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ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА У
udc 338.984 Articles
doi:10.31799/1684-8853-2019-2-2-9
Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators
N. N. Vassilieva b, PhD, Tech., Senior Researcher, orcid.org/0000-0002-0841-1168, vasiliev@pdmi.ras.ru I. N. Parasidisc, PhD, Associate Professor, paras@teilar.gr E. Providasd, PhD, Associate Professor, providas@teilar.gr
aSaint-Petersburg Department of V. A. Steklov Institute of Mathematics of the RAS, 27, Fontanka, 191023, Saint-Petersburg, Russian Federation
bSaint-Petersburg Electrotechnical University «LETI», 5, Prof. Popov St., 197376, Saint-Petersburg, Russian Federation
cDepartment of Electrical Engineering, Technological Educational Institute of Thessaly, 41110, Larissa, Greece dDepartment of Mechanical Engineering, Technological Educational Institute of Thessaly, 41110, Larissa, Greece
Introduction: In Part 7 of this article, a direct method was presented for examining the solvability and uniqueness problem, and for obtaining a closed-form solution of boundary value problems which incorporate an mth order linear ordinary Fredholm integro-differential operator, or a differential operator, along with multipoint and integral boundary conditions. Here, we focus on a special class of boundary value problems including the composite square of an integro-differential operator and the corresponding non-local boundary conditions. Purpose: To investigate the construction of the unique solution of 2mth order boundary value problems in the special case of an operator which can be presented as composite squares of lower mth order ones, and to develop an algorithm for constructing an exact solution for this special case. Results: By decomposition and applying the extension method explicated in Part 1, we provide a formula for obtaining an exact solution of boundary value problems for squared integro-differential operators, or differential operators, with multipoint and integral boundary conditions. This method is simple to use and can be easily incorporated to any Computer Algebra System.
Keywords — differential and Fredholm integro-differential equations, multipoint and non-local integral boundary conditions, decomposition of operators, correct operators, exact solutions.
For citation: Vassiliev N. N., Parasidis I. N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2019, no. 2, pp. 2-9. doi:10.31799/1684-8853-2019-2-2-9
Introduction
In the article [1], we presented the development, the applications and the necessity for studying boundary values problems encompassing mth order linear ordinary Fredholm integro-differential operators and general nonlocal boundary conditions such as multipoint and integral boundary conditions. We proposed a direct constructive method for examining the existence and uniqueness of the solution and obtaining it in closed-form. The method was based on the extension theory of linear operators in Banach spaces, in particular on the technique developed in [2] and [3] for solving exactly linear and nonlinear, respectively, integro-dif-ferential equations subject to initial and classical boundary conditions.
In this paper, which is a sequel to [1], we study separately a specific type of boundary value problems involving the composite square of an mth order linear ordinary Fredholm integro-differential,
or differential, operator, and analogous multipoint and integral boundary conditions. We establish the requirements under which there exists a unique solution and show how to construct it in closed-form by decomposing and utilizing the extension method described in [1].
The decomposition, or factorization, method for problems embracing integro-differential operators and unperturbed conventional boundary conditions is studied in [4]. Therefore, the current work can be seen also as an advancement of [4] where perturbed boundary conditions are considered. Factorization techniques find applications in several areas in sciences and engineering, see, for example, in [5, 6].
The organization of the paper is as follows. We first describe the decomposition-extension method and then we apply the method to solve second and fourth-order differential and integro-differential problems which can be formulated as composite squares of first and second-order problems, respectively. Lastly, some conclusions are quoted.
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Decomposition-extension method
Let X be a complex Banach space, usually
on
X = C[a, b] (or X = Lp(a, b), p > 1), and A: X^X an mth order linear ordinary differential operator, namely:
Au = a0u(m) + a1u(m-1) + ... + amu,
(1)
where at e M and u = u(x) e X^, where Xm = Cm [a, b] (or Xm = (a, b)). Let the space kerA be finite dimensional and z = (zv ..., zm) be a basis of it. Let A be a correct restriction of A, specifically Au = Au for all u in
d(A) = {u e D(A) : ®(u) = 0},
(2)
where ® = col(®1, ..., ®m) is a vector of m bounded linear functionals on Xl, which are biorthogonal to Zi, ..., zm and describe some boundary conditions.
