CC BY
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ТЕОРЕТИЧЕСКАЯ И ПРИКЛАДНАЯ МАТЕМАТИКА У
UDC 338.984
doi:10.31799/1684-8853-2019-4-2-8
An exact solution method for Fredholm integro-differential equations
K. D. Tsilikaa, PhD, Assistant Professor, orcid.org/0000-0002-9213-3120, ktsilika@uth.gr aLaboratory of Operations Research, Department of Economics, University of Thessaly, Volos, 38221, Greece
Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom considered with integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocal integral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocal integral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existence criterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditions and obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for the special case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for the existence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct method analytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations. If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existence and uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by two examples. The article uses computer algebra system Mathematica to demonstrate the results.
Keywords — boundary value problems, ordinary differential equations, Fredholm integro-differential equations, multipoint boundary conditions, nonlocal integral boundary conditions, correct operators, exact solutions.
For citation: Tsilika K. D. An exact solution method for Fredholm integro-differential equations. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2019, no. 4, pp. 2-8. doi:10.31799/1684-8853-2019-4-2-8
Articles
Introduction
Consider the boundary value problem of the type Bu = Au - gF(Au) = f, D(B) = {u e D(A):
O(u) = NT(u)}, u e X, f e Y, (1)
where A is an ordinary differential operator; F(Au) = col(i,1(Au), ..., Fn(Au)) is a functional vector representing the integral part of the integro-differen-tial equation; g = (gp ..., gn) is a vector; N — a constant m x l matrix; O = col(®1, ..., ®m); Y = col(T1, ..., Y) are functional vectors with Y(u) standing for the multipoint or integral part of the boundary conditions.
Note that problem (1) describes the particular case of the multipoint boundary value problem with integro-differential equation (IDE) (see [1]) given below
,(m)
(x )+X m=1 Pi (x )u(m (x )+ +XГ=о¡^ (x, y)u{m~i)(y)dy = f (x);
X^oXi^kij^1 (xj ) + ( Vk ) = 0
k = 1, ..., s, (a = x0 < x1 < ... < xv+l = b),
(2)
where (u, <pk) is an integral part of the boundary conditions. IDEs like (2) are investigated also in [2]. In [1, 2] the existence theorem for (2) has been proved. This problem is reduced to problem (1) if
s = m, v = 0, K(x, y) = g(x)h(y)pi(y), i = 0, 1, p0(x) = 1 and ||aMo| I™ =x = Im-Indeed, then we can take
m,
Au = u(m)(x) + Xi=iPi (x)u 1 )(x);
ФДц) = u(i-1)(x0), i = 1, ..., m, T =^u(i"1)(x), i = 1, ..., m 1 l(w) "[(u, 9i_m ), i = Г +1, ..., 2mj
F(Au) = ¡У (y)[u(m)(y) + Xm=1 Pi (У)u(m_i)(y)"
d y and
5ki is a Kronecker delta func-
N = HIaki1 , ,
tion.
In this paper, we study the solvability of problem (1) and analytically construct its solution. The present work lies in the field of boundary value problem for linear integral equations [3, 4]. In this setting, problems are often too complex to be solved analytically and therefore numerical solutions are proposed [5, 6]. Analytical solutions of a class of boundary value problems, involving the governing differential equation or slight generalizations of the one considered in (1), but with different and/or simpler differential operators, have been proposed in [7-10]. Existence theorems for nonlocal multipoint boundary value problem and IDEs via analytical methods are given in [11-13]. Boundary value problems B: X ^ X of the type of (1) for the specific case of l = n
have been studied by Vassiliev, Parasidis, Providas in [14, 15], using the extension method. The extension method is a generalization of the direct method, which is presented in [4]. Here, problem (1) is investigated and solved for the case l * n, X * Y mainly by functional transformations and by employing the properties of linear operators. The results of this method coincide with the results of [14, 15] for l — n.
