Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extention method Текст научной статьи по специальности «Математика»

udc 338.984 Articles

doi:10.31799/1684-8853-2018-6-14-23

Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extention method

N. N. Vassilieva-b, PhD, Tech., SeniorResearcher, orcid.org/0000-0002-0841-H68, [email protected] I. N. Parasidisc, PhD, Associate Professor, [email protected] E. Providasd, PhD, Associate Professor, [email protected]

aSaint-Petersburg Department ofV.A. Steklov Institute of Mathematics of the RAS, 27, Fontanka, 191023, Saint-Petersburg, Russian Federation

bSaint-Petersburg Electrotechnical UniversityETU "LETI", 5, Professora Popova St., 197376, Saint-Petersburg, Russian Federation

cDepartment ofElectrical Engineering, Technological Educational Institute ofThessaly, 41110, Larissa, Greece

dDepartment ofMechanical Engineering, Technological Educational Institute ofThessaly, 41110, Larissa, Greece

Introduction: Boundary value problems for differential and integro-differential equations with multipoint and non-local boundary conditions often arise in mechanics, physics, biology, biotechnology, chemical engineering, medical science, finances and other fields. Finding an exact solution of a boundary value problem with Fredholm integro-differential equations is a challenging problem. In most cases, solutions are obtained by numerical methods. Purpose: Search for necessary and sufficient solvability conditions for abstract operator equations and their exact solutions. Results: A direct method is proposed for the exact solution of a certain class of ordinary differential or Fredholm integro-differential equations with separable kernels and multipoint/integral boundary conditions. We study abstract equations of the form Bu = Au - gF(Au) = f and

with non-local boundary conditions $>(u) = N¥(u) and ®(u) = N¥(u), ®(Au) = DF(Au) + N¥(Au), respectively, where A is a differential operator, q and g are vectors, D and N are matrices, and F, O and Y are functional vectors. This method is simple to use and can be easily incorporated into any Computer Algebra System (CAS). The upcoming Part 2 of this paper will be devoted to decomposition method for this problem where the operator B, is quadratic factorable.

Keywords — differential and Fredholm integro-differential equations, multipoint and non-local integral boundary conditions, correct operators, exact solutions.

Citation: Vassiliev N. N., Parasidis I. N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extension method. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2018, no. 6, pp. 14-23. doi:10.31799/1684-8853-2018-6-14-23

Introduction

Boundary value problems (BVP) for differential and integro-differential equations (IDE) with multipoint and nonlocal boundary conditions arise in various fields of mechanics, physics, biology, biotechnology, chemical engineering, medical science, finance and others [1-14]. More precisely these are elasticity, heat and mass transfer, diffraction, underground water flow and population dynamics problems. Perhaps the first known problem which was reduced to the IDE aiylV (*) + y(t) = ~a21-1K(f, x)ylv(x)dx is Proctor's problem of Equilibrium of an elastic beam in XVII century. Fredholm integro-differential equations with nonlocal integral boundary conditions and ordinary differential operators, probably, first were considered by J. D. Tamarkin [15]. Problems with nonlocal boundary conditions for elliptic equations first were investigated by A. V. Bitsadze, A. A. Samarskii [16], while BVP for parabolic equa-

tions with nonlocal integral boundary conditions were studied by J. R. Cannon [5], L. I. Kamynin [7], N. I. Ionkin [6] and others. Later such investigations for Laplace, Poisson and heat equations were explored by V. A. Il'in and E. L. Moiseev [17] and others [18-20]. Nonlocal BVP involving integral conditions for hyperbolic equations were studied in [21]. Multipoint and nonlocal BVP with integral boundary conditions for ordinary differential equations were considered in [22, 23]. Fractional IDE with integral boundary conditions were given in [24]. The problem of the existence of solutions for nonlocal BVP was the subject of many papers [19, 20, 23, 25-28]. Exact solutions of BVP with Fredholm IDE were considered in [29] and [30]. In most cases numerical methods are employed. Here, the necessary and sufficient solvability conditions of the abstract operator equations:

Bu =Au - Qu, Qu = gF(Au), S(B) = {u e S(A) : O(u) = NT(u)|; (1)