Consider the integro-differential operator B: X ^ X:
Bu = Au - gF(Au), D(B) = {u e DD(A): ®(u) = N¥(u)},
(3)
and the more complex integro-differential operator
B1: X ^ X:
B1u = A2u - qF(Au) - gF(A2u), D(B1) = {u e D(A2): O(u) = NT(u), ®(Au) = DF(Au) + N^(Au)},
(4)
where A2 is meant to be the composite product A2 = A(A), T = col(T1, ..., x¥n) is a vector of n bounded linear functionals on Xm, F = col(F1, ..., Fn) is a vector of n bounded linear functionals on
X g = te^ gnX q = (<?l, In) 6 Xn, ^ In are linearly independent vectors, and D, N are m x n constant matrices. The equations ®(u) = NT(w) and ®(Au) = DF(Au) + NT(Au) symbolize general boundary conditions including multipoint and integral boundary conditions.
The boundary value problems Bu = f and B1u = f, for any f 6 X, were studied and solved exactly by utilizing the extension method in [1].
We contemplate here the special case of the boundary value problem B1 = f, Vf 6 X, when B1 = B2; B2 is understood to be the composite product: B2 = B(B). For this case we prove the following theorem which provides solvability conditions and describes the decomposition-extension procedure for obtaining the solution in closed form.
Theorem. (i) The operator B1 is decomposed in B1 = B2 in the case if
g 6 D(A)n, q = Ag - gF(Ag), D = ®(g) - NT(g). (5)
The operator B2 is defined by B2u = A2u - [Ag - gF(Ag)]F(Au) - gF(A2u), D(B2) = {u 6 D(A2): ®(u) = NT(u), ®(Au) = [®(g) - NT(g)] F(Au) + NT(Au)}. (6)
(ii) If the vectors q, g and matrices D, N satisfy (5), then the operator B1 is injective if and only if
detV = det[In - T(z)N] * 0;
detW = det[In - F(g)] * 0. (7)
(iii) If the vectors q, g and matrices D, N satisfy (5) and det V * 0, det W * 0, then the operator B1 is correct and the unique solution of the problem B1u = f is
u = B"lf = A~2f + Yf(Â-1/)
+ zNV"
^(("f ) + A_1Y + YF(Y) + zNV"llp(i"1Y) F(f) + A_lz + YF(z) + zNV_lvp((_lz) NV"1ip("V|, (8)
or
where
= B"lf = A ~1f + YF (f ) + zNV"1^!"1/), f = A ~lf + YF (f ) + zNV_lT(i_lf ),
(9)
Y =
A_lg + zNV
W"
(10)
Proof: (i) First we prove the second formula in (6). Denote by
D = {u 6 d(a2 ): ®(u) = ), ®(Au) = = [®(g) - NT(g )] F (Au) + NT(Au)}.
Let u 6 D(B2) and g 6 D(A)n. Then by definition, u 6 D(B) and Bu 6 D(B), which since (3) implies u 6 D(A), ®(u) = NT(u) and Bu 6 D(A), ®(Bu) = NT(Bu). From Bu = Au - gF(Au) 6 D(A) it follows that u 6 D(A2). Further from the equation ®(Bu) = NT(Bu) is implied that u 6 D.
Conversely, let u 6 D, then u 6 D(A2), ®(u) = NT(u) and ®(Au) - [®(g) - NT(g)]F(Au) = NT(Au). Then u 6 S(B), Bu 6 S(A) and ®(Au) - ®(g)F(Au) = NT(Au) + + NT(g)F(Au), which implies ®(Bu) = NT(Bu) or Bu 6 D(B). Hence u 6 D(B2), and so (6) holds. Now we prove the first formula in (6). Let u 6 D(B2), y = Bu, g 6 D(A)n. Then
B2u = By = Ay - gF(Ay) = ABu - gF(ABu) = = A[Au - gF(Au)] - gF(A[Au - gF(Au)]) =
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= A2u - AgF(Au) - gF(A2u) + gF(Ag)F(Au). Hence, B1u = B2u.
(ii) Let (5) holds and detV, detW * 0. By statement (i), B1 = B2 and so D(B1) = D(B2). Since O(z) = = Im, the relations in (6) can be written as
0(u - zNT(u)) = 0,
®(Au - [g - zNT(g)]F(Au) - zNT(Au)) = 0,
which taking into account (2) imply
u - zNT(u )e d(A ),
Au - [ g - zNT(g )] F (Au) - zNT(Au ) e d(A ).
