t K ) of T K is an isolated eigenvalue of T K with finite algebraic multiplicity. Recall that a complex number λ is called an eigenvalue of T K if there exists an element u C[, ] (u ) such that T K u = λ u ; the element u is called an eigenelement (eigenfunction) of T K. For any eigenvalue λ of T K both the eigenspace (λ I T K ) = {u C[, ] : (λ I T K )u = } and the generalized eigenspace W = W (λ, T K ) = ((λi T K ) γ ) of T K corresponding to the eigenvalue λ are finite dimensional. Here γ = γ(λ, T K ) is the rank of the eigenvalue λ, that is, the least positive integer such that ((λ I T K ) γ ) = ((λ I T K ) γ+ ). The dimension µ of W is called the algebraic multiplicity of λ : µ = µ(λ, T K ) := dim W (λ, T K ). Theorem 3 Let K W,ν,ρ,ρ, < ν <, ρ < ν, ρ < ν. Furthermore, assume that the grid points (9)-() of the grid (r) are used. Then for every nonzero eigenvalue λ of T K there exists a sequence {λ } of eigenvalues λ of P T K such that λ λ for. Conversely, if {λ } is a sequence of eigenvalues λ of P T K, and λ is an accumulation point for {λ }, then λ is an eigenvalue of T K. Proof: The assertions of Theorem 3 follow from the convergence (22) and the general results about the convergence of approximate eigenvalues, proved under the assumptions of regular or compact approximation of operators (see, for example, [2, 3,, 2]). In particular, we can apply Theorem 5. from [2], p. 68. Let Λ be some compact in the complex plane C so that Λ and λ is the unique eigenvalue of T K in Λ. It follows from Theorem 3 that for all sufficiently large, say ( 2), at least one eigenvalue λ of P T belongs to Λ. Moreover, on the basis of (22) we obtain (cf. [2], p. 68) that the sum of the algebraic multiplicities µ(λ, P T K ) of all eigenvalues λ of P T K in Λ is equal to the algebraic multiplicity µ(λ, T K ) of the eigenvalue λ of T K : λ :λ σ(p T K ) Λ µ(λ, P T K ) = µ(λ, T K ), with. Theorem 4 Let K W 4,ν,ρ,ρ, < ν <, ρ < ν, ρ < ν, and assume that the grid points (9)-() of the grid (r) are used. Let λ be an eigenvalue of T K and let Λ be some compact in C such that Λ, and λ is the unique eigenvalue of T K in Λ. Finally, let {λ } be a sequence of eigenvalues λ of P T K, such that λ λ for. Then for all sufficiently large, say ( 2), the following estimates hold: λ λ c (ɛ (r,ν,ρ) Here γ = γ(λ, T K ) is the rank of λ, ) /γ, λ λ c 2 ɛ (r,ν,ρ). (35) k λ = l (i) k λ(i) / l (i) i= i= ISB:

6 is the average of λ (i) (i =,..., k ) weighted by their algebraic multiplicities l (i) = µ(λ(i), P T K ) = dim W (λ (i), P T K ), k is the number of the different eigenvalues λ (),..., λ(k ) of P T K in Λ, ɛ (r,ν,ρ) is defined by the formula (26), c and c 2 are some positive constants not depending on. Proof: On the basis of the convergence (22) it follows from the corresponding results of [2, 3], [, p. 84], [2, pp. 68-7] that λ λ c (θ ) /γ, λ λ c θ, (36) θ = sup u P u (37) u W (λ,t K ), u = with some positive constants c and c not depending on,. Using Lemma 3 with f = and induction by i, we conclude that ((λ I T K ) i ) C 4,ρ +ν,ρ +ν (, ), i =, 2,.... Therefore, W (λ, T K ) C 4,ρ +ν,ρ +ν (, ), and we obtain from (25) and (37) that θ cɛ (r,ν,ρ),, (38) c is a positive constant not depending on. The estimates (35) now follow from (36)-(38). 5 Conclusion We have considered a class of integral equations of the form () with kernels K(x, y) which, in addition to a weak diagonal singularity at x = y, may have some weak singularities as y approaches to the boundary of the interval of integration. We have established conditions which will guarantee the convergence of numerical solutions obtained by the cubic spline collocation method on graded grids. We have also derived error estimates to the approximate solutions for the nonhomogenous problem and to the approximate eigenvalues for the corresponding eigenvalue problem. In particular, we have shown that the proposed method is of imal possible order if the grid is chosen appropriately. The results obtained generalize and complement the corresponding results of [6, 7, 2]. References: [] K. E. Atkinson, The umerical Solution of Integral Equations of the Second Kind, Cambridge Univ. Press, Cambridge, 997. [2] O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions I, umber. Funct. Anal. and Optimiz. 7, 996, pp [3] O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions II, umber. Funct. Anal. and Optimiz. 7, 996, pp [4] R. Pallav, A. Pedas, Quadratic spline collocation method for weakly singular integral equations and corresponding eigenvalue problem, Mathematical Modelling and Analysis 7(2), 22, pp [5] R. Pallav, A. Pedas, Quadratic spline collocation for the smoothed weakly singular Fredholm integral equations, umerical Functional Analysis and Optimization 3(9 & ), 29, pp [6] A. Pedas, Piecewise polynomial collocation method for weakly singular integral equations, Lipitakis, Elias A. (ed.), HERCMA 2. Proceedings of the 5th Hellenic-European conference on computer mathematics and its applications, Athens, Greece, September 2-22, 2, 22, pp [7] A. Pedas and E. Timak, The cubic splinecollocation method for weakly singular integral equations, Differential Equations 37 (), 2, pp (Translated fom Differentsial nye Uravneniya 37 (), 2, pp. 45-4). [8] A. Pedas and G. Vainikko, Integral equations with diagonal and boundary singularities of the kernel, Z. Anal. Anwend. 25, 26, pp [9] Yu. Zav yalov, B. Kvasov and V. Miroshnichenko, Methods of Spline Functions, Moscow, 98 (in Russian). [] G. Vainikko, Analysis of Discretization Methods, University of Tartu, Tartu, 976 (in Russian). [] G. Vainikko, Multidimensional Weakly Singular Integral Equations, Springer-Verlag, Berlin, 993. [2] G. Vainikko, A. Pedas and P. Uba, Methods for Solving Weakly Singular Integral Equations, Tartu, 984 (in Russian). Acknowledgements: The research was supported by the Estonian Science Foundation (Research Grant r. 94). ISB:

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