Reducing Initial Value Problem and Boundary Value Problem to Volterra and Fredholm Integral Equation and Solution of Initial Value Problem

Transcription

1 Reducing Initil Vlue Prolem nd Boundry Vlue Prolem to Volterr nd Fredholm Integrl Eqution nd Solution of Initil Vlue Prolem Hee Khudhur Kdhim M.Sc Applied Mthemtics Student, Ntionlity of Irq Deprtment of Mthemtics College of Sciences Osmni University, Hyderd, Indi. I. Introduction Origins of Integrl Equtions Integrl nd integro-differentil equtions rise in mny scientific nd engineering pplictions. Volterr integrl equtions nd Volterr integro-differentil equtions cn e otined from converting initil vlue prolems with prescried initil vlues. However, Fredholm integrl equtions nd Fredholm integrodifferentil equtions cn e derived from oundry vlue prolems with given oundry conditions. It is importnt to point out those converting initil vlue prolems to Volterr integrl equtions, nd converting Volterr integrl equtions to initil vlue prolems re commonly used in the literture. This will e eplined in detil in the coming section. However, converting oundry vlue prolems to Fredholm integrl equtions, nd converting Fredholm integrl equtions to equivlent oundry vlue prolems re rrely used. The conversion techniques will e emined nd illustrted emples will e presented. In wht follows we will emine the steps tht we will use to otin these integrl nd integrodifferentil equtions. Preliminries An integrl eqution is n eqution in which the unknown function u() ppers under n integrl sign. A stndrd integrl eqution in u() is of the form: u() = f() + λ h() g() K(, t)u(t)dt (I..) Where g() nd h() re the limits of integrtion, λ is constnt prmeter, nd K(, t) is function of two vriles nd t clled the kernel or the nucleus of the integrl eqution. The function u() tht will e determined ppers under the integrl sign, nd it ppers inside the integrl sign nd outside the integrl sign s well. The functions f() nd K(, t) re given in dvnce. It is to e noted tht the limits of integrtion g() nd h() my e oth vriles, constnts, or mied. An integro-differentil eqution is n eqution in which the unknown function u() ppers under n integrl sign nd contins n ordinry derivtive u n () s well. A stndrd integrodifferentil eqution is of the for u n () = f() + λ h() g() K(, t)u(t)dt (I..2) Whereg(), h(), f(), λ nd the kernel K(, t) re s prescried efore. Integrl equtions nd integrodifferentil equtions will e clssified into distinct Pge 2

2 types ccording to the limits of integrtion nd the kernelk(, t). All types of integrl equtions nd integro-differentil equtions will e clssified nd investigted in the forthcoming chpters. In this chpter, we will review the most importnt concepts needed to study integrl equtions. The trditionl methods, such s Tylor series method nd the Lplce trnsform method, will e used in this tet. Moreover, the recently developed methods tht will e used thoroughly in this tet will determine the solution in power series tht will converge to n ect solution if such solution eists. However, if ect solution does not eist, we use s mny terms of the otined series for numericl purposes to pproimte the solution.the more terms we determine the higher numericl Accurcy we cn chieve. Furthermore, we will review the sic concepts for solving ordinry differentil equtions. Other mthemticl concepts, such s Leinitz rule will e presented.. Tylor Series Let f() e function with derivtives of ll orders in n intervl [, ] tht contins n interior point. The Tylor series of f() generted t = is: f n () f() = n= ( ) n (I..3) n! Or equivlently f() = f() + f () ( ) + f () ( ) 2 +! 2! f () ( ) fn () ( ) n +.. (I..4) 3! n! The Tylor series generted y f() t = is clled the Mclurin series nd given y: f() = f n () n= n n! Tht is equivlent to (I..5) f() = f() + f () + f () 2 +! 2! f () fn () n + (I..6) 3! n! In wht follows, we will discuss few emples for the determintion of the Tylor series t =..2 Ordinry Differentil Equtions In this section we will review some of the liner ordinry differentil equtions tht we will use for solving integrl equtions. For proofs, eistence nd uniqueness of solutions, nd other detils, the reder is dvised to use ordinry differentil equtions tets..2. First Order Liner Differentil Equtions The stndrd form of first order liner ordinry differentil eqution is u + p()u = q() (I..7) Where p() nd q() re given continuous functions on < <. We first determine n integrting fctor μ() y using the formul: μ() = e ( o p(t) )dt (I..8) Recll tht n integrting fctor μ() is function of tht is used to fcilitte the solving of differentil eqution. The solution of (I..7) is otined y using the formul: u() = μ() [ μ(t)q(t)dt + c](i..9) Where c is n ritrry constnt tht cn e determined y using given initil condition.2.2 The Series Solution Method For differentil equtions of ny order, with constnt coefficients or with vrile coefficient, with = is n ordinry point, we cn use the series solution method to determine the series solution of the differentil eqution. The otined series solution my converge the ect solution if such closed form solution eists. If n ect solution is not otinle, we my use truncted numer of terms of the otined series for numericl purposes. Although the series solution cn e used for equtions with constnt coefficients or with vrile coefficients, where = is n ordinry point, ut this method is commonly used Pge 2