The novelties of the present paper are the following. First we address the local boundary value problem of the type of (1) without restrictions to the dimensions l, n, m; second, we assume multipoint or nonlocal integral boundary conditions, which allows us to consider a very large class of problems for the equation (1). The ultimate result is the exact solution of problem (1).
We describe now the setting of our approach. 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. D(A) and R(A) are, respectively, the domain and the range of the operator A. The following definitions are extremely important in the analysis of the next section.
Definition 1. 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 c A2, if D(A2) 3 D(A1) and A x — A2x, for all x e D(A1).
Definition 2. An operator A: X ^ Y is called closed if for every sequence xn e D(A) converging to x0 with Axn ^ f0 follows that x0 e D(A) and
Ax0 = fo.
Definition 3. A closed operator A is called maximal if R(A) — Y and kerA * 0. ^
Definition 4. An operator A: X ^ Y is called correct if R(A) — Y and the inverse A_1 exists and is continuous on Y.
Definition 5. An operator A is called a correct restriction of the maximal operator A if it is a correct operator and A c A.
Basic notations. If Y£ e X*, i — 1, ..., l, then we consider the functional vectors Y — col(Yp ..., Y^ and Y(x) — col^x), ..., Yl(x)). Let g — (g1, ... gn) be a vector of Xn. Hereafter we will denote by Y(g) the l x n matrix whose i, j-th entry Y^(gj) is the value of functional Y on element g. Note that Y(gC) — Y(g)C, where C is a n x k constant matrix. We will also denote by In the identity n x n. By 0 we will denote the zero column vector. We will denote by kerA the kernel of an operator A.
Exact solution method for Fredholm IDEs
Suppose that A: X ^ X is an ordinary m-order differential operator
Au(x) = a0u(m)(x) + alu(m-v>(x) + ... + amu(x), at e R, (3)
X = C[a, b] or X = Lp(a, b), p > 1, z = (z1, z2, ..., zm) a basis of kerA and A be a correct restriction of A defined by
A c A, D(A) = {u e D(A): O(u) = 0}, (4) where the components of the functional vector
O =
col(®i, ..., Om), belong to [ xm ]
and are biortho-
gonal to z1, ..., zm. Everywhere below we denote by Xm — Cm [a, b] if X — C[a, b], or X™ — W^ (a, b) if X — Lp(a, b) and by |X™ J we denote the adjoint to XmA space, i. e. the space of all linear bounded
that the operator A: C[a, b]^ C[a, b] defined by
A u (x ) = a0u(m) (x ) + ai«(m 1) (x ) + ... + amu(x ) = f, at e R, x e [a, b], (5)
D (A ) =
= {u(x) e Cm [a, b]: u(a) = u'(a) =... = u(m-1) (a) = o}, a0 ^ 0
is a correct restriction of A and the unique solution of (5) for a0 = 1, a1 = ... = am = 0, a = 0 is
u (x ) = A-1f (x ) = (-L) £ (x - t )m-1 f (t )d t,
f(x) e C[0, b].
(6)
Before we address the issue of finding the exact analytic solutions of the Fredholm IDE, we will mention briefly some properties of the linear operators of these equations.
Lemma 1. Let A be a differential operator defined by (3), z — (zp ..., zm) a basis of kerA, N a m x l matrix with constant elements, the components of a vector Y — col(Yp ..., Yl) be functionals on X™m, the matrix V — Il - Y(z)N and the operator C: Xm ^ Xm be defined by
Cu — u - zNY(u).
Then:
(i) The operator C is invertible if and only if detV * 0;
(ii) the inverse operator C-1v — v + zNV-1Y(v).
Proof: Let detV * 0 and u e kerC. Then
u - zNY(u) — 0 and Y(u - zNY(u)) — 0, Y(u) -- Y(z)NY(u) — 0, [Il - Y(z)N]Y(u) — 0. From the last equation since detV * 0 follows that Y(u) — 0. Substituting this value into u - zNY(u) — 0 we get u — 0. Consequently, kerC — {0} and C is inver-tible.
functionals defined on Xm. It is a well-known fact
Conversely, let detV = 0. Then there exists a vector с * 0 such that Vc = 0.
