The Method of Successive Approximations (Neumann s Series) of Volterra Integral Equation of the Second Kind

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1 Pure and Applied Matematics Journal 016; 56): ttp:// doi: /j.pamj ISSN: Print); ISSN: Online) Te Metod of Successive Approximations Neumann s Series) of Volterra Integral Equation of te Second Kind Tesome Bayleyegn Matebie College of Natural Science, Department of Matematics, Arba Minc University, Arba Minc, Etiopia address: sbayleyegn130@gmail.com To cite tis article: Tesome Bayleyegn Matebie. Te Metod of Successive Approximations Neumann s Series) of Volterra Integral Equation of te Second Kind. Pure and Applied Matematics Journal. Vol. 5, No. 6, 016, pp doi: /j.pamj eceived: October 1, 016; Accepted: October 8, 016; Publised: December 3, 016 Abstract: In tis paper, te solving of a class of bot linear and nonlinear Volterra integral equations of te first kind is investigated. Here, by converting integral equation of te first kind to a linear equation of te second kind and te ordinary differential equation to integral equation we are going to solve te equation easily. Te metod of successive approximations Neumann s series) is applied to solve linear and nonlinear Volterra integral equation of te second kind. Some examples are presented to illustrate metods. Keywords: Volterra Integral Equation, First Kind, Second Kind, Kernel, Metod of Successive Approximations 1. Introduction Te integral equation originates from te conversion of a boundary-value problem or an initial-value problem associated wit a partial or an ordinary differential equation, but many problems lead directly to integral equations and cannot be formulated in terms of differential equations [7], [10] & [11]. Tere as on for doing tis is tat it may make solution of te problem easier, or sometimes enable us, to prove fundamental results on te existence and uniqueness of te solution. An integral equation is an equation in wic te unknown function ) to be determined appears under te integral sign [3] & [7].A typical form of an integral equation in ) is of te form. ) = )+λ ),)) 1) ) Were Kx,t) is called te kernel of te integral equation, and αx) and βx) are te limits of integration. It can be easily observed tat te unknown function fx) appears under te integral sign. It is to be noted ere tat bot te kernel Kx,t) and te function φx) in equation are given functions; and λ is a constant parameter [1], [3], [5], [7]&[9]. Te prime objective of tis paper is to determine te unknown function fx)tat will satisfy equation 1) using a number of Numerical tecniques. It needs considerable efforts in exploring tese metods to find solutions of te unknown function. Te teory and application of integral equations is an important subject witin applied matematics, pysics, and engineering. In particular, tey are widely used in mecanics, geopysics, electricity and magnetism, kinetic teory of gases, ereditary penomena in biology, quantum mecanics, matematical economics, and queuing teory [], [7], [8] & [11]. Integral equations are used as matematical models for many and varied pysical situations, and integral equations also occur as reformulations of oter matematical problems. Now begin wit a brief classification of integral equations, and ten in later sections, by considering volterra integral equations of te first and te second kind. By doing tis, it elps te researcers to prepare temselves for more callenging problems tat will be considered in subsequent topics.. Significance of te Study Integral equations are often easier to solve, more elegant and compact tan a corresponding differential equation. Because it does not require supplementary initial or boundary conditions and te contribution of tis study is tat: Distinguis te importance of integral equation over te differential equation. Initiate oter researcers for dept study about integral equation using different numerical tecniques.