3 for ordinry differentil equtions with vrile coefficients where = is n ordinry point. The series solution method ssumes tht the solution ner n ordinry point = is given y u() = n= n n, (I..) Or y using few terms of the series u() = o (I..) Differentiting term y term gives u () = u () = u () = ) (I. And so on. Sustituting u() nd its derivtives in the given differentil eqution, nd equting coefficients of like powers of gives recurrence reltion tht cn e solved to determine the coefficients n, n. Sustituting the otined vlues of n, n in the series ssumption (I..) gives the series solution. As stted efore, the series my converge to the ect solution. Otherwise, the otined series cn e truncted to ny finite numer of terms to e used for numericl clcultions. The more terms we use will enhnce the level of ccurcy of the numericl pproimtion. It is interesting to point out tht the series solution method cn e used for homogeneous nd inhomogeneous equtions s well when = is n ordinry point. However, if = is regulr singulr point of n ODE, then solution cn e otined y Froenius method tht will not e reviewed in this tet. Moreover, the Tylor series of ny elementry function involved in the differentil eqution should e used for equting the coefficients. The series solution method will e illustrted y emining the following ordinry differentil equtions where = is n ordinry point. Some emples will give ect solutions, wheres others will provide series solutions tht cn e used for numericl purposes..3 Leinitz Rule for Differentition of Integrls One of the methods tht will e used to solve integrl equtions is the conversion of the integrl eqution to n equivlent differentil eqution. The conversion is chieved y using the well-known Leinitz rule for differentition of integrls. Let f(, t) e continuous nd f e continuous in domin of the -t plne tht t includes the rectngle, t t t, nd let F() = h() g() f(, t) dt (I..3) then differentition of the integrl in (.3) Eists nd is given y F ()= df = d h() f(,t) g() dh() f(, h()) d dt (I..4) f(, g()) dg() d If g() = nd h() = where nd re constnts, then the Leinitz rule (I..4) reduces to F ()= df = d f(,t) dt (I..5) Which mens tht differentition nd integrtion cn e interchnged such s d d et dt t e t dt (I..6) is interested to notice tht Leinitz rule is not pplicle for the Ael s singulr integrl eqution: F() = u(t) o ( t) α d t <α< (I..7) The integrnd in this eqution does not stisfy the conditions tht f(, t) e continuous nd continuous, ecuse it is unounded t = t. We illustrte the Leinitz rule y the following emple. Emple f t + = e Pge 22

4 Find F () for the following: F() = cos sin + t 3 dt We cn set g() = sin nd h() = cos. It is lso cler tht f(, t) is Function of t only. Using Leinitz rule (I..4) we find tht F () = sin + cos 3 cos + sin 3.4 Lplce Trnsform In this section we will review only the sic concepts of the Lplce trnsform method. The detils cn e found in ny tet of ordinry differentil equtions. The Lplce trnsform method is powerful tool used for solving differentil nd integrl equtions.the Lplce trnsform chnges differentil equtions nd integrl equtions to polynomil equtions tht cn e esily solved, nd hence y using the inverse Lplce trnsform gives the solution of the emined eqution. The Lplce trnsform of function f(), defined for, is defined y F(s) =L { f() } = e s f() d (I..8) where s is rel, nd L is clled the Lplce trnsform opertor. The Lplce trnsform F(s) my fil to eist. If f() hs infinite discontinuities or if it grows up rpidly, then F(s) does not eist. Moreover, nimportnt necessry condition for the eistence of the Lplce trnsform F(s) is tht F(s) must vnish s s pproches infinity. This mens tht lim s F(s) =. (I..9) In other words, the conditions for the eistence of Lplce trnsform F(s) of ny function f() re:. F() is piecewise continuous on the intervl of integrtion < A for ny positive A, 2. f() is of eponentil order e s, i. e. f() Ke, M where is rel constnt, nd K nd M re positive constnts. Accordingly, the Lplce trnsform F(s) eists nd must stisfy lim F(s) =. (I..2) s 2. Fredholm Integrl Equtions For Fredholm integrl equtions, the limits of integrtion re fied. Moreover, the unknown function u() my pper only inside integrl eqution in the form: f() = K(, t) u(t) dt (I.2.8) This is clled Fredholm integrl eqution of the first kind. However, for Fredholm integrl equtions of the second kind, the unknown function u() ppers inside nd outside the integrl sign. The second kind is represented y the form: u()= f() + λ K(, t) u(t) dt (I.2.9) Emples of the two kinds re given y sin cos 2 And u() = + 2 Respectively. = sin(t) u(t)dt ( t)u(t)dt (I.2.) (I.2.) 2.2 Volterr Integrl Equtions In Volterr integrl equtions, t lest one of the limits of integrtion is vrile. For the first kind Volterr integrl equtions, the unknown function u() ppers only inside integrl sign in the form: f() = K(, t) u(t) dt. (I.2.2) However, Volterr integrl equtions of the second kind, the unknown function u() ppers inside nd outside the integrl sign. The second kind is represented y the form: u()= f() + λ K(, t) u(t) dt (I.2.3) Pge 23