Consider the element u0 = zNc. Note that u0 * 0, otherwise the linear independence of the components of the vector z implies Nc = 0 and then Vc = [Il - Y(z)N]c = c - Y(z)Nc = c - 0 = c = 0. This contradicts the hypothesis that c * 0. So, u0 * 0. We will show that u0 e kerC. Observe that Cu0 = u0 -- zNY(u0) = zNc - zNY(z)Nc = zN[Il - T(z)N]c = = zNVc = zN0 = 0. Thus u0 e kerC and kerC * {0}. Hence C is not invertible. So C is invertible if and only if detV * 0. Let Cu = v, i. e. u - zNY(u) = v. Then Y(u) - T(z)NT(u) = Y(v), or [Il - Y(z)N]Y(u) = Y(v), which implies Y(u) = V-1Y(v). From the last equation and u - zNY(u) = v follows that u = v + zNV-1Y(v), which yields u = C-1v. The lemma is proved.
The next theorem is analogous to theorem which has been proved in [8] when F1, ..., Fn are linearly independent. This theorem is already proved for the linearly independent set g1, ., gn, though using a different method. In addition, it is proved here that the criterion for correctness of the operator B in [8] coincides with the criterion for injectivity of B.
Theorem 1. Let X, Y be complex Banach spaces, A:: X ^ Y be a correct linear operator, the functional vector F = col(F1, ..., Fn) e [Y*]n, a vector g = (g1, ..., gn) e Yn and g1, ..., gn is a linearly independent set. Then:
(i) The operator B defined by
Bu = Au - gF(Au) = f, D(B) = D(.A), f e Y (7)
is injective if and only if
detW = det[In - F(g)] * 0.
(8)
(ii) If B is injective, then B is correct and the unique solution of (7) is given by
u =
B~1f = AA~1f + (A"1g)[I„ -F(g)]-1 F(f), f e Y. (9)
Proof: (i) Let detW = 0 and u = kerB, i. e. Bu = A u - gF (Au ) = 0. Since gi e R (A ) = Y, i = 1, ..., n, by using (7) we have F (Au)- F (g )F (A u ) = 0 and [l„ - F(g)]F(Au) = 0. Taking into account that detW * 0, from the last equation we get F (A u) = 0. Then Bu = Au = 0 and u = 0. The last follows from
ker A = {0}. So kerB = {0} and B is an injective operator.
Conversely. Let detW * 0. Then there exists a vector c = col(c1, ..., cn) * 0 such that Wc = 0. Note that gc * 0 because g1, ..., gn is a linearly independent set. Then u0 = A _1gc * 0 and Buo = Au0 - gF((u0) = gc - gF(g)c = g[I„ - F(g)]c = = gWc = g0 = 0.
Hence u0 = A 1gc e ker B *{0} and B is not injective. So statement (i) holds.
(ii) Let detW * 0. Then from (7) we get
F (A u)- F (g )F (A u) = F (f), [I„ - F(g)]F(Au) = F(f), F ((Au) = W-1F(f) for all f e Y; (10)
u - A_1 gF (Au ) = A_1 (f ). (11)
Substituting (10) into (11), we obtain the unique solution (9) of (7) for all f e Y. Then R(A) = Y. The boundedness of B 1 on Y results from the bound-edness of the operator A_1 and the components of vector F. Hence, B is correct.
Now we generalize this theorem for the case of perturbed boundary conditions and dim kerA = m. The next theorem is a generalization of the Theorem 1 of [14] for the case l * n and X * Y.
Theorem 2. Let X, Y be complex Banach spaces, A: X ^ Y be an operator from (3) with finite dimensional kernel z = (zv ..., zm) which is a basis of kerA, the operator A defined by (4) and the components of the functional vectors O = col(O1, ..., Om), Y = col(Y1, ..., Y), and F = col(F1, ..., Fn) belong to
[x™ J and Y* respectively. Suppose also that O1,
..., Om are biorthogonal to z1, ..., zm and the components of vector g = (g1, ..., gn) e Yn are linearly independent and N is a m x l matrix.