Then from (6), since z e [kerA]m, we obtain A (au - gF (Au ) + zN [T(g )F (Au )-T(Au )]) +
+ g
F (Ag )F (Au )- F (A2u ) = f,
Au - gF (Au ) + zN [T(g )F (Au) - T( Au )] -
+ A "ig
F (Ag )F (Au )- F (a2u ) = A _1f, A (u - zNT(u )) - gF(Au) + zN [T(g )F (Au ) -T(Au )] -F (Ag )F (Au)-F ((2u) = A_1f,
+ A-g
and hence
u - zNT(u )- A 1gF (Au ) + +A _1zN [T(g )F (Au )-T(Au )] +
+A 2 g
F (Ag )F (Au ) - F (a 2 u )) = A - 2 f
and then
A2u = [Ag - gF(Ag)]F(Au) + gF(A2u) + f, Au = gF (Au ) - zN [T(g )F (Au ) -T(Au) -F (Ag )F (Au)-F (a2u ) + A~lf, u = zNT(u ) + A-1gF( Au ) -- A _1zN [T(g )F (Au )-T(Au )]-F (Ag )F (Au ) - F (a 2 u ) ) + A- 2 f.
- A _1g
- A " 2 g
Acting by the vectors F and T we get F (Au ) = F (g )F (Au) - F (z) x x N [T(g )F (Au )-T(Au)]- F (A _1g ) x
F (Ag )F (Au)-F (a2u ) + F (A_1f),
T(Au) = T(g )F (Au)-T(z) x x N[T(g)F(Au)-Y(Au)]-Y(A_1g) x
x F (Ag )F (Au)-F (a2u ) + Y(A_1f),
VT(u ) =
t(A _1g )-t(A _1z)NT(g )-t(A "2g )F (Ag )]: x F (Au ) + t(A _1z)nY(âu) + + t(A "2g )f (â2u) + t(A "2f ),
F(A2u) = [F(Ag) - F(g)F(Ag)] F(Au) + + F(g)F(A2u) + F(f),
or
n - F (g ) + F (z )NT(g ) + F (A "1g )f (Ag )]x x F (Au)-F (A_1g )F (a2u )-- F (z)NT(Au ) = F (A ~1f ),
VT(Au)- VT(g)-t(A_1g)F(Ag)f(Au)-
-t(A "1g )f (a2u ) = t(A "1f ),
VT(u ) - t(A _1z)nY(âu ) -
- ^(A^g)-t(A"1z)NT(g)-t(A"2g)f(Ag)] :
x F (Au )-t(A "2g )F (a2u ) = t(A "2f ),
[F(Ag) - F(g)F(Ag)]F(Au) + + [F(g) - IJF(A2u) = -F(f). Denoting
D1 = -VT(g ) + t(A "1g )F (Ag ); D2 = W + F (z)NT(g ) + F (A "1g )f (A g ); D3 = -t(A "1g ) + t(A "1z)NT(g ) + + t(A "2g )F (Ag ),
we get
' -F (z)N D2 -F( AÂ-1g ))
0n V D1 -T "1g )
V -t(a _1z)n D3 -T (^"2g )
V n 0n WF (Ag ) -W j
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TEOPETMHECKA51 M nPMKAAAHAI MATEMATMKA
f i(u) ^ l(Au) F (Au)
F (2uj
Designating the matrix the left by L2, we have
(
F (A-1f))
i(A-1/) i(A "2/)
-f(f) ,
'On -F(z)N D2 -F(i"1g)
On V D1
det V 8/
V -l(i_1z)N D3 -l(i"2g)
V n On WF(Ag) -W /
= det
f V-1 0 0
* "n "re
On V-
On On
On On 0,
V On On On W
-F(z)N D2 -F(i_1g ^
On V
Di
V -l(i_1z)N D3 On On WF(Ag) -W
-l(i"2g)
V 3 W =
= det
O„
In V
On
-V-1F(z)N V-1D2 -V-1f(( _1g^
In V-1D1 -V-1l(i"1g)
1i(("1z)n V-1D3 -V-1l(i"2g)
On F(Ag) -In
V 3 W = ±det
-V-1F(z)N V-1D2 -V-1F(i_1g)^
In V-1D1 -V-1l(i"1g) On F(Ag) -In
WW .