5 Emples of the Volterr integrl equtions of the first kind re: e = e t u(t) dt (2.4) nd = ( t)u(t)dt(I.2.5) However, emples of the Volterr integrl equtions of the second kind re u()= And u()= + ( t) u(t) dt (I.2.7) u(t) dt (I.2.6) 2.3 Volterr-Fredholm Integrl Equtions The Volterr-Fredholm integrl equtions rise from prolic oundry vlue prolems, from the mthemticl modeling of the sptio-temporl development of n epidemic, nd from vrious physicl nd iologicl models. The Volterr- Fredholm integrl equtions pper in the literture in two forms, nmely u()=f() + λ K (, t) λ 2 K 2 (, t) u(t)dt (I.2.8) And u(, t) = f() + t u(t)dt + λ F(, t, ε, τ, u(ε, t))dε dτ, (, t) εω [, T] Ω (I.2.9) Wheref(, t) nd F(, t, ξ, τ, u(ξ, τ)) re nlytic functions on D =Ω [,T], nd Ω is closed suset of R n,n =,2,3. It is interesting to note tht (I.2.8) contins disjoint Volterr nd Fredholm integrl equtions, wheres (I.2.9) contins mied Volterr nd Fredholm integrl equtions. Moreover, the unknown functions u() nd u(, t) pper inside nd outside the integrl signs. This is chrcteristic feture of second kind integrl eqution. If the unknown functions pper only inside the integrl signs, the resulting equtions re of first kind, ut will not e emined in this tet. Emples of the two types re given y u() = u(t) dt t u (t) dt, (I.2.2) u(, t) = + t t2 t 2 (τ ε) dε dτ (I. 2. 2) 2.4 Singulr Integrl Equtions Volterr integrl equtions of the first kind h() f() = λ K(, t)u(t) dt (I.2.22) g() or of the second kind u() = f() + h() g() (I.2.23) K(, t) u(t) dt re clled singulr if one of the limits of integrtion g(), h() or oth re infinite. Moreover, the previous two equtions re clled singulr if the kernel K(, t) ecomes unounded t one or more points in the intervl of integrtion. In this tet we will focus our concern on equtions of the form: f() = ( t) α or of the second kind: u() = f() + u(t) dt < α < (I.2.24) ( t) α u(t)dt, < α < (I.2.25) t Pge 24

6 The lst two stndrd forms re clled generlized Ael s integrl eqution nd wekly singulr integrl equtions respectively. Forα = 2 f()= t u(t)dt (I.2.26), the eqution: is clled the Ael s singulr integrl eqution. It is to e noted tht the kernel in ech eqution ecomes infinity t the upper limit t =. Emples of Ael s integrl eqution, generlized Ael s integrl eqution, nd the wekly singulr integrl eqution re given y = 3 = And t ( t) 3 u(t)dt u()= + + Respectively. u(t)dt ( t) 3 (I.2.27) (I.2.28) u(t)dt, (I.2.29) 2.5 Integro-eqution: 2.5. Fredholm Integro-Differentil Equtions Fredholm integro-differentil equtions pper when we convert differentil equtions to integrl equtions. The Fredholm integro-differentil eqution contins the unknown function u() nd one of its derivtives u n (),n inside nd outside the integrl sign respectively. The limits of integrtion in this cse re fied s in the Fredholm integrl equtions. The eqution is leled s integro-differentil ecuse it contins differentil nd integrl opertors in the sme eqution. It is importnt to note tht initil conditions should e given for Fredholm integro-differentil equtions to otin the prticulr solutions. The Fredholm integro-differentil eqution ppers in the form: u n ()= f() + λ K(, t) u(t) dt (I.2.3) Where u n indictes the nth derivtive of u(). other derivtives of less order my pper with u n t the left side. Emples of the Fredholm integrodifferentil equtions re given y u ()= - 3 And + u(t) dt, u() = (I.2.3) u"()+u () = sin u()= (I.2.32) ( π 2 ) tu(t)dt, u()=, Volterr Integro-Differentil Equtions Volterr integro-differentil equtions pper when we convert initil vlue prolems to integrl equtions.the Volterr integro-differentil eqution contins the unknown function u() nd one of its derivtives u n (), n inside nd outside the integrl sign. At lest one of the limits of integrtion in this cse is vrile s in the Volterr integrl equtions. The eqution is clled integro-differentil ecusedifferentil nd integrl opertors re involved in the sme eqution. It is importnt to note tht initil conditions should e given for Volterr integrodifferentil equtions to determine the prticulr solutions. The Volterr integro-differentil eqution ppers in the form: u n () = f() + λ K(, t) u(t), dt (I.2.33) Whereu n indictes the nth derivtive of u(). Other derivtives of less order my pper with u n t the left side. Emples of the Volterr integro-differentil equtions re given y Pge 25