2 Pure and Applied Matematics Journal 016; 56): It gives clue to extend te concept of integral equation to many interdisciplinary areas instead of using differential equations. It invites oter researcers interested on te rest part of integral equations. 3. esearc Metods In tis researc, we ad used successive approximation metod of integral equations in transforming from ordinary differential equation ODE). Integral equations of te first kind are often extremely ill-conditioned. Applying te kernel to a function is generally a smooting operation, so te solution, wic requires inverting te operator, will be extremely sensitive to small canges or errors in te input. Te treatment of te equation will depend on te smootness of Kx,t) and fx). Integral equations of te first kind linear or nonlinear) are generally suspected of being ill-conditioned or illposed. In suc circumstances small canges in fx) or Kx,t) may ave a large effect in te numerical solution of fx) of te problem. If te original problem ad a solution te perturbed equation may ave no solution, and vice-versa.. Integral Equations and Teir elationsip to Differential Equations Te teories of ordinary and partial differential equations are a fruitful source of integral equations. Te researcer sall sketc ere one of te ways in wic integral equations can arise from ordinary differential equations. Most ordinary differential equations can be expressed as integral equations, but te reverse is not true [10] & [11]. To investigate te relationsip between integral and deferential equations, Te researcer will need te following lemma wic will allow us to replace a double integral by a single one. Lemma 1: eplacement Lemma) suppose tat [,!] I is continuous. Ten ' && ) ) = & )), [,!] Proof:Define F [a,b] I by 0) = & )), [,!] Asx t)ft) and 1 [x t)ft)] are continuous for all x 1 and t in [a, b], we can use [Leibniz rule] to differentiate F: 0 ) ) = ))3 5 +& 6 6 [ ))] = &). 9 Since, again by [Leibniz rule], 78) ;8 and ence <=, are : <> continuous functions of > on [:,?], we may now apply te fundamental teorem of calculus I to deduce 0 ) ) = 0 ) ) 0 )= ' 0 ) ) ' = ) ) Swapping te roles of x and x we ave te result as stated. Alternatively, define, for t,x ) [a,x A, ) ) =B ),CEF ), 0,CEF ). Te function G = Gt,x ) is continuous, except on te line given by t =x, and ence integrable. Using Fubini s Teorem ' && ) ) = &J&A, ) )K ) 3) = L A, ) ) ) M = L ) ) M = ' )) Hence proved. Now considering te first-order differential equation. ) NO N =P) =,P) 5) Wit te initial condition y 0) = y if say, fx,y) is continuous function of x,y), integrate 5) from 0 tox, obtaining &S P T = &,P)&P = &L,P)MP) P0) = &L,P)M P) = P L,P)M,CEUEP0) = P 6) Tis illustrates te general fact tat, by going over to integral equations, it includes bot te differential equation and te initial conditions in a single equation. Again consider te second-order differential equation. V W X V W = y)) =fx,y), Wit initial conditions.

3 13 Tesome Bayleyegn Matebie: Te Metod of Successive Approximations Neumann s Series) of Volterra Integral Equation of te Second Kind Ten integrate 7) from 0 to x. Ten y0) = y,y ) = y Y 7) & SP T = &[L,P)M] P ) ) P ) 0) = &[L,P)M] P ) ) = P ) 0)+ &[L,P)M]![,P ) 0) = P Y P ) ) = P Y + [L,P)M] 8) Wence te second integration &S P T =&P Y +&\&[L[,P[)M] [] P) P0)= P Y + &&L[,P[)M[,![P0) = P P) = P +P Y + &&L[,P[)M[ P) = P +P Y + &[L,P)M] &[ P) = P +P Y + )^L,P)M_ 9) Te argument is reversible, so tat ere again te differential equation 7), togeter wit te initial conditions, is equivalent to te single integral equation 9). We see also tat any solution of 7) satisfies an integral equation of te form. P) = `+a+ )^L,P)M_ 10) Te constant A and B being determined by te initial conditions. Tey may also be determined in oter ways suppose, for instance, tat yx) is required to satisfy a two - point boundary condition, sayp0) =d,pe) =fg =e) substituting in 10), we obtain. d =P0) =`f =Pe) ` = d,a = i Y =`+ae+ &e )[L,P)M] e )[, P)] 11) Hence, te function yx) must terefore satisfy te integral equation P) =d+ f d e + & )[,P)] e &e )[,P)] Tis can be written in te form P) = j), )[, P)] Were j) = d+ i e ),mu 0 k,) =l e e ),mu e e 1) K x,t) is te kernel of te equation te argument is again reversible, so tat 1) is equivalent to 7) togeter wit te boundary conditions. If te differential equation is linear, we are led in tis way to a linear integral equation of te second kind. Example: 1 educe te initial value problem P )+P) =qgf, wit initial conditions P0) = 0,P 0) =0 =0to volterra integral equation of te second kind. Solution: Volterra equation can be obtained in te following manner: Let Integrate 13) from 0 to. We ten ave & P )) ) =) 13) S P T = &) P ) ) P ) 0)=&) P ) ) = P ) 0)+&) P ) ) = 0+&) = &) Wence te second integration & P = &&[)[ = &)&[ = & ))