Multiplying from the right the third column of the determinant by the matrix F(Ag) and adding to the second column, we obtain
(
det L2 = ±det
V-1F(z)N V-1D2 - V-1F(A_1g)F(Ag) -V-1F(A In V-1D1 -V-1l(A _1g )f (Ag) -V-1l(A _1g)
V 3 W =
= det
-V-1F (z)N V-1 [W + F (z)Nl(g )] -V-1F (A _1g^
-v-1i(A "1g)
-^(g) On
V 3 W =
= det
-V-1F (z )N V-1 [W + F (z )N1(g)]
In -^(g)
V|3 |W|.
Finally, multiplying from the right the first column of the determinant by the matrix 1(g) and adding to the second column we get
(
detL2 = ±det
-V-1F (z)N V-1W^
V 3 W =
n
= ± V-1 |W|Iv|3\W\ = ±|V|2 |W|2.
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So detL2 = +|V|2|W|2 * 0. Let u e kerBr Then in (11) f = 0 and L2col(T(u), Y(Aw), F(Au), F(A2u)) = 0, which since detL2 * 0, yields Y(u) = Y(Au) = F(Au) = = F(A2u) = 0. Substitution of these values into (6) imply B1u = B2u = A2u = 0, ®(u) = ®(Au) = 0. Taking into
account (2) we acquire ue D|A j and B1u = A2u = 0.
By hypothesis A is correct and so u = 0. Thus kerB1 = {0} and B1 is injective.
Conversely, let detV = 0, then there exists a vector c = col(c1, ..., cn) * 0 such that Vc = 0. Note that u0 = zNc * 0, otherwise, since the components of z are linearly independent, we have Nc = 0 and from Vc = 0 follows that c = 0 wich contradicts the hypothesis c * 0. Substituting u0 into the first boundary condition (6), we get ®(u0) - N¥(u0) = Nc - N¥(z)Nc = = N[In - T(z)N]c = NVc = 0. Substituting u0 into the second boundary condition, we obtain ®(Au0) -- [®(g) - NT(g)] F(Au0) - NT(Au0) = 0, since z e kerA. So u0 e S(B2). It is evident that u0 e kerB2. Hence u0 e DD(B2) and u0 e kerB2. So kerB2 = kerB1 * {0} and B2 = B1 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 gv ..., gn is a linearly independent set and that the element
c * 0, otherwise c = 0.
uo =
A _1g + zNV-1T
(A "1g )
For u0 we obtain O(u0) - N¥(u0) = NV -1¥( A _1g)c - N¥( A _1g)c -- NY(z)NV-1Y( A _1g)c =
= N[I„ - ¥(z)N]V-1¥(A_1g)c - NY(A_1g)c = = [NT( A_1 g) - NY( A _1 g)]c = 0.
So u0 satisfies the first boundary condition (6). For u0 we also obtain
O(Au0) - [®(g) - N^(g)]F(Au0) - N^(Au0) = = ®(g)c - [®(g) - NT(g)]F(g)c - Nf(g)c = = ®(g)[I„ - F(g)] c - N*F(g)[I„ - F(g)] c = = ®(g)Wc - N¥(g)Wc = 0.
So u0 e D(B2). Moreover
B2u0 = A2u0 - [Ag - gF(Ag)]F(Au0) - gF(A2u0) = = Agc - [Ag - gF(Ag)]F(g)c - gF(Ag)c = = Ag[I„ - F(g)] c - gF(Ag)[I„ - F(g)] c = = AgWc - gF(Ag)Wc = 0.
Hence u0 e D(B2) and u0 e kerB2. So kerB2 * {0} and B2 is not injective. So we proved that B1 is injec-tive if and only if detV * 0, detW * 0. The statement (ii) holds.