7 u ()= e - tu(t)dt, u()=, (I.2.34) And u"()+u () = -(sin + cos ) tu(t)dt, u() =, u () = (I.2.35) Volterr-Fredholm Integro-Differentil Equtions The Volterr-Fredholm integro-differentil equtions rise in the sme mnner s Volterr-Fredholm integrl equtions with one or more of ordinry derivtives in ddition to the integrl opertors. The Volterr- Fredholm integro-differentil equtions pper in the literture in two forms, nmely u n () = f()+ λ 2 K 2 (, t) u(t)dt (I.2.36) And u n (, t) = f(, t) + t λ K (, t) u(t)dt + λ F(, t, ε, τ, u(ε, τ)) dεd τ, (, t) Ω Ω [, T](I.2.37) Where f(, t) nd F(, t, ξ, τ, u(ξ, τ)) re nlytic functions on D = Ω [, T], nd Ω is closed suset of R n,n=,2,3 It is interesting to note tht (I.2.36) contins disjoint Volterr nd Fredholm integrl equtions, wheres (I.2.37) contins mied integrls. Other derivtives of less order my pper s well. Moreover, the unknown functions u() nd u(, t) pper inside nd outside the integrl signs. This is chrcteristic feture of second kind integrl eqution. If the unknown functions pper only inside the integrl signs, the resulting equtions re of first kind. Initil conditions should e given to determine the prticulr solution. Emples of the two types re given y u ()= tu(t)dt, u() ( t)u(t)dt = (I ) And u (, t)= + t t2 t 2 (τ ε)dεdτ, u(, t) = t 3 (I.2.39) II. VOLTERRA THEORY Volterr opertor It ws stted in Chpter tht Volterr integrl equtions rise in mny scientific pplictions such s the popultion dynmics, spred of epidemics, nd semi-conductor devices. It ws lso shown tht Volterr integrl equtions cn e derived from initil vlue prolems. Volterr strted working on integrl equtions in 884, ut his serious study egn in 896. The nme integrl eqution ws given y du Bois- Reymond in 888. However, the nme Volterr integrl eqution ws first coined y Llesco in 98. Ael considered the prolem of determining the eqution of curve in verticl plne. In this prolem, the time tken y mss point to slide under the influence of grvity long this curve, from given positive height, to the horizontlisisequltoprescriedfunctionoftheheight. Aelderivedthe singulr Ael s integrl eqution, specific kind of Volterr integrl eqution, tht will e studied in forthcoming chpter. Volterr integrl equtions, of the first kind or the second kind, re chrcterized y vrile upper limit of integrtion. For the first kind Volterr integrl equtions, the unknown function u() occurs only under the integrl sign in the form: f() = k(, t) u(t) dt (II.) However, Volterr integrl equtions of the second kind, the unknown function u() occurs inside nd outside the integrl sign. The second kind is represented in the form: t u()= f() + λ k(, t)u(t) dt (II.2) Pge 26

8 The kernel K(, t) nd the function f() re given rel-vlued functions, nd λ is prmeter. A vriety of nlytic nd numericl methods, such s successive pproimtions method, Lplce trnsform method, spline colloction method, Runge-Kuttmethod nd others hve een used to hndle Volterr integrl equtions. In this tet we will pply the recently developed methods, nmely, the A domin decomposition method (ADM), the modified decomposition method (madm), nd the vritionl itertion method (VIM) to hndle Volterr integrl equtions. Some of the trditionl methods, nmely, successive pproimtions method, series solution method, nd the Lplce trnsform method will e employed s well. The emphsis in this tet will e on the use of these methods nd pproches rther thn proving theoreticl concepts of convergence nd eistence. The theorems of uniqueness, eistence, nd convergence re importnt nd cn e found in the literture. The concern will e on the determintion of the solution u() of the Volterr integrl eqution of first nd second kind. 2. Volterr Integrl Equtions: 2.2. Volterr Integrl Equtions of the Second Kind We will first study Volterr integrl equtions of the second kind given y: u() = f() + λ k(, t)u(t)dt (II.3) The unknown function u(), tht will e determined, occurs inside nd outside the integrl sign. The kernel K(, t) nd the function f() re given rel-vlued functions, nd λ is prmeter. In wht follows we will present the methods, new nd trditionl, tht will e used The Adomin Decomposition Method The Adomin decomposition method (ADM) ws introduced nd developed y George Adomin nd is well ddressed in mny references. A considerle mount of reserch work hs een invested recently in pplying this method to wide clss of liner nd nonliner ordinry differentil equtions, prtil differentil equtions nd integrl equtions s well. The Adomin decomposition method consists of decomposing the unknown function u() of ny eqution into sum of n infinite numer of components defined y the decomposition series u() = n= u n () (II.4) or equivlently u() = u () + u () + u 2 () +. (II.5) Where the components u n (), n > re to e determined in recursive mnner. The decomposition method concerns itself with finding the components. u,u,u 2,...individully. As will e seen through the tet, the determintion of these components cn e chieved in n esy wy through recurrence reltion tht usully involves simple integrls tht cn e esily evluted. To estlish the recurrence reltion, we sustitute (II.5) into the Volterr integrl eqution (II.4) to otin n= u n () = f() + λ k(, t)( u n (t) dt(ii.6) or equivlently n= ) u () + u ()+ u 2 ()+ f() + λ k(, t) u (t) + u (t) dt u () = f()(ii.7) Pge 27

