A Computational Method for Solving Linear Volterra Integral Equations

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1 Applied Mthemticl Sciences, Vol. 6, 01, no. 17, A Computtionl Method for Solving Liner Volterr Integrl Equtions Frshid Mirzee Deprtment of Mthemtics, Fculty of Science Mlyer University, Mlyer, , Irn f.mirzee@mlyeru.c.ir, mirzee@mil.iust.c.ir Abstrct The im of the present pper is to introduce numericl method for solving liner Volterr integrl equtions of the second kind.the min ide is bsed on the dptive Simpson s qudrture method.the technique is very effective nd simple.we show tht our estimtes hve good degree of ccurcy. Keywords: Volterr integrl equtions; Qudrture;Simpson s qudrture method 1. Introduction Modified Simpson s method for solving integrl x i+ x i f(x)dx is s followes: xi+ x i f(x)dx = h [f i +4f i+1 + f i+ ] + h4 [f i f i+ 180 ] h7 160 f (6) (ζ i ); ζ i (x i,x i+ ). (1) In generl for integrl [,b] we hve: b f(x)dx = N 1 xi+ + h x i f(x)dx h f()+4h N 1 i=1 N 1 f i+1 f i + h h4 f(b)+ 180 [f () f (b)], () where N is even.

2 808 F. Mirzee. Development of modified Simpson s method Consider liner Volterr integrl equtions of the second kind: y(t) =x(t)+ k(t, s)y(s)ds; t b, () where k(t,s) nd x(t) re known functions, but y(t) is n unknown function [1-4]. Now,for solving the equtions () with repeted modified Simpson s method,we consider two cses. Cse 1.The prtil derivtives k(t,s) does not exist: In this cse,we solve equtions () with repeted Simpson s method, so we hve: y(t) = x(t)+ h j 1 [k(t, s i )y i +4k(t, s i+1 )y i+1 + k(t, s i+ )y i+ ]; (4) Hence for t = t 0,t 1,,t N,we get the following system of equtions: y j = x j + h j 1 [k j,i y i +4k j,i+1 y i+1 +k j,i+ y i+ ]; j = 1(1)( N ). (5) Set then we hve y i+1 y i + y i+, (6) j 1 y j = x j + h(k j,0 +k j,1 )y 0 + h i=1 (k j,i 1 + k j,i + k j,i+1 )y i 1 h(k ; j,j 1 + k j,j ) j = 1(1)( N ), (7) where y() =y 0 = x() =x 0.

3 Solving liner Volterr integrl equtions 809 Cse.The prtil derivtives k(t,s) exist: In this cse,we solve equtions () with repeted modified Simpson s method, so we hve: y(t) = x(t)+ h j 1 [k(t, s i )y i +4k(t, s i+1 )y i+1 + k(t, s i+ )y i+ ] + h4 180 [J (t, s 0 )y 0 + k(t, s 0 )y 0 +J (t, s 0 )y 0 +J(t, s 0 )y 0 J (t, s j )y j k(t, s j )y j J (t, s j )y j J(t, s j)y j ]; j = 1(1)(N ), (8) where k(t, s) J(t, s) =,J (t, s) = k(t, s),j (t, s) = k(t, s) s s s must exist. By using eqution (6) nd for t = t 0,t 1,,t N,we get the following system of equtions: y j = x j + h j 1 [(k j,i +k j,i+1 )y i +(k j,i+1 + k j,i+ )y i+ ] + h4 180 [k 0,0y 0 +J 0,0 y 0 +J 0,0y 0 + J 0,0y 0 k j,j y j J j,j y j J j,jy j J j,jy j ]; j = 1(1)( N ). (9) By tking three derivtive from eqution () we obtin y (t) = x (t)+ H(t, s)y(s)ds + k(t, t)y(t); t b, (10) y 0 = y () = x ()+k(, )y(), (11) y (t) = x (t)+ H (t, s)y(s)ds + H(t, t)y(t) +T (t, t)y(t)+k(t, t)y (t); t b, (1)