B1u =A2u - Q1u, Q1u = qF(Au) + gF(A2u), ©(B^ = {u e S(A2) : O(u) = Nf(u)}, O(Au) = DF(Au) + NT(Au), (2)

and their exact solutions are obtained in closed form. This formalism is applied to solve Fredholm IDE with multipoint or nonlocal integral boundary conditions, when A is a differential operator and Q, Q1 are integral operators with separable kernels. The problems (1), (2) arise naturally from A. A. Dezin, R. O. Oinarov extensions of linear operators [31, 26], which are not restrictions of a maximal operator, unlike the classical M. G. Krein, J. Von. Neu-man extensions [32, 33] in Hilbert space and in Banach space [34]. This work is a generalization of the papers [26-28, 35], where integral boundary conditions have not been considered. Solving differential or Fredholm IDE with integral boundary conditions is a complicated problem, since the operators B and B1 in (1), (2) are obtained by perturbations of boundary conditions and the action of an operator A. Whereas in [26-28, 35] the operators B = A + Q,

) = ) and Bl = A2 + Ql, B1 ) = 2)

are obtained only by perturbation of the action of a correct operator A which is a restriction of a maximal operator A.

Terminology and notation

Let X, Y be complex Banach spaces and X* the adjoint space of X, i. e. the set of all complex-valued linear and bounded functionals on X. We denote by f(x) the value of f on x. We write S(A) and R(A) for the domain and the range of the operator A, respectively. An operator A2 is said to be an extension of an operator A1, or A1 is said to be a restriction of A2, in symbol A1 cA2, if S(A2) 3 S(A1) and A1 for all x £ S(A1). An operator A: X ^Y is called closed if for every sequence xn in S(A) converging to x0 with Axn ^ f0, it follows that x0 £ S(A) and Ax0 = f0. A closed operator A is called maximal if R(A) = Y and ker A ^ {0}. An operator A: X ^Y is called correct if R(Aj = Y and the inverse A_1 exists and is continuous on Y. An operator A is called a correct restriction of the maximal operator A if it is a correct operator and A <z A. If ^ £ X*, i = 1, ..., n, then we denote by ¥ = coZ(T1, ..., TJ and ¥(x) = col(^(x), ..., Tn(x)). Let g = (g1, ..., gn) be a vector of Xn. We will denote by ¥(g) the n x n matrix whose i, J-th entry ^(g.) is the value of functional ^ on element g.. NNote that T(gC) = ¥(g)C,

i J

where C is a n x k constant matrix. We will also denote by 0n the zero and by In the identity n x n matrices. By 0 we will denote the zero column vector.

Extension methods for ordinary differential and Fredholm IDE

on

Let A: X^X be an ordinary mth order differential operator

Au(x) = a0u(m)(x) + a1u(m-1)(x) + ... + amu(x),

at £ M (3)

and X be a Banach space. Usually X = C[a, b] or X = Lp(a, b), p > 1. In the sequel we denote by

X™ ={d (A), |-|xm j theBanach space of all m times

differentiable functions with norm ||u(x)||

m t-\

= E )

1=0

Xn.

.. 4

and by X™ 1 the Banach space of all

x

m - 1 times differentiable functions with norm

m-1

"1 "

\\u

(x)||XT1 = £ u(l\x) i=0

(4)

Note that for X = C [a, b] the spaces , Xf_1 are defined by Cm[a, b], Cm-1[a, b], respectively. it is a well-known fact that the operator defined by

A u(x ) = a qU^"1 ^ (x ) + +a + + a mu (x) = f,

ateM, x £ [a, b], (5)

®(Â ) =

= {«(x) £ Cm [a, b] : u(a) = u'(a) =... = um~1 (a) = o|

is a correct restriction of A and the unique solution of (5)is

- 1 1 -x m_1

«(*) = A f (x ) = Jo (x -1 ) f (t

f(x) £ C[a, b].