9 The zeroth component u () is identified y ll terms tht re not included under the integrl sign. Consequently, the components u j (), j > of the unknown function u() re completely determined y setting the recurrence reltion: u n+ = λ k(, t)u n (t)dt, n tht is equivlent to u () = f(), u () = λ k(, t)u u 2 () = λ k(, t)u λ k(, t)u 2 (t)dt (II.9) (II.8) (t)dt (t)dt, u 3 = nd so on for other components. In view of (II.9), the components u (), u (), u 2 (), u 3 (),... re completely determined. As result, the solution u() of the Volterr integrl eqution (II.3) in series form is redily otined y using the series ssumption in (II.4). It is clerly seen tht the decomposition method converted the integrl eqution into n elegnt determintion of computle components. It ws formlly shown y mny reserchers tht if n ect solution eists for the prolem, then the otined series converges very rpidly to tht solution.the convergence concept of the decomposition series ws thoroughly investigted y mny reserchers to confirm the rpid convergence of the resulting series. However, for concrete prolems, where closed form solution is not otinle, truncted numer of terms is usully used for numericl purposes. The more components we use the higher ccurcy we otin. 3. Volterr Integro-Differentil Equtions Volterr studied the hereditry influences when he ws emining popultion growth model. The reserch work resulted in specific topic, where oth differentil nd integrl opertors ppered together in the sme eqution. This new type of equtions ws termed s Volterr integro-differentil equtions, given in the form u (n) () = f() + λ K(, t)u(t)dt, Whereu (n) () = dn u d n Becuse the resulted eqution in comines the differentil opertor nd the integrl opertor, then it is necessry to define initil conditions For the determintion of the prticulr solution u() of the Volterr integro-differentil eqution. Any Volterr integro-differentil eqution is chrcterized y the eistence of one or more of the derivtives u(),...outside the integrl sign. The Volterr integrodifferentil equtions my e oserved when we convert n initil vlue prolem to n integrl eqution y using Leinitz rule. The Volterr integrodifferentil eqution ppered fter its estlishment y Volterr. It then ppered in mny physicl pplictions such s glss forming process, Nno hydrodynmics, het trnsfer, diffusion process in generl, neutron diffusion nd iologicl species coeisting together with incresing nd decresing rtes of generting, nd wind ripple in the desert. More detils out the sources where these equtions rise cn e found in physics, iology nd engineering pplictions ooks. To determine solution for the integro-differentil eqution, the initil conditions should e given, nd this my e clerly seen s result of involving u() nd its derivtives. The initil conditions re needed to determine the ect solution. 4. Systems of Volterr Integrl Equtions 2.4. Introduction Systems of integrl equtions, liner or nonliner, pper in scientific pplictions in engineering, physics, chemistry nd popultions growth models Studies of systems of integrl equtions hve ttrcted much concern in pplied sciences. The generl ides nd the essentil fetures of these systems re of wide pplicility. The systems of Volterr integrl equtions pper in two kinds. For systems of Volterr integrl equtions of the first kind, the unknown functions pper only under the integrl sign in the form: f () = K (, t)u(t) + K (, t)v(t) + dt, Pge 28

10 f 2 () = dt,..., K 2 (, t)u(t) + K 2(, t)v(t) + (II.45) However, systems of Volterr integrl equtions of the second kind, the unknown functions pper inside nd outside the integrl sign of the form: u() = f () + K (, t)u(t) + K (, t)v(t) + dt, v() = f 2 () + K 2(, t)v(t) + dt,... K 2 (, t)u(t) +,(II.46) The kernels K i (, t) ndk i(, t), nd the functions f i (), i =,2,..., nre given rel-vlued functions. A vriety of nlyticl nd numericl methods re used to hndle systems of Volterr integrl equtions.the eisting techniques encountered some difficulties in terms of the size of computtionl work, especilly when the system involves severl integrl equtions. To void the diffculties tht usully rise from the trditionl methods, we will use some of the methods presented in this tet. The Adomin decomposition method, the Vritionl itertion method, nd the Lplce trnsform method will form resonle sis for studying systems of integrl equtions. The emphsis in this tet will e on the use of these methods rther thn proving theoreticl concepts of convergence nd eistence tht cn e found in other tets Systems of Volterr Integrl Equtions of the Second Kind We will first study systems of Volterr integrl equtions of the second kind given y u() = f () + K (, t)u(t) + K (, t)v(t) + dt, v() = f 2 () + K 2 (, t)u(t) + K 2(, t)v(t) + dt., (II.47) The unknown functions u(), v(),..., tht will e determined pper inside nd outside the integrl sign. The kernels K i (, t) nd K i(, t), nd the function f i ()re given rel-vlued functions. In wht follows we will present the methods, new nd trditionl, tht will e used to hndle these systems The Adomin Decomposition Method The Adomin decomposition method ws presented efore.the method decomposes ech solution s n infinite sum of components, where these components re determined recurrently. This method cn e used in its stndrd form, or comined with the noise terms phenomenon. Moreover, the modified decomposition method will e used wherever it is pproprite. It is interesting to point out tht the VIM method cn lso e used, ut we need to trnsform the system of integrl equtions to system of integro-differentil equtions tht will e presented lter in this chpter. 5. Nonliner Volterr Integrl Equtions 2.5..Introduction It is well known tht liner nd nonliner Volterr integrl equtions rise in mny scientific fields such s the popultion dynmics, spred of epidemics, nd semi-conductor devices. Volterr strted working on integrl equtions in 884, ut his serious study egn in 896. The nme integrl eqution ws given y du Bois-Reymond in 888. However, the nme Volterr integrl eqution ws first coined y Llesco in 98. The liner Volterr integrl equtions nd the liner Volterr integrodifferentil equtions were presented in previocesection respectively. It is our gol in this chpter to study the nonliner Volterr integrl equtions of the first nd the second kind. The nonliner Volterr equtions re chrcterized y t lest one vrile limit of integrtion. In the nonliner Pge 29