(iii) Let the vectors q, g, v, w and matrices D, N satisfy (5) and detV * 0, detW * 0. Then, by the statement (ii), the operator B1 = B2 is injective and the problem B1u = f has a unique solution. We recall that by Theorem [1] the unique solution of the equation Bu = f for all f e X is given by
u = B~1f = A ~1f + + [ A _1g + zNV-1Y(A _1g)]W-1F(f) +
+ zNV-1T( A ~lf). (12)
Let B1u = B2u = f, where f e X. Denoting by Y = [A-1g + zNV-1T(A_1g)]W-1 and f = Bu, we get Bf = f. Then by means of (12) the solution of this equation is given by
f = B ~lf = A ~lf + YF(f) + zNV-1 A ~lf). (13)
Applying again (12) we find the solution of the problem Bu = f or B1u = f
u = B-1f = B~lf = A~lf + YF(f) + zNV_1Y(A~lf), (14)
which is equation (9). Substituting the value of f from (13) into (14), we get
u = B-1f = B"2f = B~1f = A "2f + A _1YF(f) + +A _1zNV-1T( A _1f) + Y[F( A ~1f) + F(Y)F(f) + + F(z)NV-1T( A _1f)] + zNV-1 [T( A "2f) + + A _1Y)F(f) + A _1z)NV-1T( A _1f)] =
= A "2f + YF( A _1f) + zNV-1¥( aA-2f) + + [ A _1Y + YF(Y) + zNV-1^(aA _1 Y)]F(f) + + [ aA _1z + YF(z) + zNV-1¥(A "1z)]NV"1T( A"1 f),
which is the solution (8). In the above solutions f
is arbitrary, consequently, R(B1) = X. Since the
* - 2 -1
operators A , A and the functionals F and T are bounded, from (8) or (9) follows the boundedness of Bf1 = B"2, i. e. the operator B1 is correct. The theorem is proved.
The next corollary follows from the above theorem in the case q = g = 0 and it is useful for solving a class of differential equations with multipoint and nonlocal boundary conditions.
Corollary. The differential operator B1: X ^ X be defined by
B1u = A2u = f, D(B1) = {u e D(A2): ®(u) = NT(u),
®(Au) = NT(Au)}. (15)
Then the operator B1 is correct if and only if
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^ TEOPETMHECKAI M nPMKAAAHAI MATEMATMKA X.
detV = det[I„ - 1(z)N] * 0 (16)
and the unique solution of (15) is given by
u = B-1f = A "2f + zNV-11( A-2f) + + [ A _1z + zNV-11( A _1z)]NV-11(A _1f). (17)
Proof: For g = 0 from (7) immediately follows detW = detIn = 1 ^ 0. Then by Theorem, the operator B1 is correct if and only if detV ^ 0. From (8) for g = q = 0 follows the solution of this problem.
Examples
In this section we consider some examples boundary value problems to explain the application of the decomposition-extension method and to demonstrate its efficiency.
First, we recall some known results. The problem
Au = u(m) (x) = f (x), DD(A) = {u e Cm [0, 1]: u(0) = = u'(0 ) =... = um-1 (0 ) = 0},
is correct and its exact solution is given by
u(x) = A"1f (x) = L) j0x(x -1)m-1 f (t)dt. (18)
If the function u(x) e Cm[a, b] and x0 e [a, b], then the functionals Tk(u) = u(k-1)(x0), k = 1, ..., m, and T (u ) = ^ r—=1aku(k-11 (x0) are linear and bounded on Ck[a, b] and Cm[a, b], respectively.
Example 1
Consider the differential boundary value problem u"(x) = f(x), x e [0, 1]:
u(0) = vu(1), u'(0) = vu'(1). (19)
By taking X = C [0, 1] and
Au = u'(x), D(A) = {u e C1[0, 1]}, A2u = u"(x), DD(A2) = {u e C2[0, 1]}; Au = Au, D(A) = {ueD(A):u(0) = 0}, A2u = A2u, DD(A2) = {u e DD(A2): u(0) = u'(0) = 0},
we can put the problem (19) as in Corollary, equation (15), namely:
B1u = A2u = f, D(B1) = {u e DD(A2): u(0) = vu(1), u'(0) = vu'(1)},
where ®(u) = (u(0)), Y(u) = (u(1)) and N = (v). Then ®(Au) = (u'(0)) and Y(Au) = (u'(1)). Let z = (1) and notice that ®(z) = z(0) = 1. If detV = det(1 - v) * 0 then the unique solution u = B-1/ follows from Corollary where
A~lf = j0f(t)dt, A~2f = JQ(x-1)f(t)dt, by means of (18). Example 2
Let the fourth-order differential boundary value problem with multipoint boundary conditions u(4)(x) = f(x), x e [0, 1]:
u (0)=viiu | 1 j+vi2u (1);
u(0) = v21u ^ 2 ] + v22u (1); u"(0) = vnu"^ 1 j + v^i);
u"'(0) = v2iu"[ 2 ^j + v22u"(1).