(6)

Lemma 1. Let Ai, B^ Ci, D are n x n matrices,

where i = 1, 2, 3, and G =

f A1 A2 A3 ^ Bl B2 B3

v C1 C2 C3

. Then the

next properties of determinants hold true:

= det

f Ai A2 A3

det Bi B2 Bg =

I Ci Cg j

f A1 ± DBX A2+DB2 A3 + DBgN

Bi B2 B3

I Ci C2 C3 J

(7)

f A! A2 A3 ^ f A! A2 + A3D A3 ^

det Bi B2 B3 = det Bi B2 + B3D B3 . (8)

I Ci C2 C3, I Ci C2 + C3D C3 ,

f 1« -D 0„ ^

Proof: Let H = o„ 1» 0„

10„ o„ J

Then H"1 =

( h D 0„ ^

0„ 1» 0„ , detH = detH1 = 1, |HG| = |H||G|

10„ o„

= |G| and |H_1G| = |H-1||G| = |G|. So (7) holds.

Let now H =

I n 0„ 0„ ]

o„ In 0„

0„ D J

Then H"1 =

' I,

o„

v 0„

o,A 0,

Kj

H| = IH-1! = 1, |GH| = |G||H| = |G|

and IGH-1! = |G||H-1| = |G|. So (8) holds and Lemma 1 is proved.

Remark 1. Consider a n2 ( Ai

matrix G =

11

Ai„ ^

, where Aj, i, j = 1, ..., n are n :

n

'■nn y

matrices. Let Г be the matrix obtained from G by multiplying from the left a row by the n x n matrix D and then adding it to another row, or by multiplying from the right a column of G by the matrix D and then adding it to another column of G. Then detG = detR

Theorem 1. Let X be a complex Banach space,

on

A: X ^ X an operator from (3) with finite dimensional kernel z = (zv ..., zm) which is a basis of ker A, and let A be a correct restriction of A defined by

A c A, ) = {u e ®(A): ) = 0},

(9)

the components of the functional vectors O =col (®1, ..., ®m), ¥ =col (T1, ..., x¥n) and F = = col (F1, ..., Fn) belong to Xm-1 and respectively.

Suppose also that ®1, ..., Om biorthogonal to z1, ..., zm and that the components of vector g = (g^, ..., gn) e Xn are linearly independent and N is a m x n matrix. Then:

(i) The operator B defined by

Bu =Au - gF(Au) = f, feX;

S(B) = {u e S(A) : O(u) = NT(u)}

(10)

is injective if and only if

detV = det[In - ¥(z)N] * 0 and detW = det[In - F(g)] * 0.

(11)

(ii) If B is injective, then B is correct and for all feX the unique solution of (10) is given by

u = B~lf = A ~lf +

A _1g + zNV _1g) W_1F(/) + ZNV_1T(A ~lf).

(12)

Proof: (i). Let detW * 0, detV * 0 and u £ kerB. Then Bu =Au - gF(Au) = 0, O(u) = N¥(u) and [In - F(g)]F(Au) = 0, O(u - zNT(u)) = 0. The last

equation, since (9), implies u -

From [In - F(g)]F(Au) = 0, since deW * 0, follows F(Au) = 0. Then Bu =Au = 0 which yelds A (u - zNY(«)) = 0 and so u = zN^(u). Then

T(u) = ¥(z)N¥(u) or [In - T(z)N]T(u) = 0. The last, since detV ^ 0 implies T(u) = 0 and so from u = zNY(u) we get u = 0, i. e. kerB = {0} and B is an injective operator.

Conversely. Let detV = 0. Then there exists a vector c = col (c1, ..., cn) = 0 such that Vc = 0.

Consider the element u0 = zNc ^ 0, otherwise Nc = 0 and from [In - ¥(z)N]c = 0 follows c = 0, which contradicts the hypothesis c ^ 0. Note that u0 £ S(B), since O(u0) = Nc, Y(u0) = T(z)Nc, O(u0) - N¥(u0) = Nc - N^(z)Nc = N[In - ¥(z)N]c = = NVc = 0. It is evident that u0 £ kerB. So u0 £ kerB. Hence ker B ^ {0} and B is not injective. Let now detV ^ 0, but detW = 0. Then there exists a vector c = col(c1, ..., cn) ^ 0 such that Wc = 0. Note that gc ^ 0 because of g1, ..., gn is a linearly independent set and