Then: (i) The operator B defined by
Bu = Au - gF(Au) = f, f e Y;
D(B) = {u e D(A) : O(u) = NY(u)} (12)
is injective if and only if
detV = det[Il - Y(z)N] * 0; detW = det[In - F(g)] * 0.
(13)
(14)
(ii) If B is injective, then B is correct and for all f e Y the unique solution of (12) is given by
u = B 1f = A 1f +
A-1g + zNV-1y(A _1g )]x x W_1F(f ) + zNV-1y(A_1/). (15)
Proof: (i) The problem (12) can be rewritten as Bu = A(u - zNY(u)) - gF(A[u - zNY(u)]) = f, f e Y, D(B) = {u e D(A) : O(u - zNY(u)) = 0}. (16)
Then u-zNY(u)eD(A), Bu = A(u-zNY(u))-- gF (A [ u - zNY(u )]) = f for every u e D(B). We introduce the substitution v = u - zNY(u) or since
Lemma 1, v — Cu, where we take C: X ^ X, D(C) — — D(B). It is evident that C is a linear operator and R (C) c D (A). From (16) it follows that problem (12), by the substitution v — Cu, is reduced to the problem
Bu — B1v — Av - gF(Av) — f, v e D (B1) — D(A). (17)
Since v — Cu, from (17) we get Bu — Bfiu for all u e D(B). Then B is injective if and only if Bx and C are also inj ective. Note that the operator C, by Lemma 1, is injective if and only if detV — det[Il - Y(z)N] * 0. The operator B1, by Theorem 1, where instead of B, u we have B1, v respectively, is injective if and only if (8) holds. So the operator B is injective if and only if (13), (14) hold true.
(ii) The unique solution of (17) by (9) for all f e Y is given by
v =
A ~1f + (i _1g )w-1F (f). (18)
Then
Y(v) = y(A _1f ) + y(A _1g )w-1F (f) (19)
and since detV ^ 0, v = u - zNY(u), by Lemma 1, we get
u = C_1v = v + zNV -1 Y(v) =
= A "1f + (A "1g )w-1F (f) + y(A _1f ) + y(A _1g )w-1F (f). (20)
+ zNV"
The last equation easily implies the unique solution of (12) defined by (15). The operator B1, by Theorem 1, is correct. Then R(B1) — Y, which since Bu — Bxv gives R(B) — Y. Since the operator A_1 and the functionals Fp ..., Fn, Y1, ..., Yl are bounded, the boundedness of B-1 follows from (15). Hence, the operator B is correct. This completes the proof of Theorem 2.
Now, we implement the solution method of Theorem 2 on two numerical examples, the first involving a linear second order differential operator and the other a linear forth order differential operator A.
Example 1. Consider the multipoint problem for loaded differential equation on C[0, 1]
u" - (3t3 - 2t)[u'(1) - u(1) + u(0)] = = -6t3 + 4t + 4, u(0) = 2u(1) + u(0.5) - 2u'(1), u'(0) = -u(1) - 2u(0.5) + 3u'(1).
(21)
All the assumptions of Theorem 2 hold and the unique solution of (21) is
u(t) = 2t2 - 5t + 3.
(22)
Proof: If we compare (21) with (12), it is natural to take X = Y = C[0, 1],
Bu = u" - (3 - 2t)^u'(1) - u(1) + u(0)^ =
= u"-(3t3 -2t))u"(x)dx, D(B)=0 ........... (1)^
u (x) e D (A):
,(0) '(0)
^ 2 1 -2 -1 -2 3
u( u( u'
/J
(23)
(0.5) '(1)
Au = u"(t), D(A) = {u e C2[0, 1]}, xm = c2 [0,1], X = C[0, 1], m = 2, l = 3, n = 1, z = (zv z2) = (1, t), Au = Au, D(A) = {u e D(A): u(0) = u'(0) = 0}.