(20)
We recast the problem (20) into the form (15) where X = C [0, 1] and
Au = u"(x), DD(A) = {u e C2[0, 1]}, A2u = u(4)(x), DD(A2) = {u e C4[0, 1]}, Au = Au, D(A) = {u e D( A): u (0) = u'(0) = 0}, A2u = A2u,
DD(A2) = {u e DD( A2): u (0) = u'(0) = u"(0) = u"'(0) = 0};
0(u ) =
i,,(c\W
(0)
V u'(0)y
; ^(u ) =
( i i il ul2
v u (1) ,
0(Au) =
(
(0 )
V u"'(0)y
; Y(Au ) =
u|2
u
"(1)
®(u) = N¥(u) = (vi1 vi2 |T(u); ®(Au) = NT(Au).
Vv21 v22 .
Let z = (1, x) and notice that ®(z) = z(0) = I2. If
(
det V = det
1 1/2 1 1
N
* 0,
then the unique solution u = B1 1f follows from Corollary by making use of (18).
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Example 3
Contemplate the fourth-order Fredholm inte-gro-differential boundary value problem with general integral boundary conditions
u(4)(x) + (x2 - 1))xu"(x)dx-(x2 + 1)x x jo xu(4)(x)dx = x - 2,
x e [0, 1],
15 i-1
u(0)= 22 j0 u(x)d x; u'(0) = 0;
u"(0) = —j xu"(x)dx + —j u"(x)dx;
11J0 v ' 2^0 u""(0) = 0.
(21)
We formulate the problem (21) as in (4). We take X = C [0, 1] and
Au = u"(x), D(A) = {u e C2[0, 1]}, A2u = u(4)(x), D(A2) = {u e C4[0, 1]}, A u = Au, D( A) = {u e D( A) : u (0 ) = u'(0 ) = 0}, A 2u = A2u,
D(A2) = {u e D(A2) : u(0) = u'(0) = u"(0) = u"'(0) = 0}; f(x) = x - 2; q = (1 - x2); g = (x2 + 1);
F (Au) = V ft xu"(x )d x J ; F (a2u ) = i J0xu(4) (x )d x J ;
0(u ) = 0(Au) =
iu(0)J
v u'(0)y
iu"(0\J
T(u ) = V J^u (x )d x
(0)) T(Au) = (io1 u"(x)dx];
References
1. 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. Informatsion-no-upravliaiushchie sistemy [Information and Control Systems], 2018, no. 6, pp. 14-23. doi:10.31799/ 1684-8853-2018-6-14-23
2. Parasidis I. N., Providas E. Extension operator method for the exact solution of integro-differential Equations. In: Pardalos P., Rassias T. (eds). Contributions
®(u ) = NY(u ) =
15/22
Y(u );
0( Au ) = DF (Au ) + NY(Au ) = ^ ' F (Au ) + NY (Au ).
Observe that g = (x2 + 1) e S(A) and q = Ag -- gF(Ag),
D = O(g) - NY(g).
(22)
Let z = (1, x). Then O(z) = I2, Y(z) = 1 1, - I and
det V = det(l1 - Y(z)N) = — * 0; det W = det (I1 - F (g )) = 1 * 0.
(23)
Bacause of (22) and (23), Theorem applies. Hence the operator B1 is correct which means that the problem (21) admits a unique solution. By substituting into (8) or (9) and making use of (18), we get
u(x) =--1-(2352x6 -2646x5 +
v ' 317520V
+ 212 387x4 +1169 427x2 + 926103).