4 Pure and Applied Matematics Journal 016; 56): Ten te given ODE becomes )+& )) =qgf Tis is te Volterra integral equation. 5. Integral Equation Definition: Any equation in wic te unknown function fx) appears under te integral sign and integrals of tat function to be solved for fx) is known as integral equations. Te general linear integral equation for te unknown function fx) is s)) = 0) + λ ), )) ) 1) Kx,t) is called te kernel, λ te parameter of te integral equation and αx) and βx) are constants orαx) is a constant and βx) = x. If Fx) = 0, ten te equation is referred to as omogeneous. Weng x) = 0, te equation is of te first kind; oterwise, it is of te second kind. Te kernel is always defined and continuous on u ={,):d) f),d) f)} classification of integral equations. An integral equation can be classified as a linear or nonlinear integral equation as we know in te ordinary and partial differential equations [], [6] & [7]. In te previous section, Tese ave noticed tat te differential equation can be equivalently represented by te integral equation. Terefore, tere is a good relationsip between tese two equations. Te most frequently used integral equations fall under two major classes, namely Volterra and Fredolm integral equations. Of course, Tese ave to classify tem as omogeneous or non omogeneous; and also linear or nonlinear. In some practical problems. Tis researc is focusing on te Volterra integral equations and its solution by te metod of successive approximations Neumann s series) Te classification of integral equations centers on tree basic caracteristics wic togeter describe teir overall structure and it is useful to set tese down briefly before entering into greater detail. I Te kind of an equation refers to te location of te unknown function. First kindequations ave te unknown function present under te integral sign only. Secondkind equations also ave te unknown function outside te integral. II Te istorical descriptions Fredolm and Volterra are concerned wit te integration.interval. In a Fredolm equation te integral is over a finite interval wit fixed endpoints. In a Volterra equation te integral is indefinite. III Te adjective singular is sometimes used wen te integration is improper, eiter. because te interval is infinite, or because te integrand is unbounded witin te. given interval. Obviously an integral equation can be singular on bot counts. 6. Volterra Integral Equation Definition: Volterra equations are written in a form were te upper limit of integration βx) = x independent variable). Te most standard form of Volterra linear integralequations is given by te form. s)) = 0) + λ, )) 15) ) Volterra integral equation of te first kind. Ifgx) = 0, ten 15) yields 0= 0)+λ,)) 16) ) Tis equation is called Volterra first kind integral equations. Homogeneous volterra integral equations. If gx) = 1and Fx)=0, ten eqn. 15)gives. ) = λ, )) 17) ) Volterra integral equation of te second kind, If te function gx) = 1, ten 15)yields. ) = 0)+λ,)) 18) ) Tis equation is called Volterra integral equations of second kind Non omogeneous volterra integral equations. If gx)=1 and Fx) 0, en eqn.8) gives ) = 0)+λ,)) 19) ) Tis equation is called non omogeneous Volterra integral equations of second kind For non omogeneous Volterra integral equations λ is numerical parameter, wereas for omogeneous Volterra integral equations λ is an eigen value parameter because in suc a case te integral equation presents an eigen value problem in wic te objective is to determine tose values of λ, called te eigenvalues for wic te integral equation possesses nontrivial solutions called eigen functions Kernel of an Integral Equation Wen considering numerical metods for integral equations, particular attention sould be paid to te caracter of te kernel, wic is usually te main factor governing te coice of an appropriate quadrature formula or system of approximating functions. Various commonly occurring types of singularity call for individual treatment. Likewise provision can be made for cases of symmetry, periodicity or oter special structure, were te solution may ave special properties and/or economies may be affected in te solution process. We note in particular te following cases to wic we sall often ave occasion to refer in te description of individual algoritms. Te presence of te kernel under te operator makes te

5 15 Tesome Bayleyegn Matebie: Te Metod of Successive Approximations Neumann s Series) of Volterra Integral Equation of te Second Kind beavior of tese equations less transparent tan differential equations. Consider te apparently benign kernel clearly te form of te kernel is crucial to nature of te solution, indeed, to its very existence. A linear integral equation wit a kernel kx,t) = kt,x) is said to be symmetric. Tis property plays a key role in te teory of fredolm integral equations. If Kx,t) =Ka+b x,a+b t) in a linear integral equation, te kernel is called centro-symmetric If te equations of te kernel as te form Kx,t) = KLx,t,yt)M=Kx t)glt,yt)m, te equation is called a convolution integral equation; in te linear case Kx t)glt,yt)m=yt). If te kernel as te form KLx,t,yt)M=K Y Lx,t,yt)M,a t x. KLx,t,yt)M Lx,t,yt)M,x t b Were te functions Y are well beaved. 