that the element u0 =

c * 0,

c * 0,

A _1g + zNV"1T^"1g) otherwise g = 0. For u0 we obtain

u0 = A _1g + zNV'^A^g) O (u0 ) - N¥(u0 ) = NV_1 ^(A_1g- NT (A_1g-- N¥(z)NV_1g= = N[l„ -Y(z)N]V_1g- Nt(A_1g= = Nt(A _1g - NT^A _1g )c = 0, Bu0 = Au0 - gF (Au0 j = = gc - gF(g)c = g[l„ - F(g)]c = gWc = go = 0.

So u0 £ kerB. Consequently kerB ^ {0} and B is not injective. Hence B is injective if and only if detV * 0, detW * 0. The statement (i) holds.

(ii) Let detW ^ 0 and detV ^ 0. By statement (i), the operator B is injective. Since z e [ker A]m, O(z) = Im, the problem (10) is written as

Bu =A(u - zNT(u)) - gF(Au) = f,

f e X;

S(B) = {u e S(A) : 0(u - zNY(u)) = 0}. (13)

Then, applying Equation (9) and relation Bu = A (u - zNT(u))- gF (Au ) = f we obtain u - zNf(u) £ 1 Bu = (u - zN¥(w)) - gF (Au) = f and for every u e \D(B), f eX using (10), (13) we obtain

[l„ -F(g)]F(Au) = F(/), F (Au) = W "1F (/), u - zNW(u) = A _1gF(Au) + A _1/, W(u )-f(z)Nf(w) = _1g )f (Au ) + _1/),

[l„ -T(z)N] ) = ^(i~1g) W^F(/) + ~1f), ¥(«) = V"1 Y^A _1g jw_1F (/) + t(A_1/) u = B~1f = A_1/ + A _1gW_1F (/) +

+zNV'

-1

From the last equation for every f e X follows the unique solution (12) of (10). Because f in (12) is arbitrary, we obtain R(B) = X. Since the operator A_1 and the functionals F1, ..., Fn, T1, ..., x¥n are bounded, from (12) follows the boundedness of B-1. Hence, the operator B is correct if and only if (11) holds and the unique solution of (10) is given by (12). The theorem is proved.

From the previous theorem for g = 0 follows the next corollary which is useful for solving some classes of differential equations with nonlocal boundary conditions.

Corollary 1. Let a complex Banach space X, the operators A, A, the vector z and functional vectors O, ¥ and the matrix N be defined as in Theorem 1. Then:

(i) The operator B defined by

Bu =Au = f, feX;

S(B) = {u £ S(A) : O(u) = NT(u)}

(14)

is correct if and only if detV = det[In - T(z)N] ^ 0 and for all feX the unique solution of (14) is given by

u = B~1f = Â ~1f + zNV"1^! ~1f ). (15)

Theorem 2. Let a Banach space X, the vectors z, O, F, the operators A, A be defined as in Theorem 1 and the operator B1: X^X by

B1u =A2u - qF(Au) - gF(A2uj = f; (16) S(B1) = {u £ S(A2) : O(u) = N¥(u),

O(Au) = DF(Au) + N¥(Au)}. (17)

Suppose also that the vectors q and g are linearly independent, q= (q1, ..., qn), g=(g1, ..., g„) £ Xn, and D, N are m x n matrices. Then:

(i) The operator B1 corresponding to the problem (16), (17) is injective if and only if

= det

det L =

0re -F(z)N Ki -F^g)

Y -^Â^zjN -Kg

-K2 -^fi^g) -F(q) W

V

*0, (18)

where

Kx = I„ - F(z)D - F(i^q), K2 =Y(z)D + ),

K3 =T(JL_1z)d + Y(l"2q),

W = I„ - F(g), V = I„ - T(z)N. (19)

(ii) If the operator B1 is injective, then it is correct and the unique solution of (16), (17) is given by

u = B[1f = i ~2f + +(zN, A_1zN, i_1zD + A_2q, i"2g)x

x IT1 col(F(i"V), ~2f ), ^(i"V),F(/)). (20)