Conclusion
By means of decomposition and the extension method, we provided a ready to use formula for constructing the solution in closed form of boundary value problems involving the composite square of an mth order integro-differential operator of Fredholm type and nonlocal boundary conditions such as appropriate multipoint and integral conditions. The method is also applicable to boundary value problems for the composite squared mth order linear ordinary differential operators.
in Mathematics and Engineering. Springer, Cham, 2016, pp. 473-496. doi:10.1007/978-3-319-31317-7_23
3. Parasidis I. N., Providas E. On the exact solution of nonlinear integro-differential equations. In: Rassias T. (eds). Applications of Nonlinear Analysis. Series: Springer Optimization and Its Applications, Springer, Cham, 2018, vol. 134, pp. 591-609. doi:10.1007/978-3-319-89815-5_21
4. Parasidis I. N., Providas E., Tsekrekos P. C. Factorization of linear operators and some eigenvalue problems of special operators. Vestnik Bashkirskogo uni-versiteta, 2012, vol. 17, no. 2, pp. 830-839.
MHOOPMAIiMOHHO-ynPABAfiroWME CMCTEMbl / № 2, 2019
8
^ ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА X.
5. Barkovskii L. M., Furs A. N. Factorization of inte-gro-differential equations of optics of dispersive ani-sotropic media and tensor integral operators of wave packet velocities. Optics and Spectroscopy, 2001, vol. 90, iss. 4, pp. 561-567. doi:10.1134/1.1366751
6. Barkovsky L. M., Furs A. N. Factorization of inte-gro-differential equations of the acoustics of dispersive viscoelastic anisotropic media and the tensor integral operators of wave packet velocities. Acoustical Physics, 2002, vol. 48, iss. 2, pp. 128-132. doi:10.1134/ 1.1460945
Авторы статьи 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, 2018, № 6 приносят извинения за опечатки.
Страница Столбец Строка Напечатано Следует читать
15 правый 8 снизу of (5) is of (5) for о = 1, a1 = ... = am = 0, a = 0 is
15 правый 6 снизу f(x) e С [a, b]. f(x) e С[0, b].
16 левый 9 снизу Xmand respectively [Xm-1]* and X* respectively
/
УДК 338.984
doi:10.31799/1684-8853-2019-2-2-9
Метод нахождения точных решений для интегро-дифференциальных уравнений Фредгольма с мультиточечными и интегральными краевыми условиями. Часть 2. Метод разложения-расширения квадратичных операторов
Н. Н. Васильева>б, канд. физ.-мат. наук, старший научный сотрудник, orcid.org/0000-0002-0841-1168, vasiliev@pdmi.ras.ru И. Н. Парасидисв, PhD, доцент, paras@teilar.gr Е. Провидасг, PhD, доцент, providas@teilar.gr
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, Профессора Попова ул., 5, Санкт-Петербург, 197376, РФ
вКафедра электротехники, Технологический институт Фессалии, 41110, Лариса, Греция гКафедра машиностроения, Технологический институт Фессалии, 41110, Лариса, Греция
Введение: в первой части статьи представлен прямой метод исследования проблемы разрешимости и единственности и получения в замкнутой форме решения краевых задач, включающих линейный обыкновенный интегро-дифференциальный оператор Фредгольма или дифференциальный оператор m-го порядка, а также многоточечные и интегральные граничные условия. Здесь мы сосредоточимся на специальном классе краевых задач, включающих квадрат интегро-дифференциального оператора и соответствующих нелокальных граничных условий. Цель: исследование построения единственного решения краевых задач 2-го порядка в частном случае оператора, который может быть представлен в виде композиции квадратов операторов более низких порядков, а также разработка алгоритма построения точного решения в этом частном случае. Результаты: с помощью декомпозиции и метода расширения, описанного в первой части, нами разработан алгоритм для получения точного решения краевых задач для квадро-интегро-дифференциальных операторов или дифференциальных операторов с многоточечными и интегральными граничными условиями. Этот метод прост в использовании и может быть легко имплементирован в большинство современных систем компьютерной алгебры.
Ключевые слова — дифференциальные и фредгольмовы интегро-дифференциальные уравнения, многоточечные и нелокальные интегральные граничные условия, разложение операторов, корректные операторы, точные решения.
Для цитирования: Vassiliev N. N., Parasidis I. N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators. Информационно-управляющие системы, 2019, № 2, с. 2-9. doi:10.31799/1684-8853-2019-2-2-9
For citation: Vassiliev N. N., Parasidis I. N., Providas E. Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2019, no. 2, pp. 2-9. doi:10.31799/1684-8853-2019-2-2-9
№ 2, 2019
ИНФОРМАЦИОННО-УПРАВЛЯЮЩИЕ СИСТЕМЫ 9