6.. Te Conversion First Kind Integral Equations to te Second Kind Integral Equations Tus it would appear tat Volterra integral equations of te second kind are more well beaved tat Volterra integral equation of te first kind [7]. To te extent tat tis is true, we may replace any Volterra integral equation of te first kind wit Volterra integral equations of te second kind. Te relation between Volterra integral equations of te first and te second kind can be establised in te following manner. Te first kind Volterra equation is usually. ) = k, ) [) 0) N{ if te derivatives = N ), N = N,) F N = N,) exists and are continuous, ten te equation can be reduced to one of te second kind into ways. Te first and te simplest way are to differentiate bot sides of equation 11) wit respect to x and we obtain by using te Leibnitz rule. Leibniz General ule If }) Ix) = & fx,t)dt, ten I ) x) = d dx [ & fx,t)dt ] ~) }) = & x fx,t))dt ~) }) ~) ) ) = & 6 6 k,)[)) +,)[) ) 0,)[)0) ),)[)+,)[) = ) ) 1) If Kx,x) 0, ten dividing trougout by tis we obtain.,)[)+,)[) = ) ) ) [)+,),) [) = {' ),) 3) And te reduction is accomplised. Tus, we can use te metod already given above. If te kernel in 1) is squareintegrable, and [ { ),) will ave a unique solution in [0,1]. Te second way to obtain te second kind Volterra integral equation from te first kind is by using integration by parts, if we set. Or equivalently [) [Š)Š Ten by using integration by parts,,)&[š)š) Œ wic reduces to 5 = ) ) = ) 5) &,) J&[Š)Š) K = ) [,) )] 5 &,) ) = ) And finally we get,) ),0) 0),) ) = ) 6) It is obvious tat φ 0) = 0, and dividing out by Kx, x) we ave ) = were { ' ) Ž+,) Ž,),) 0) = )0) A, ) ) 7) ),),A,)=,), ) = ƒ) ) {,) +[),)[ ) ) ),) ) ), N{) N = k,)[)) For tis second process it is apparently not necessary for fx) to be differentiable However, te function ux) must finally be calculated by differentiating te function φx) given by te formula. ) = { ' ),) Ž+,;1) { ' ),) Ž 8)

6 Pure and Applied Matematics Journal 016; 56): Were Hx,t 1 is te resolvent kernel corresponding to,. To do tis fxmust be differentiable, 6.3. Solution of Volterra Integral Equation In te previous sections, we ave clearly defined te integral equations wit some useful illustrations. Tis section deals wit te Volterra integral equations and teir solution tecniques. Te approac of Volterra equations in muc te same way as it as been done Fredolm equations, but tere is te problem tat te upper limit of te integral is te independent variable of te equation. For tus coose a quadrature sceme tat utilizes te endpoints of te interval; oterwise we will not be able to evaluate te functional equation at te relevant quadrature points. One could adopt te view tat Volterra equations are, in general, just special cases of Fredolm integral equation equations. λ, 9) Were Kx,t is te kernel of te integral equation, φ x a continuous function of x,and λ a parameter. Here, φ xand Kx,t are te given functions but fx is an unknown function tat needs to be determined. Te limits of integral for te Volterra integral equations are functions ofx. Te nonomogeneous Volterra integral equation of te first kind is defined as, 30) Te important class of integral equations in wic many features of te general teory already appeared in te introduction. Tere are a ost of solution tecniques to deal wit te Volterra integral equations Te Metod of Successive Approximations for Linear Volterra Integral Equations In tis metod, replace te unknown function fx under te integral sign of te Volterra equation by any selective real-valued continuous function f t, called te zerot approximation [10]. Tis substitution will give te first approximation f Y t by. Y λ, 31) It is obvious tat f Y x is continuous if φx, Kx,t, and f x are continuous. Te second approximation x can be obtained similarly by replacing f x in equation 31) by f Y x obtained above. And we λ, Y 3) Continuing in tis manner, we obtain an infinite sequence of function f x,f Y x,f x,f x, Tat satisfies te recurrence relation λ, iy 33) for n1,,3,... and f x is equivalent to any selected realvalued function. Te most commonly selected function forf x are 0,1,and x. Tus, at te limit, te solution fx of te equation 9) is obtained as eg 3) so tat te resulting solution fx is independent of te coice of te zerot approximation f x. Tis process of approximation is extremely simple. However, if we follow te Picard s successive approximation metod, it needs to set f x =φx, and determine f Y x and oter successive approximation as follows: Y λ, Y iy λ, i@ λ, iy Te last equation is te recurrence relation. Consider * Y λ,λ, " * λ, Were,, 36) Tus, it can be easily observed from equation 36) tat If g xφx, and furter tat 37) š5 λ š s š 38) s š,s šiy 39) Were m= 1,, 3, and ence g Y x Kx,tφtdt Te repeated integrals in equation 37) may be considered as a double integral over te triangular region indicated in Figure 1; tus intercanging te order of. Figure 1. Double integration over te triangular region saded area). Integration, we obtain.