Proof: (i) Let detL ^ 0. Since O(z) = Im, the relations (17) can be represented as

0(u - zN¥(u)) = 0,

0(Au - zDF(Au) - zNT(Au)) = 0,

which taking into account (9) imply

u - zN*¥(u)e s(i); (21)

Au - zDF (Au)-zNT(Au)e ©(A ). (22)

Then, since z £ [kerA]m, A c A and A is correct, from (16) we obtain

 [Au - z[DF(Au) + N*¥(Au)])-

- qF(Au)-gF(a2u) = /,

Au - z[DF(Au) + NY(Au)] - A_1qF(Au) -

- A - XgF (a2u ) = A"V, A (u - zN¥(« )) - z [DF (Au ) + N W(Au )] -

- A _1qF (Au) - A _1gF (a2u ) = A_1/,

u - zNT(u) - A _1z [ DF (Au ) + NY( Au)] -

- A "2qF (Au ) - A "2 gF {a 2 u ) = A "2/. Then taking into account (16) we get

A2u = qF(Au) + gF(A2u) + f, Au = z [DF (Au ) + N¥( Au )] + +A _1qF (Au) + A _1gF (a2u ) + A "V,

u = zNT(u ) + A _1zNT(Au ) + + ^A_1zD + A"2q)F(Au) + A"2gF(a2u) + A~2f. (23)

Further acting by functionals F and T we get the next system

F (Au) = F (z)[DF (Au ) + NY (Aw)] +

+ F (i _1q )F (Aw) + F (a _1g )f (a2u) + F (a ~lf ),

Y(w) = Y(z)NY(w) +^(A _1z)N¥(4u ) + + y(a_1z)d + Y(i~2q)] F (Aw) + + "2g )F ^A2w ) + ~2f ), Y(Aw) = Y(z)[DF (Aw ) + NY(Aw)] + + y(a _1q )f (Aw ) + y(a _1g )f (a2u) + y(a "V F(A2u) = F(q)F(Au) + F(g)F(A2u) + F(f), or -F(z)NY(Aw)+ I„ -F(z)D-F(A_1q) F(Aw)-

- f(A"1g)f(A2w) = f(A"v),

VY(w ) - Y^A _1z )NY(Aw) -- y(a_1z)d + y^a_2qj f(Aw)--^(a "2g )F (a2w ) = y(a ~2f ), VY(Aw)- Y(z)D + F(Aw)-

- ^(A_1g)F^A2u) = Y^A~lf ),

- F(q)F(Au) + [In - F(g)] F(A2u) = F(f).

Using the notations (19) from the above equations we get the system

0„

V

0„ 0„

-F (z)N Kx

-y(a_1z)n -Kg

V -K2

0„ -F(q)

f T(w) л Y(Aw )

-F (a ^g)

-^(a "2 g)

Л

(A

w

(

F (Aw ) F ^A2uj

F (i-1/)

^(a ~2f )

^(a ~lf ) F (/) ,

Л

(24)

Let u e kerBr Then in the systems (23), (24) f = 0 and from (24) we get Lcol(¥(u), Y(Au), F(Au), F(A2u)) = = 0, which since detL ф 0, yields Y(u) = Y(Au) = = F(Au) = F(A2u) = 0. Substitution of these values into (16), (17) imply B1u =A2u = 0, O(u) = O(Au) = 0.

Taking into account (9) we acquire и e S^A2 j and

л о л

BjU = А -и = 0. By hypothesis A is correct and so u = 0. Thus ker B1 = {0} and B1 is injective.