4 810 F. Mirzee y 0 = y () = x ()+H(, )y()+t (, )y() y (t) = x (t)+ +k(, )y (), (1) H (t, s)y(s)ds + H (t, t)y(t)+v (t, t)y(t) +H(t, t)y (t)+t (t, t)y(t)+t (t, t)y (t) +k(t, t)y (t); t b, (14) y 0 = y () = x ()+H (, )y()+v (, )y()+h(, )y () +T (, )y()+t (, )y () where H(t, s) = k(t,s), H (t, s) = k(t,s) t t +k(, )y (); t b. (15), H (t, s) = k(t,s) t, T (t, t) = dk(t,t), T (t, t) = d T (t,t), V (t, t) = dh(t,t),x (t),x (t),x (t) must dt dt dt exist. We solve equtions (10),(1) nd (14) with repeted modified Simpson s method.by using (6) nd for t = t 0,t 1,,t N, we obtin y j = x j + h j 1 [(H j,i +H j,i+1 )y i +(H j,i+1 + H j,i+ )y i+ ] + h4 180 [H 0,0y 0 + L 0,0 y 0 +L 0,0 y 0 +L 0,0y 0 L j,j y j L j,j y j L j,jy j H j,jy j ] +k j,j y j ; j = 1(1)( N ). (16) y j = x j + h j 1 [(H j,i +H j,i+1 )y i +(H j,i+1 + H j,i+ )y i+]

5 Solving liner Volterr integrl equtions h4 [M 180 0,0y 0 +M 0,0y 0 +M 0,0 y 0 + H 0,0y 0 M j,j y j M j,j y j M j,jy j H j,j y j ] +H j,j y j + T j,j y j + k j,j y j ; j = 1(1)(N ). (17) y j = x j + h j 1 [(H j,i +H j,i+1 )y i +(H j,i+1 + H j,i+ )y i+] + h4 180 [D 0,0 y 0 +D 0,0 y 0 +D 0,0y 0 + H 0,0 y 0 D j,jy j D j,jy j D j,j y j H j,jy j] +H j,j y j + V j,j y j + H j,j y j + T j,j y j +T j,j y j + k j,jy j j = 1(1)(N ). (18) Where L(t, s) = k(t,s) t s, L (t, s) = k(t,s) s t L (t, s) = 4 k(t,s) s t M(t, s) = k(t,s), M (t, s) = 4 k(t,s), M (t, s) = 5 k(t,s), s t s t s t D(t, s) = 4 k(t,s),d (t, s) = 5 k(t,s),d (t, s) = 6 k(t,s) must exist. s t s t s 4 t For i = 1(1)( N ) from systems (9),(16),(17) nd (18) we obtin system with N equtions nd N unknowns. By solving system, the pproximte solution of eqution (),is obtined.. Numericl exmples In this section, we intend to compre this method with other methods such s repeted Simpson s (S), repeted modified trpezoid (MT), nd Pouzet (P) methods (Tble 1).We solve these exmple by using MATLAB v7.1. exmple 1.In this exmple we solve eqution [5]: y(t) =t tsy(s)ds; 0 t, (19)

6 81 F. Mirzee where exct solution is y(t) =te t 15 nd numericl results re shown in Tble. Tble 1 Solution of exmple 1 with S,MT,P methods (methods of Ref.[5]) Tble Nodes t Exct solution S MT P t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t=

7 Solving liner Volterr integrl equtions 81 Tble Solution of exmple 1 with modified Simpson s method Nodes t Exct solution h=0.1 h=0.05 t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= t= Conclusions In this work, we pplied n ppliction of modified Simpson s method for solving the liner Volterr integrl equtions.according to the numericl results which obtining from the illustrtive exmples, we conclude tht for sufficiently smll h we get good ccurcy, since by reducing step size length the lest squre error will be reduced.in ith eqution of qudrture system in (9) (for using Cse ) the error of pproximtion of integrl given in liner integrl eqution with repeted modified Simpson s method is i 160 h7 f (6) (ζ), but

8 814 F. Mirzee i this, for instnce, by using repeted Simpson s method is 180 h5 f (4) (ζ), nd by i using repeted modified trpeziod method is 70 h5 f (4) (ζ). This method will be developed by uthors for solving two-dimensionl Volterr integrl equtions nd their systems. References [1] C.T.H.Bker, G.F.Miller, Tretment of Integrl Equtions by Numericl Methods, Acdemic Press Inc., London, 198. [] L.M.Delves, J.L.Mohmed, Computtionl Methods for Integrl Equtions, Cmbridge University Press, [] A.J.Jerri, Introduction to Integrl Equtions with Applictions, Second ed.,jhon Wiley nd Sons, [4] R.Kress,Liner Integrl Equtions, Springer-Verlg,Berlin Heidelberg,1989. [5] J.Sberi-Ndjfi,M.Heidri, Solving Liner Integrl Equtions of the Second Kind with repeted modified trpezoid qudrture method, Appl. Mth. Comput. 189(007) Received: August 011