11 Volterr integrl equtions of the second kind, the unknown function u() ppers inside nd outside the integrl sign. The nonliner Volterr integrl eqution of the second kind is represented y the form u() = f() + K(, t)f(u(t))dt. (II.55) However, the nonliner Volterr integrl equtions of the first kind contins the nonliner function F(u()) inside the integrl sign.the nonliner Volterr integrl eqution of the first kind is epressed in the form f() = K(, t)f(u(t))dt. (II.56) For these two kinds of equtions, the kernel K(, t) nd the function f() re given rel-vlued functions, ndf(u()) is nonliner function of u() such s u 2 (), sin(u()), nd e u() Eistence of the Solution for Nonliner Volterr Integrl Equtions In this section we will present n eistence theorem for the solution of nonliner Volterr integrl equtions. However, in wht follows, we present rief summry of the conditions under which solution eists for this eqution. We first rewrite the nonliner Volterr integrl eqution of the second kind y u() = f() + G(, t, u(t))dt. (II.57) The specific conditions under which solution eists for the nonliner Volterr integrl eqution re: (i) (ii) The function f() is integrle nd ounded in The functionf() must stisfy the Lipschitz condition in the intervl (, ). This mens tht f() f(y) < k y. (II.58) (iii)the function G(, t, u(t)) is integrle nd ounded G(, t, u(t)) < K (iv) The function G(, t, u(t)) must stisfy the Lipschitz condition G(, t, z) G(, t, z < M z z,(ii.55) The emphsis in this chpter will e on solving the nonliner Volterr integrl equtions rther thn proving theoreticl concepts of convergence nd eistence. The theorems of uniqueness, eistence, nd convergence re importnt nd cn e found in the literture. The concern in this tet will e on the determintion of the solution u() of the nonliner Volterr integrl eqution of the first nd the second kind. 6. Nonliner Volterr Integrl Differentil Eqution 2.6..Introduction It is well known tht liner nd nonliner Volterr integrl equtions rise in mny scientific fields such s the popultion dynmics, spred of epidemics, nd semi-conductor devices. Volterr strted working on integrl equtions in 884, ut his serious study egn in 896. The nme integrl eqution ws given y du Bois-Reymond in 888. The liner Volterr integrodifferentil equtions were presented. It is our gol in this chpter to study the nonliner Volterr integrodifferentil equtions of the first nd the second kind. The nonliner Volterr integro-differentil equtions re chrcterized y t lest one vrile limit of integrtion. The nonliner Volterr integrodifferentil eqution of the second kind reds u (n) () = f() + K(, t)f(u(t))dt (II.6) And the stndrd form of the nonliner Volterr integro-differentil eqution of the first kind is given y Pge 2

12 K (, t)f(u(t))dt + K 2 (, t)u (n) (t)dt = O f(), K 2 (, ) (II.6) Where u (n) () is then th derivtive of u(). For these equtions, the kernels K(, t), K (, t) nd K 2 (, t), nd the function f() re given relvlued functions. The function F(u()) is nonliner function of u() such s u 2 (), sin(u()), nd e u() Nonliner Volterr Integro-Differentil Equtions of the Second Kind The liner Volterr integro-differentil eqution, where oth differentil nd integrl opertors pper together in the sme eqution, hs een studied. In this section, we will etend the work presented in this section to nonliner Volterr integro-differentil eqution. The nonliner Volterr integro-differentil eqution of the second kind reds u n () = f() + K(, t)f(u(t))dt(ii.62) where u (i) () = di u function of u(). d i, nd F(u()) is nonliner Becuse the eqution in (II.62) comines the differentil opertor nd the integrl opertor,then it is necessry to define initil conditions for the determintion of the prticulr solution u() of the nonliner Volterr integro-differentil eqution. The nonliner Volterr integro-differentil eqution ppered fter its estlishment y Volterr. It ppers in mny physicl pplictions such s glss-forming process, het trnsfer, diffusion process in generl, neutron diffusion nd iologicl species coeisting together with incresing nd decresing rtes of generting. More detils out the sources where these equtions rise cn e found in physics, iology nd ooks of engineering pplictions. We pplied mny methods to hndle the liner Volterr integrodifferentil equtions of the second kind. In this section we will use only some of these methods. However, the other methods presented i cn e used s well. In wht follows we will pply the comined Lplce trnsform-adomin decomposition method, the vritionl itertion method (VIM), nd the series solution method to hndle nonliner Volterr integrodifferentil equtions of the second kind (II.62). III. Fredholm theory Introduction It ws stted in Chpter tht Fredholm integrl equtions rise in mny scientific pplictions. It ws lso shown tht Fredholm integrl equtions cn e derived from oundry vlue prolems. Erik Ivr Fredholm ( ) is est rememered for his work on integrl equtions nd spectrl theory. Fredholm ws Swedish mthemticin who estlished the theory of integrl equtions nd his 93 pper in ActMthemtic plyed mjor role in the estlishment of opertor theory. As stted efore, in Fredholm integrl equtions, the integrl contining the unknown function u() is chrcterized y fied limits of integrtion in the form u() = f() + λ K(, t)u(t)dt, where nd re constnts. For the first kind Fredholm integrl equtions, the unknown function u() occurs only under the integrl sign in the form f() = K(, t)u(t)dt. However, Fredholm integrl equtions of the second kind, the unknown function u() occurs inside nd outside the integrl sign. The second kind is represented y the form u() = f() + λ K(, t)u(t)dt. The kernel K(, t) nd the function f() re given rel-vlued functions, nd λ is prmeter.when Pge 2