Conversely. Let detL = 0. Then there exists a vector c = col(c1, c2, c3, c4), where ci = col(ciV ..., cin), i = 1, ..., 4 such that c Ф 0 and Lc = 0, which since (24) yields

-F(z)Nc2 + KlCg - F(l^g)c4 = 0; (25)

vc! -^(a"1z)nc2 - K3C3 -^a^g)c4 = 0; (26)

~1g )c4 = 0; (27)

(28)

Vc2 - K2Cg -

-F(q)c3 + Wc 4 = 0. Consider the element

Uq = zNcx + A 1 z(Nc2 + Dcg)+ A 2 (qc3 + gc4). (29)

Note that u0 ^ 0, otherwise because of the linear independence of the vectors q, g, z and D (a )n ker A = {0} [18], we get Nc1 = Nc2 = c3 =

= c4 = 0. Then from (27) follows that c2 = 0 and from (26) we obtain c1 = 0. Thus ci = 0, i = 1, ..., 4 and c = 0. But the last contradicts the hypothesis c ^ 0. So u0 ^ 0. From (29), since O(z) = Im,

K3 = Y^A_1z|d +Y(i"2q) and (26) we get

Au0 = z (Nc2 + Dc3) + A_1 (qc3 + gc4), A2u0 = qc3 + gc4, ®(u0)-NY(u0) = Ncx -NY(z)Ncx -Ny(a_1z)x X (Nc2 + Dc3 )- NW^A"2q)c3 - NT^.4"2g)c4 =

18 У информаиионно-управляюшие системы

"7 № 6, 2018

= N

VCl -T(^Î-1Z)Nc2 - Kgс3 g)c4

= N0 = 0.

Then 0(u0) = NT(«0) and so u0 satisfies the first boundary condition (17). We will show, using (27) and (25), that u0 satisfies the second boundary condition (17) 0

®(Au0 ) - DF(Au0 ) - Au0 ) = Nc2 + Dc3 -- DF(z)(Nc2 + Dc3 )- DF(.4"1q)c3 - DF(.4~1g)c4 -- N¥(z)(Nc2 + Dc3 )-Ny(Â_1q)c3 -Ny(Â_1g)c4 = = N[Vc2 -K2c3 -T(a~1g)c4 -F(z)Nc2 + K1c3 - F(â~1g)c4

+D

= NO + DO = 0,

where K1, K2 from (19). So u0 e S(B1). Now, using (25) and (28) we will show that u0 e kerB1

Biu0 = A2u0 -qF(Au0)-gF(a2u0 ) = qc3 + gc4 -- q[F(z)(Nc2 + Dc3 ) + F(A^qjcg + F(i^g)c4 -- gF (q)c3 - gF (g)c4 = = q[-F(z)Nc2 + KlC3 - F(A~1g)c4 ] + +g [-F(q )c3 + Wc4 ] = qO + gO = 0.

So there exists a nonzero element u0 e S(Bj) and u0 e kerB1. This means that B1 is not injective. Hence the operator B1 is injective if and only if detL * 0.

(ii) Since detL * 0, the system (24) for all feX has an unique solution

coZ^(u), Au), F(Au), F=

= L~1col(F(a"V), ^(a~2f), ^(a"V), F(/)) (30)

and the operator B1, by statement (i), is injective. Substituting (30) into (23) we obtain the unique solution (20) of the problem (16), (17). In the above solution an element f is arbitrary. Consequently, R(B1) = X. Since the operators A ~2, A_1 and the functional vectors F and T are bounded, from (20) follows the boundedness of Bf1, i. e. the operator B1 is correct. The theorem is proved.

The next corollary follows from the above theorem for q = g = 0 and is useful for solving some classes of differential equations with nonlocal boundary conditions.

Corollary 2. Let the operators A, A, the vectors z, O, T, F, V and matrices D, N be defined as in Theorem 2 and the operator B1 : X ^ X be defined by

Blu =A2u = f, SBi) = {u e S(A2) : Ф(и) = N¥(u), Ф(Аи) = DF(Au) + NY(Au)}.

(31)

Then:

(i) The operator B1 corresponding to the problem (31) is injective if and only if

det Li = det

0re -F(z)N In -F(z)D^ V -*p|Â_1zjN -Y(Â_1Z)d -¥(z)D

0 „ V

ф 0. (32)

(ii) If the operator B1 is injective, then it is correct and the unique solution of (31) is given by

и = Bf1/ = À~2f + (zN, A_1zN, A"

x LfW(F(i"1/), ^(A ~2f), ^A"1/)). (33)