7 17 Tesome Bayleyegn Matebie: Te Metod of Successive Approximations Neumann s Series) of Volterra Integral Equation of te Second Kind & &,), ) = ) ) =,), ). Similarly, we find in general. s š ) = š, ) ), =1,,3, 0) Were te iterative kernels Y, ),. are defined by te recurrence formula šžy,) ) š,) Tus, te solution forf x) can be written as ) =)+ š5y š, ) ) =)+ { λ š š5yλ š 1) ) š, ) } ) 3) Hence it is also plausible tat te solution of equation 9) will be given be as n lim f x) = fx) =)+& { λ š š, )} ) = )+λ, ; Were š5y λ) L)+λ Ei ) M ), ;λ) = š5y λ š š, ) 5) is known as te resolvent kernel. Example 1 Solve te following Volterra integral equation of te second kind of te convolution type using successive approximation metod. ) = )+λ E i ) Solution: using successive approximation metod Let us assume tat te zerot approximation is 6) ) = 0 7) ) =)+λ&e ) =) +λ&e i \)+λ&e i ) ] = )+λ&e i &&E i ) = )+λ&e i & )E i ) In te double integration te order of integration is canged to obtain te final result. In a similar manner, te fourt approximation f x) can be at once written as. ) = )+λ&e i & )E i ) + λ & )@ E i )! Tus, continuing in tis manner, we obtain as n ) = eg ) = ) +λ &E i L1+λ + M) = )+λ&e i E λi) )) +λ&e Yžλ)i) ) Ten te first approximation can be obtained as Y ) = ) 8) Using tis information in equation 6), te second approximation is given ) =)+λ&e i Y ) = )+λ E i ) 9) Proceeding in tis manner, te tird approximation can be obtained as Here, te resolvent kernel is,; ) =E Yžλ)i) Anoter metod to determine te solution by te resolvent kernel Te procedure to determine te resolvent kernel is te following: Given tat ) =)+λ&e i )

8 Pure and Applied Matematics Journal 016; 56): Here, te kernel is,) =E i. Te solution by te successive approximation is. =+λ&, ;λ were te resolvent kernel is given by In wic, ;λ= λ žy, 5 žy,=+λ&,,,f=1,,3 It is to be noted tatk x,t=kx,ttus, we obtain x,t=&e i e i dτ=e i«&dτ=x te i««similarly, proceeding in tis manner, we obtain K x,t=&e i e i τ tdτ «K x,t=e i«x t 3! «=e i«x t@! = K žy x,t=e i«x t n! Hence te resolvent kernel is Hx,τ;λ= λ K žy x,t=e i«x t Hx,t; λ n! 5 =e Yžλi«Once te resolvent kernel is known terefore succeeded in inverting te integral equation because te rigt- and side of te above formula is a known quantity Te Metod of Successive Approximations for Non-Linear Integral Equations Nonlinear integral equations yield a considerable amount of difficulties. However, due to recent development of novel tecniques it is now possible to find solutions of some types of nonlinear integral equations if not all. In general, te solution of te nonlinear integral equation is not unique. However, te existence of a unique solution of nonlinear integral equations wit specific conditions is possible. We first define a nonlinear integral equation in general, and ten cite some particular types of nonlinear integral equations. In general, a nonlinear integral equation is defined as given in te following equation: 5 fx= Fx+λ& Kx,tGLftMdt = Fx+λ& Gx,t,ftdt orfx Were αx and βx are constants and te equations are called non linear Fredolm integral equations. fx= Fx+λ& Kx,tGLftMdt orfx = Fx + λ& GLx,t,ftMdt Were αx is constant and xis independent variable te equations are called non linear Volterra integral equations. Te function G f t is non-linear except G =a constant or G fx =fx in wic case G is linear. GLfxM= f x,forn, ten function G is non linear. Example 1 and fx= x+λ& x tdt x=1+λ&kx,tlnlftmdt Are non linear volterra integral equations. By contrast, solving nonlinear Volterra equations usually involves only a sligt modification of te algoritm