The operator A is correct and, since (6), its solution is A_1f(t) —JQ(t - x)f(x)dx. Further comparing (21), (23) with (12), we take g(t) — 3t3 - 2t,
f(t) = -6t3 + 4t + 4, N — ^_21 -12 "g2 F (Au) — — xu"(x)d x.
The solution method of Theorem 2 for problem (21) is implemented in computer algebra system Mathematica (v. 11.3) [16]. The Mathematica notebook containing the solution method is available upon request.
Even though Mathematica has a dedicated function to symbolically solve an integro-differential equation (solves also Fredholm integral equations), DSolveValue (new feature in Mathematica v. 11, see https://www.wolfram.com/language/11/symbol-ic-and-numeric-calculus/solve-an-integro-differ-ential-equation.html?product — language), in this example fails to give an output. Trying to take the analytic solution with NDSolveValue instead, the procedure gave a bunch of errors. Built-in function NDSolve failed also.
In the Mathematica codes given below, we provide an overview of the symbolic computations we employed for the illustrative Example 1. The computer codes give to the reader a computerized approach along with the analytical one. All of the commands have quite similar analogues in the case of any order integro-differential equation. It is then possible to create a script for solving this type of integro-differential equations. Here, only the main ideas and procedures for the solution are given and the basic knowledge of how to use Mathematica is assumed.
In Mathematica's environment, we begin by writing the functions that represent the structural elements of the operator equation (21). Then we de-
fine the functional vectors of the boundary conditions. Functions and functional vectors are written in a way to take various elements as arguments. In Example 1, t takes values from the interval [0, 1]. The criterion for injectivity of B that is tested and verified is the only requirement to apply Theorem 2 and formulate the unique solution as in (15). Mathematica code in[*]: = (*definitions*)
in[*]: = (*Give the order m of the differential operator A or the number of conditions or the dimension m of the functional vector O*) in[*]: = m = 2;
in[*]: = (*Give the dimension l of the functional vector Y*) in[*]: = l = 3;
in[*]: = (*Give the dimension n of the functional vector F*)
in[*]: = n = 1;
in[*]: = (*Define the structural elements of the operator equation Au - gF(Au) = f*)
in[*]: = (*F is the integral part of the IDE*)
ire[*] = F[function_] ^{j^x*functiondx
in[*]: = f[t_]: = {-6*t3 + 4*t + 4} in[*]: = g[t_]: = {3*t3 - 2*t}
in[*]: = (*Give the values of the variables in the boundary conditions*) in[*]: = ti = {1, 0.5, 1};
in[*]: = (*Give the m x l N matrix "nmatrix" such that O(u) = NY(u)*)
in[*]: = nmatrix = {{2, 1, -2}, {-1, -2, 3}}; in[*]: = (*Give the functional vector Y such that O(u) = NY(u)*)
in[*]: = Y[function_]: = {function/.t ^ ti[[1]], function/.t ^ ti[[2]], D[function, t]/.t ^ ti[[3]]}
in[*] in[*] in[*] in[*]
= (*The solution method*) = W: = IdentityMatrix[n] - F[g[x]] = V: = IdentityMatrix[i] - Y[z].nmatrix = z: = Table[ii/i!, {i, 0, m - 1}];
in\*\ = inverse A [function_] : = 1
10 с - -)"
* functiond 2
(m -1)
in[*]: = (*Verify the assumptions of Theorem 2*) in[*]: = (*Testing necessary and sufficient conditions for operator Bu = Au - gF(Au) to be injective*) in[*]: = Det[W]
г n 16 out\*\ = — L J 15
in[*]: = Det[V]
oui[*]: = 1.5
in[*]: = (*Testing the existence and uniqueness criterion*)
in[*]: = If[Det[W] ф 0 && Det[V] ф 0, "The IDE has a unique solution", "The solution is not unique"] oui[*]: = The IDE has a unique solution
2t2 - 5t + 3 3.0
2.5 2.0 1.5 1.0 0.5 0.0
0 0.2 0.4 0.6 0.8
■ Values for u(t) over the domain [0, 1]
1.0 t
in[*]: = (*Here is the unique solution by the exact solution method*)
in[*]: = solution: = Simplify[inverseA[f[x]] + (in-verseA [ g [ x ]] + z.nmat rix.Inverse[ V ]. Y[inverseA[g[x]]]).Inverse[W].F[f[x]] + z.nmatrix. Inverse[V].Y[inverseA[f[x]]]]
in[*]: = Print["The exact solution of the IDE is" Flatten[solution]]
{The exact solution of the IDE is (3.-5.Î + 2i2)} The graphical representation of the analytical solution to this problem is shown in Figure:
in[*] = Plot[3 - 5i + 2t2, {t, 0, 1}, AxesLabel ^ {t, 3 - 5t + 2t2}].