Proof: (i) For g = q = 0 from (18) and (19) immediately follows

det L = det

0re -F(z)N In -F(z)D 0re^

V -y(Â_1z)N -Y(A_1Z)D 0re

0re V -T(z)D 0re

\0„ v n

o„

o„

I

. (34)

n у

It is evident that detL = detL1. From (20) for g = q = 0 follows the solution of (31)

и = Bf1/ = Â ~2f +(:

zN, Â_1zN, Â_1zD, 0

x L~1col(f(a"1/), t(a"2/ J,^!"1/ ),F(/)). (35) It is easy to verify that

^zN, A_1zN, A_1zD, 0)x x L~1col(f(a-1/), ^(a~2f ), -1/), F(/)) = = (zN, A_1zN, A_1zD)x x LfW(F(A"1/), T(A"2/), -1/)).

Hence, from (35) follows (33). Examples

In the next example we use the extension method from Theorem 1.

Example. The multipoint problem for loaded in-tegro-differential equation on C [0, 1]

u" - 3t

jV u"(x )d x + - (i2 +1 )[«'(! )- «'(0 )] =

= 8 r + 2t +12,

(36)

¿(0) = I u(l/2) + ^(l^ „'(0) = I« (1)

is correct and the unique solution of (36) is given by u(t) = 4t3 + 2t2 + 2t + 1. (37)

Proof: If we compare (36) with (10), it is natural to take Au = u"(t), 3(A) = {ueC2[0, 1]}, xl = C2IXl], x\ = C1 [0, l], m=n = 2, z = (z1, z2) = (1, t), A u = Au,

) = {u e ®(A) : u(0) = u'(0) = 0},

Bu = u" - 3tj\2u"(x)dx +1(i2 + l)[w'(l) - w'(0)] =

1

2'

= u"- 3t

J*xV'(x )d x + - (i2 + l)jV'(x )d x,

= \u (x )eS(A ):

'u(o)Wi/6 V18"if«(V2)

'(0)A 0 2/9 J[ u(1)

\YI

>. (38)

Since (5), the operator A, is correct and its solution is A ~lf (i ) = Jp(i - xjf (x)d x. Further comparing (36), (38) with (10), we take g1 = 3t, g2 =--(t2 +1), f= 8t2 + 2t + 12,

N =

F2 (Au) = J*«"(x)d x.

Then

F1 (f ) = J*x2f (x)dx, F2 (f ) = jV(x)dx,

0 ' V ' ■ v 7 JO

^ (u)W u(0)^

®2 (W)J = [ «'(0)

^ (u)Wu(l/2)^

0(u ) = x¥(u) =

(«) J [ „(1)

The set z = (1, t) is biorthogonal to (®1, ®2). From |T1(u)| = |u(1/2)| < ||u|| c + ||u'|| c = ||u|| ci follows that

Yj e C1* = _1* = . By analogy T2, ^ e C1*, i=1, 2. Further from \Fi (f )|= J1 x2f (x)dx

* /

it follows that e C[0,1J = X . By analogy it is

proved that F2 £ X . We can apply Theorem 1. Now we calculate

À ^ (t) = Jq(î - x(x)dx = Jq(î - x)3xd

t°

^ (t ) = --Jo (t -x )(x2 + l)d x = Compute

A"1g = (i^g!, À~1g2) =

f 2 3 r

'(t2 + 6) 24 .

24

Further we find T1(z1) = z1(1/2) = 1, T1(z2) =

= 02(1/2) = 1/2, ^2(0!) = ^(1) = 1, ^2(^2) = Z2(1) = 1.

, N fl

Then Y(z)= '

1 1

. Further compute Y^Â =

= 1/16, T2 (i ~1g1 ) = 1/2, Yj (i ~lg2 ) = -25/384, T2(A ~lg2 ) = -7/24, then

TU-1«) ~T41-

v [ 1/2 "V24 J

Now we find

(«1 ) = J013x3 d x = ^4, (*2 ) = ^ Jo^2 +l)d x =-^15, ^2 (äi ) = J013x d x = ^2, F2 (Ä2) = Jo(^2 +l)dX =-2/3.

