A SHOOTING METHOD FOR SINGULAR NONUNEAR SECOND ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

The method of parallel shooting will be employed to solve nonlinear second order singular 
Volterra integro-differential equations with two point boundary conditions.

assumed that S(x, t) satisfies: i) IS(x,t)ldt _< L < oo, for all 0 _< x _< a and ii) for every > 0 there exists a 6 6() > 0 independem ofx and a such that for all a such that 0 < a < x .
With these additional conditions, the singular problem has a unique solution.Boundary valued problems for ordinary differential equations have been considered by many authors.Two textbooks containing material on numerical solutions are given in the references [6,8].References [7,9,10] consider some problems of the Volterra integro-differential equation type.singular problems are considered in [2,3,5,7].
In Na [8], ordinary differential equations with two point boundary conditions are solved by a method of parallel shooting.The general idea here was to divide the interval of integration into a number of subintervals and to find approximations to the solution of the differential equation on each subinterval.Continuity of the solution from subinterval to subinterval was imposed.In [11], this method was emended for application to second order two point boundary value problems of the Volterra type.In contrast to this approach, which involves expressing the problem as an initial value problem, other approaches such as that in [4] for linear problems and in [9] for nonlinear problems exist.In this article, the parallel shooting method in 11 is reworked for application to problems which have at most a finite number of singularities in the kernel.A numerical example illustrates the procedure and the reader is able to compare the results of this indirect method of solution with the work in [5].
The methods to be considered are comprised of three distinct parts: (p, a) denotes a pair of polynomials which characterize a linear multistep method for numerically solving a second order ordinary differential equation; Q will denote a set of quadrature weights associated with the numerical integration of the singular imegrand; and PS will denote a parallel shooting method associated with solving a Volterra integro-differential problem with its boundary conditions.k k In particular, to establish the method (p,a) let p(z)= cz', a(z)= /,z' and for a z=0 z=0 problem ofthe form y" o(x, y), we write k k E c,yt+, h2 E/,ot+,, g 0,1, n k. ( Here yz denotes an approximation to y(z) and o qo(x, y).
To develop quadrature rules for the approximation of an integral with a singular integrand, consider the integral where is assumed to be continuous on [0, ] and is assumed to have at most a finite number of singularities on [0, ] but can be simply integrated with respect to all polynomials of all degrees.Thus ' (t)(t)dt w,() where with x3 w, ,(t)(t)dt (t)g(,) :(t) I] (t- g,(t) (t-z,)Ra(z,)' ,=0 For each j, j k, k + 1, n, a set of quadrature weights is determined.For the integrand (2), these are collectively denoted by Q.
2. Set + 1 and integrate the systems of the first order integro-differential equation ( 6), taking D, D and C, C[. 3. Integrate the systems ofintegro-differential problems given by the system (9) 4. Using the values obtained in steps 2 and 3, solve the system (7) to obtain C,t+l, 0,1, m and D +1, 0,1, m 1.If the desired accuracy between successive iterations is reached, then stop.Otherwise, repeat starting with step 2.