Example 2. Consider the multipoint problem for differential equation on C[0, 1]
u(4)(t)-(t2 - 4t + 1)x
x|1xu(4)(x)d x = -12t2 + 48t +12, u(0) = 3u(1), u'(0) = -5u'(1) + u"(1),
¿"(0) = -^u"(1), u"'(0) = 24u"(1) - 10u"'(1). 5
(24)
The unique solution of (24) is u(t) = t4 - t2.
(25)
We solve problem (24) using the previous Mathematica script, by changing only the input parameters m, l, n, F, f(t), g(t), N, Y(u). The program checks if all the assumptions of Theorem 2 hold and the unique solution is determined explicitly in finitely many steps of computation, following (15). Mathematica code in[*] = (*definitions*)
in[*] = (*Give the order m of the differential operator A / number of conditions or the dimension m of the functional vector O*) in[*] = m = 4;
in[*] = (*Give the dimension l of the functional vector Y*) in[*] = l = 4;
in\
in[*] = (*Give the dimension n of the functional vector F*) in[*] = n = 1;
in[*] = (*Define the structural elements of the operator equation Au = gF(Au) = f*)
in[*] = (*F is the integral part of the IDE*)
i[*] = F [function _ ]: = jj* x*function d x
in[*] = f[t_]: = {-12*t2 + 48*t + 12} in[*] = g[t_]: = {t2 - 4*t + 1}
in[*] = (*Give the values of the variables in the boundary conditions*) in[*] = ti = {1, 1, 1, 1};
in[*] = (*Give the m x l N matrix "nmatrix" such that Ф(и) = NY(u)*)
in[*] = nmatrix = {{3, 0, 0, 0}, {0, -5, 1, 0}, {0, 0, -1/5, 0}, {0, 0, 24, -10}};
in[*] = (*Give the functional vector Y such that Ф(и) = NY(u)*)
in[*] = Y[function_]: = {function/.t ^ ti[[1]]; D[function, t]/.t ^ ti[[2]], ^[function, {t, 2}]/.t ^ ^ ti[[3]], ^[function, {t, 3}]/.t ^ ti[[4]]} in[*] = (*The solution method*) in[*] = W: = IdentityMatrix[n] - F[g[x]] in[*] = V: = IdentityMatrix[l] - Y[z].nmatrix in[*] = z: = Table[t;/il, {i, 0, m - 1}];
in[*] = inverse A [function_ ]: =
= -—i1 (t - x)m functiondx (m -1)!J0V >
in[*] = (*Verify the assumptions of Theorem 2*) in[*] = (*Testing necessary and sufficient conditions for operator Bu = Au - gF(Au) to be injective*) in[*] = Det[W]
References
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2. Tamarkin J. D. The notion of the Green's function in the theory of integro-differential equations. Transactions of the American Mathematical Society, 1927, vol. 29, no. 4 pp. 755-800.