Then

Since

F(*) =

f Z/A -A/lb s\ 3/2 -2/3

W = l2 - F(g) =

f y4 A/lbs\ -3/2 5/3

V = l2 -Y(z)N =

10 ^ f 1 1/2 ^^V6 V18

0 1J {1 1 ){ 0 2/9 f 5/6 -1/6 ^ -1/6 1^18

and detW * 0, detV ^ 0, the problem (36), by Theorem 1 (ii), is correct. For f = 8t2 + 2t + 12 we calculate

t t2 (212 +1 +18)

 "V (t ) = Jo (t - x )(2x + 4 )d X =—'---'-

(f ) = J* x2 ^8x2 + 2x + 12^dx = 61/10, F2 (f ) = + 2x + 12^dx = 47/3.

Then F(f) = col(61/10, 47/3). We also compute

Then

! ( A" 7) = A" ^i=l/2 = 19/12, Y 2 ( "7) = "7i=1 = 7.

y(Â "7 ) = col (19/12,7).

Substitution of these values into (12) yields the solution to the problem (36)

u (i) = A"7 + [A _1g + zNV_1Y(A _1g)]W_1F(/) +

i2 (

2r +1 +18

+ zNV" 7) =

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Conclusion

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УДК 338.984

doi:10.31799/1684-8853-2018-6-14-23

Метод нахождения точных решений для интегро-дифференциальных уравнений Фредгольма с многоточечными и интегральными краевыми условиями. Часть 1. Метод расширения

Н. Н. Васильева>б, канд. физ.-мат. наук, старший научный сотрудник, orcid.org/0000-0002-0841-1168, [email protected]

И. Н. Парасидисв, PhD, доцент, [email protected] Е. Провидасг, PhD, доцент, [email protected]

аСанкт-Петербургское отделение Математического института им. В. А. Стеклова РАН, наб. р. Фонтанки, 27, Санкт-Петербург, 191023, РФ

бСанкт-Петербургский государственный электротехнический университет «ЛЭТИ», Санкт-Петербург, ул. Профессора Попова, 5, Санкт-Петербург, 197376, РФ

вКафедра электротехники, Технологический институт Фессалии, 41110, Лариса, Греция гКафедра машиностроения, Технологический институт Фессалии, 41110, Лариса, Греция

Введение: краевые задачи для дифференциальных и интегро-дифференциальных уравнений с многоточечными и нелокальными граничными условиями возникают в различных областях механики, физики, биологии, биотехнологии, химической инже-

нерии, медицинской науки, финансов и других. Нахождение точных решений краевых задач с фредгольмовыми интегро-диффе-ренциальными уравнениями является трудной проблемой. В большинстве случаев решения получаются численными методами. Цель: поиск необходимых и достаточных условий разрешимости абстрактных операторных уравнений и метод построения их точных решений. Результаты: предложен прямой метод для точного решения некоторого класса обыкновенных дифференциальных или фредгольмовых интегро-дифференциальных уравнений с сепарабельными ядрами и многоточечными и интегральными граничными условиями. Исследованы абстрактные уравнения вида Bu =Au - gF(Au) = f и B1 u = A2u - qF(Au) - gF(A2u) = f с нелокальными граничными условиями ®(u) = NY(u) и ®(u) = NY(u), Ф(А^ = DF(Au) + NY(Au) соответственно, где q, g являются векторами, D, N — матрицами, а F, Ф, W — функциональными векторами. Предложенный метод прост в использовании и может быть легко интегрирован в любую систему компьютерной алгебры. Исследована корректность уравнений вида Bu = f и B1u = f и их точные решения. Вторая часть этой статьи будет посвящена случаю, когда оператор B1 имеет квадратичную факторизацию.

Ключевые слова — дифференциальные и фредгольмовы интегро-дифференциальные уравнения, многоточечные и нелокальные интегральные граничные условия, разложение операторов, корректность операторов, точные решения.

Цитирование: Vassiliev N. N., Parasidis I. N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extension method. Информационно-управляющие системы, 2018, № 6, с. 1423. doi:10.31799/1684-8853-2018-6-14-23

Citation: Vassiliev N. N., Parasidis I. N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 1. Extension method. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2018, no. 6, pp. 14-23. doi:10.31799/1684-8853-2018-6-14-23

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