13 f() =,the equtions sidto e homogeneous. In this chpter, we will mostly use degenerte or seprle kernels. A degenerte or seprle kernel is function tht cn e epressed s the sum of the product of two functions ech depends only on one vrile. Such kernel cn e epressed in the form K(, t) = n i= f i ()g i (t)(iii.4) Emples of seprle kernels re t, ( t) 2, 4t, etc. In wht follows we stte, without proof, the Fredholm lterntive theorem 3.. Theorem (Fredholm Alterntive Theorem) If the homogeneous Fredholm integrl eqution u() = λ K(, t)u(t)dt Hs only the trivil solution u() =, then the corresponding nonhomogeneous Fredholm eqution u() = f() + λ K(, t)u(t)dt Alwys unique solution. This theorem is known y the Fredholm lterntive theorem 3..2.Theorem (Unique Solution) If the kernel K(, t) in Fredholm integrl eqution (III.) is continuous, rel vlued function, ounded in the squre nd t, ndiff() is continuous, rel vlued function, then necessry condition for the eistence of unique solution for Fredholm integrl eqution (III.) is given y λ M( ) <, (III.7) Where K(, t) M R (III.8) On the contrry, if the necessry condition (III.7) does not hold, then continuous solution my eist for Fredholm integrl eqution. To illustrte this, we consider the Fredholm integrl eqution u() = (3 + t)u(t)dt (III.9) It is cler tht λ =, K(, t) 4 nd ( ) =. This gives λ M( ) = 4 nd( ) =. (III.) However, the Fredholm eqution (III.9) hs n ect solution given y u() = 6. (III.) A vriety of nlytic nd numericl methods hve een used to hndle Fredholm (III. integrl 5) equtions. The direct computtion method, the successive pproimtions method, nd converting Fredholm eqution to n equivlent oundry vlue prolem re mong mny trditionl methods tht were commonly used. However, in this tet we will pply the recently developed methods, nmely, the Adomin decomposition method (ADM), the modified decomposition (III. method 6) (ADM), nd the vritionl itertion method (VIM) to hndle the Fredholm integrl equtions. Some of the trditionl Methods, nmely, successive pproimtions method, nd the direct computtion method will e employed s well. The emphsis in this tet will e on the use of these methods rther thn proving theoreticl concepts of convergence nd eistence. The theorems of uniqueness, eistence, nd convergence re importnt nd cn e found in the literture. The concern will e on the determintion of the solution u() of the Fredholm integrl equtions of the first kind nd the second kind. IV. Differentil Eqution Reducing -Into Volterr nd Fredholm Integrl Eqution nd Appliction. Converting IVP to Volterr Integrl Eqution Pge 22

14 In this section, we will study the technique tht will convert n initil vlue prolem (IVP) to n equivlent Volterr integrl eqution nd Volterr integrodifferentil eqution s well.for simplicity resons, we will pply this process to second order initil vlue prolem given y y"() + p()y () + q()y() = g() (IV.) Suject to the initil conditions : y() = α, y () = β, (IV.2) Where α nd β re constnts. The functions p() nd q() re nlytic functions, nd g() is continuous through the intervl of discussion. To chieve our gol we first set y"() = u() (IV.3) Where u() is continuous function. Integrting oth sides of (IV.3)from to yields y ()- y () = u(t)dt ((IV.4) u() + p()[β + u(t)dt] + q()[α + β + ( t)u(t)dt] = g() (IV.8) The lst eqution cn e written in the stndrd Volterr integrl eqution form: u() = f() K(, t)u(t)dt (IV.9) k(, t)= p() + q()( t) (IV.) f() = g() [βp() + αq() + βq() (IV.) It is interesting to point out tht y differentiting Volterr eqution (IV.9) With respect to, using Leinitz rule, we otin n equivlent Volterr integro-differentil eqution in the form: u () + k(, )u() = f () - k(,t) dt, u() = f() (IV.2) Or equivlently y ()= β + u(t)dt (IV.5) The technique presented ove to convert initil vlue prolems to equivlent Volterr integrl equtions cn e generlized y considering the generl initil vlue prolem: Integrting oth sides of (IV.5) from to yields y() y() = β + u(t) dtdt, (IV.6) Or equivlently y() = α + β + ( t)u(t) dt (IV.7) Otined upon using the formul tht reduce doule integrl to single integrl tht ws discussed in the previous section. Sustituting (IV.3),(IV.5), nd (IV.7) into the initil vlue prolem yields the Volterr integrl eqution: y (n) + ()y n + n ()y + n ()y = g() (IV.3) suject to the initil conditions : y() = C, y () = c, y () = c 2,. y n () = c n (IV.4) We ssume tht the functions re nlytic t the origin, nd the function g() is continuous through the intervl of discussion. Let u() e continuous function on the intervl of discussion, nd we consider the trnsformtion: y n () = u() (IV.5) Pge 23