3. Polyanin A. D., Zhurov Alexei I. Exact solutions to some classes of nonlinear integral, integro-function-al and integro-differential equation. Doklady Mathematics, 2008, vol. 77, no. 2, pp. 315-319. doi:10.1134/ S1064562408020403
4. Wazwaz A. M. Linear and Nonlinear Integral Equations. Methods and Applications. Springer, Beijing, 2011. 657 p.
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boundary value problems with Fredholm integro-dif-
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r-! I9
outi*l = — L J 12 in[*] = Det[F]
r-i 648 outi*l =-
L J 5
in[*]: = (*Testing the existence and uniqueness criterion*)
in[*]: = If[Det[W] * 0 && Det[F] * 0, "The IDE has a unique solution", "The solution is not unique"] oui[*]: = The IDE has a unique solution in[*]: = (*Here is the unique solution by the exact solution method*)
in[*] = solution: = Simplify[inverseA[f[x]] + (in-verseA [ g [ x ]] + z.nmat rix.Inverse[ V ]. Y[inverseA[g[x]]]).Inverse[W].F[f[x]] + z.nmatrix. Inverse[V].*F[inverseA[f[x]]]]
in[*] = Print["The exact solution of the IDE is" Flatten[solution]]
{The exact solution of the IDE is i2(-1 + t2)} in[*] = Expand[solution[[1]]] out[*] = -t2 + t4
Conclusion
In the paper, an analytical method for giving exact solutions for Fredholm integro-differential equations is presented. This analytical method is constructive in the sense that it allows to study the unique solvability of the problem and analytically construct its solution. A couple of numerical examples were solved by this method in Mathematica software. Reproducibility of the proof is possible through the Mathematica script.
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УДК 338.984
doi:10.31799/1684-8853-2019-4-2-8
Метод точного решения для интегро-дифференциальных уравнений Фредгольма
К. Д. Тсиликаа, PhD, доцент, orcid.org/0000-0002-9213-3120, ktsilika@uth.gr ^Университет Фессалии, 38221, Волос, Греция
Введение: линейные краевые задачи с интегральными граничными условиями для дифференциальных и интегро-дифферен-циальных уравнений Фредгольма редко рассматриваются в литературе. Представляется важным исследование интегро-диффе-ренциальных уравнений с многоточечными или нелокальными интегральными граничными условиями. Даже для дифференциальных уравнений получение точных решений с подобными граничными условиями является непростой задачей. Цель: поиск критерия существования и единственности решений для обыкновенных интегро-дифференциальных уравнений Фредгольма с сепарабельным ядром и многоточечными или нелокальными интегральными граничными условиями; разработка математической методики, ведущей к точным аналитическим решениям заданного уравнения. Результаты: для одного класса абстрактных операторных уравнений, частным случаем которых являются интегро-дифференциальные уравнения Фредгольма с многоточечными или нелокальными интегральными граничными условиями, получены критерий существования и единственности точного решения и его аналитическое представление; предложен прямой метод, символически решающий подобные задачи, в котором все вычисления воспроизводимы в любой программе символьных вычислений. Если пользователь устанавливает входные параметры и начальные условия задачи, выделенные компьютерные коды проверяют условия существования и единствености и генерируют аналитическое решение задачи. Этапы решения проиллюстрированы двумя примерами. В статье использовано программное обеспечение CAS Mathematica.
Ключевые слова — краевые задачи, обыкновенные дифференциальные уравнения, интегро-дифференциальные уравнения Фредгольма с сепарабельным ядром, многоточечные граничные условия, нелокальные интегральные граничные условия, корректные операторы, точные решения.
Для цитирования: Tsilika K. D. An exact solution method for Fredholm integro-differential equations. Информационно-управляющие системы, 2019, № 4, с. 2-8. doi:10.31799/1684-8853-2019-4-2-8
For citation: Tsilika K. D. An exact solution method for Fredholm integro-differential equations. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2019, no. 4, pp. 2-8. doi:10.31799/1684-8853-2019-4-2-8