for linear equations. Te Picard s metod to obtain successive algebraic approximations. By putting numbers in tese, we generally get excellent numerical results. Unfortunately, te metod can only be applied to a limited class of equations, in wic te successive integrations be easily performed. We sall treat everal examples by tese metods to enable teir merits to be compared. Consider te initial value problem given by te first-order nonlinear differential equation. df/dx = φ x, fx wit te initial condition fa = batx=a.tis initial value problem can be transformed to te nonlinear integral equation and is written as =!+&, For a first approximation, we replace tefxin φlx,fxmbyb, for a second approximation, we replace it by te first approximation, for te tird by te second, and so on. =! Y =!+&,!

9 19 Tesome Bayleyegn Matebie: Te Metod of Successive Approximations Neumann s Series) of Volterra Integral Equation of te Second ) =!+&! +&&,!) To demonstrate tis metod by examples. Example 1 Solve te integral equation by Picard s metod of successive approximation. ) = )M Solution Te given differential equation can be written in integral equation form as ) = )) Zerot approximation is: fx) = 0. First approximation: Put fx)=0in x), yielding ) =& = 1 Second approximation:put ) = yielding ) ³ + 0 Tird approximation: f= + in x+f@, giving in ), fx) =& t+ t@ t + 0 dt =&²t+ t t ty 00 ³ dt = x@ + x 0 + x¹ 10 + xyy 00 Proceeding in tis manner, Fourt approximation can be written after a rigorous algebraic manipulation as Fourt approximation fx) = x@ x x¹ xYY xY x@ , and so on. Tis is te solution of te problem in series form, and it seems from its appearance te series is convergent. 7. Conclusion In tispaper we proposed some of te numerical metods to solve volterra linear and non linear) integral equations. And fx) =φx)+λ & kx,t)ft)dt ) fx) = φx)+λ& Kx,t)GLft)Mdt ) To solve tis for fx) te coice of te initial data f 0 x), plays an essential role on te speed of te convergence of te numerical metods, successive approximation metod. Numerical metods ave been selected based on te ability to obtain rapid convergence, based on existence of reliable and ceaply computable error estimates and personal predilection Wen te kernel is simple enoug to be closely approximated by a degenerate kernel wit a few functions, tis can be very efficient. How competitive in general suc a metod is wit te more usual approaces is an unresolved question. A solution of te integral equation gives more accurate results tan a solution of te differential equation using te same step size and degree of precision in te integration procedure. eferences [1] W. V. Lovitt. 1950). Linear Integral Equations, Dover epresented. [] S.S.SASTY. 003). Introductory Metods of Numerical Analysis, Tird Edition. [3] M. aman. 007). Integral Equations and teir Applications, 1 st ed; WIT Press. [] Peter Linz. 1985). Analytical and Numerical Metods for Volterra Equations. [5] Porter, David. 1990). Integral equations A practical treatment, from spectral teory to applications, first edition. [6] F. Smities 1958). Cambridge Tracts in Matematics and Matematical Pysics. [7] Peter J. Collins. 006). Differential and Integral equations, WIT Press. [8] William Squire. 1970). Modern Analytic and Computational Metods in Science and Matematics. [9] W. Pogorzelski, Integral Equations and teir Applications, volume I. [10] Abdul-Majid Wazwaz. A First Course in Integral Equations - Solutions Manual, nd ed; World Scientific Publising Co. Pte. Ltd. [11] T. A. Burton Eds. 005). Volterra Integral and Differential Equations, nd ed; Academic Press, Elsevier.