THEORETICAL RESULTS
In this section we consider the convergence of the method.Let (p, a) denote the characteristic polynomials associated with the method for solving a first order differential equation and let Q denote the quadrature role The polynomial p(z) is given by p(z)= ckz k +... +a0 Let "A, 0,1,2 be a set of coefficients defined by 1/(ck + ck-lz +... +a0zk) "0 +/,z + "/2z + (10) DEFINITION 1.The linear k-step methods for first order ODEs and defined by the polynomials (p, a) is said to be zero-stable if no root of p has modulus greater than one and that roots of modulus one be simple Let Yz denote the exact solution of system (5) and Fxn,Yz,z(x))=(u(z),f(x,(zn),z(zn))) T at xn e lN.
DEFINITION 2. Denote the two part method of solution by ((p, a), Q).The difference operators L and M (notation taken from ]) associated with the method are defined by where {C,} are coefficients independent of the function.We define the order of L to be the unique integer p such that C, 0, 0(1)p but C,+1 # 0.
For sufficiently smooth functions F, the two operators L and M are related by OF(x,.,+,,Y(z,.,+,),z+,) M[Y(x,,); h] L[Y(x.);h] h_# , For the quadrature Q, we assume that weights w,, exist which for,all f E C[0, a] and for all 0 _< x _< a, satisfy the conditions: i) , n oo such that nh x.In addition, we assume that maxlw, O, as h O, k oo, kh xk where the maximum is over 0 < _< k and 0 < k < n; and ii) [0, a] can be partitioned into a finite number of subintervals with 0 z0 < Zl < < z a such that for any k _< n, the sum of the absolute values of the weights w, associated with points in each subinterval [z, Z,+l] can be made as small as possible.Thus for a given h sufficiently small and any e > 0 there exists a partition of [0, a] such that for [z,, Z,+x] partitioned by x,, Xn,+ 1 n,+l With the above assumptions, we now define the order of the quadrature.DEFINmON 3. Let f(t) E Col0, a], q > 0. The quadrature rule Q will be said to have order q if q is the largest positive real number such that xk S(zk, t)f(t)dt Wk, f(z,) =0 <_ Ch , C < oo, VXk e Iz.
For special forms of S(x, t), the order may be non-imeger (see, [2]) For the parallel shooting method, starting values are required on each subinterval of the partition JM.
The two part method is likewise employed on each subinterval.
DEFINITION 4. The linear two part method ((p, a), Q) is said to be convergent if for all equations of the form (5) subject to the conditions stated on f and K, we have that i) ir Y, Y(x) and nh=x ii) for all solutions {Y,) of ( 3) and ( 4) satisfying the starting conditions Y Y(h) for which limh-,O Y(h) Y0, j 0(1)k 1 and z for which limh_0 z 0, j 1(1)k 1.
REMARK.It is well known [6, p. 218], that zero-stability of (p, a) is a necessary condition for convergence.
For the function, F(z, Y, z) the following notation is introduced" i) F,, F(x, Y,, z) ii) F F x,,Y(x,), w,,K(x,,x,,y(x,)) =0 iii) and Then from ( ) subtract the same equation with approximate values replaced by exact values to get --0 where En=Yn-Y(xn).Further, we write Fn n D(n 1) +D(n 2) where D( and D 2) O, F, Fn.We define Gn (gl,, )r where Then IlGlloo < LF1 max(L, 1).Similarly, define 1 (r, r2n) T by rln =0 Then IIPlloo _< LmL3W where Lm max(L2,1) and Iw,,l _< W for all 0 _< _< N With these notations, we can now write down a difference equation for the errors To establish the convergence of this sequence (i.e.E, 0 as co, h 0 with ih x,), let us recall a iemma from Linz [7, p 368] LEMMA 1.If l,l<a<l for =r,r+l,... THEOREM 1.Let us assume that the kernel S(x,t)G(t,y(t)) satisfies the conditions stated in section and that the weights (w,,) exist which for all G(t, y(t)) e C[0, a] and for all 0 _< x _< a, satisfy conditions i) and ii) of this section.Let p(z) az k + + ao satisfy the condition of zero-stability and assume the sequence {7,) is such that suplTl F < co.Let 7-, T*,/, B* and A be constants such that k T.,, ]9,d+1 and Ir,l < 7-, T" kT, z,,. ]9,g,,,+, and ID,,.I < B', and IM.I < A for all --0 0 _< n _< N Then every solution of for which IIE, < E, 0(1)k I satisfies IIEII O(E')Vx IN where E" (NrA + ArE) 1 hrB" hrT'lwl k =0 PROOF.For j 0(1)n k, multiply equation (11) coesponding to m n k j by % d add the resulting equatiom.On the le d side, we get E plus tes, where we have us the fact that a70 1 d the tes e fll zero except for those involng E, m 0(1)k 1.Each of these non zero tes is bounded by kAY.
As a final step in organizing this equation, the terms involving llEnl] are collected and we can write whre X-1 hrB" hrT'l'.l,M* FB*/X, N* rT'/X and E" (Nrh + ArE)/X.
By assumption and for h sufficiently small, the interval [0, a] can be divided into a finite number of subintervals such that hM* / hN* < I for all n _< N and nk-1 < n Thus for all points x, in the first Since there are only a finite number of subintervals, by induction we conclude that IIE,,II O(E*) V:r,n.IN REMARK.For special forms of the function S(x, t), sharper expregsions for the error bound are possible (see, [2]) 4. NUMERICAL EXAMPLE This section contains a numerical example taken from [5].The particular method which was employed to solve the problem is described below.Table 2 contains some numerical results 2V/X + 1 4(1 + x)] , 3 ln(2z + + 2v/z(z / )) + vq + + with the boundary conditions y(O)= 1 and y(1) = The exact solution for this example is (x)--,-, METHOD: For the system (6), the Milne-Simpson method defined by (p, a) where p(z) z 1 and a(z) (z + 4z + 1)/3 is used to discretize each equation.Since the integrand in the example is such that part can be solved by usual quadrature rules and the other by product integration rules, then the EXAMPLE: weights of the fourth order Newton-Gregory rules (see Table 1) are used for the former and a set of product integration rules for the latter based on Simpson's product integration with weights {Wk) The weights {Wk), for k even, consist of a composite Simpson product integration rule with weights given by wk3 .5S0 (j, k) 1.5S, (j, k) + $2 (j, k) wk3+, So(j, k) + 2.0S, (j, k) o+ .ss0 (j, k) 0.5S (j, ) for j O(2)k where and S(j,k) S(x,t)dt.
For k odd, the above rule is used on each subinterval Ix3, x:+2] for .7"O(2)k 3 and the four weights obtained by using the Lagrange three step rule on [xk-3,xk] tO integrate S(xk, t) are given by (,4;l S(xk,t)dt.