15 Integrting oth sides with respect to gives y n () = c n + u(t)dt (IV.6) Integrting gin oth sides with respect to yields y n 2 ()= c n 2+ c n u(t)dtdt = c n 2+ c n + ( t) u(t)dt (IV.7) otined y reducing the doule integrl to single integrl. Proceeding s efore we find y n 3 () = c n 3 + c n c n 2 =c n 3 + c n 2 (.8) + u(t)dtdtdt + c 2 n ( t) 2 u(t)dt Continuing the integrtion process leds to n c k k! y() = k= k + u(t) dt (n )! Sustituting (IV.5) (IV.9) into (IV.3) gives u() = f() k(, t) u(t)dt (IV.2) m n (k )! k(, t)= k= ( t) k- (IV.2) nd f() = g() m j c j= j ( n k k= (j k)! ( t) (IV.9) where j k )(IV.22) Notice tht the Volterr integro-differentil eqution cn e otined y differentiting(iv.2) s mny times s we like, nd y otining the initil conditions of ech resulting eqution. n- The following pplictions will highlight the process to convert initil vlue prolem to n equivlent Volterr integrl eqution. V. CONCLUSION An integrl eqution is n eqution which is one of n indefinite integrl. They re importnt in severl physicl domins. Mwell's equtions re proly their most fmous representtives. They pper in the prolems of rditive energy trnsfer nd virtion prolems of rope, memrne or n is. The oscilltion prolems cn lso e solved using differentil equtions. Why we d wnt to convert differentil equtions into integrl equtions or vis vers? Finding nlyticl or numericl solutions in the former cse is often esier, lso qulittive nlysis of the symptotic nd singulrity ehviour in the phse spce.returning to sics of differentil eqution, we know tht the vlues of y() which stisfy ove differentil equtions re their solutions. Performing conversion from differentil eqution in y to integrl eqution in y is nothing ut solving the differentil eqution for y. After converting n initil vlue or oundry vlue prolem into n integrl eqution, we cn solve them y shorter methods of integrtion. This conversion my lso e treted s nother representtion formul for the solution of n ordinry differentil eqution. A differentil eqution cn e esily converted into n integrl eqution just y integrting it once or twice or s mny times, if needed.but convert differentil eqution into Volterr nd Fredholm integrl eqution it request some initil or oundry condition. For tht reson in our project which hs s title ((the solution of differentil eqution reduction into Volterr nd Fredholm integrl eqution)) The gol of this project ws finding the solution of some differentil eqution throughout the solution of integrl eqution. After seeing some sics notion of differentil eqution, Volterr nd Fredholm integrl, we Pge 24

16 discussed out converting : Converting initil vlue prolem into Volterr integrl eqution nd Converting oundry vlue prolem into Fredholm integrl eqution vis vers sed on some conditions. We cn sk two questions t the end of this work:. Cn n integrl eqution lwys e rewritten s differentil eqution? 2. Cn differentil eqution lwys e rewritten s integrl eqution? The question 2 is trivil to nswer yes ny differentil eqution cn e rewritten s integrl eqution ut the question which is rewritten ny integrl eqution s differentil, the nswer is, In generl, no. An integrl eqution cn e non-locl, wheres differentil eqution is locl (in the sense tht it cn e descried y function over the jet-undle). As n illustrtion. Let K()=δ_ ()+δ_ () e n integrl kernel, where δ_i re the Dirc delt's supported t i. Consider the integrl eqution, for some fied smooth f()= K(-y)ϕ(y)dy for the unknown ϕ. The eqution reduces to ϕ()+ϕ(+)=f(). Any continuous function g()on [,] stisfying g()+g()=f() genertes continuous solution of the eqution. We chllenge you to find differentil eqution whose solution set cn e thus generted. REFERENCES. A.J. Jerri, Introduction to Integrl Equtions with Applictions, Wiley, New York, (999). 2. A.M. Wzwz, A First Course in Integrl Equtions, World Scientific, Singpore, (997). 3. A.M. Wzwz, A First Course in Integrl Equtions, World Scientific Singpore, (997). 4. A.M. Wzwz, A relile modifiction of the Adomin decomposition method, Appl. Mth. Comput., 2 (999) A.M. Wzwz, Necessry conditions for the ppernce of noise terms in decomposition solution series, Appl. Mth. Comput., 8 (997) A.M. Wzwz, Prtil Differentil Equtions nd Solitry Wves Theory, HEP nd Springer, Beijing nd Berlin, (29). 7. A.M. Wzwz, Prtil Differentil Equtions nd Solitry Wves Theory, HEP nd Springer, Beijing nd Berlin, (29). 8. A.M. Wzwz, Prtil Differentil Equtions nd Solitry Wves Theory, HEP nd Springer, Beijing nd Berlin, (29). 9. A.M.Wzwz, A First Course in Integrl Equtions, World Scientific, Singpore, (997).. A.M.Wzwz, Necessry conditions for the ppernce of noise terms in decomposition solution series, Appl. Mth. Comput., 8 (997) A.N. Tikhonov, On the solution of incorrectly posed prolem nd the method of regulriztion, SoIVet Mth, 4 (963) B.L. Moiseiwitsch, Integrl Equtions, Longmn, London nd New York, (977). 3. C. Bker, The Numericl Tretment of Integrl Equtions, Oford University Press, London, (977). 4. D. Porter nd D.S. Stirling, Integrl Equtions: A Prcticl Tretment from Spectrl Theory to Applictions, Cmridge, (24). Pge 25

17 5. D.L. Phillips, A technique for the numericl solution of certin integrl equtions of the first kind, J. Assoc. Comput. Mch, 9 (962) G. Adomin nd R. Rch, Noise terms in decomposition series solution, Comput. Mth. Appl., 24 (992) G. Adomin, A reivew of the decomposition method nd some recent results for nonliner eqution, Mth. Comput. Modelling, 3(992) Pge 26