View of Existence and approximation of a solution of boundary value problems for delay integro-differential equations

J. Numer. Anal. Approx. Theory, vol. 44 (2015) no. 2, pp. 154–165 ictp.acad.ro/jnaat

EXISTENCE AND APPROXIMATION OF SOLUTIONS TO BOUNDARY VALUE PROBLEMS

FOR DELAY INTEGRO-DIFFERENTIAL EQUATIONS

IGOR CHEREVKO^∗and ANDREW DOROSH^∗

Abstract. The paper deals with existence, uniqueness and spline approximation of solutions to boundary value problems for delay integro-differential equations.

An iterative approximation scheme based on the use of cubic splines with defect two is presented, and sufficient conditions for its convergence are obtained.

MSC 2010. 34A45, 34K10, 34K28, 65L03, 65L10.

Keywords. Boundary value problems, integro-differential equations, delay, spline functions.

1. INTRODUCTION

Dynamic processes in many applied problems are described by delay dif- ferential and integral equations (Andreeva, Kolmanovsky and Shayhet 1992).

An analytical solution of such equation exists only in the simplest cases, so the construction and study of approximate algorithms for solutions of these equations are important.

In the present note we study an approximate method of solving boundary value problems for delay integro-differential equations based on approximation of the solution by cubic splines with defect two.

Existence and uniqueness of a solution of delay boundary value problems in various function spaces were considered by Grim and Schmitt (1968), Ka- mensky and Myshkis (1972), Biga and Gaber (2007), Athanasiadou (2013).

Applying spline functions for solving differential-difference equations was in- vestigated by Nikolova and Bainov (1981), Cherevko and Yakimov (1989), Nastasyeva and Cherevko (1999).

∗Faculty of Mathematics and Informatics, Yuriy Fedkovych Chernivtsi National Univer- sity, Ukraine, e-mail: [email protected],[email protected].

2. PROBLEM STATEMENT. EXISTENCE OF A SOLUTION

Let us consider the following boundary value problem

y⁰⁰(x) =f x, y(x), y(x−τ₀(x)), y⁰(x), y⁰(x−τ₁(x))+ (1)

+ Z b

a

g x, s, y(s), y(s−τ₀(s)), y⁰(s), y⁰(s−τ₁(s))ds, y⁽ⁱ⁾(x) =ϕ⁽ⁱ⁾(x), i= 0,1, x∈[a^∗;a], y(b) =γ,

(2) where

a^∗= minⁿ inf

x∈[a;b](x−τ0(x)), inf

x∈[a;b](x−τ1(x))^o, γ∈R, τ0(x)≥0, τ1(x)≥0.

Letf(x, u₀, u₁, v₀, v₁), g(x, s, u₀, u₁, v₀, v₁) be continuous functions inG= [a, b]×G²₁×G²₂ and Q = [a, b]×G, whereG1 ={u ∈R: |u|< P1}, G2 = {v ∈R: |v| ≤P₂}, P₁, P₂ are positive constants, ϕ(t) ∈ C¹[a^∗;a], delays τ₀(x) and τ₁(x) are continuous functions on [a, b], and additionally, τ₁(x) is such that the setE ={x_i∈[a, b] : x_i−τ₁(x_i) =a, i= 1, k}is finite.

We introduce the notations:

P = sup

f(x, u, u1, v, v1)+ Z b

a

g(x, s, u, u1, v, v1)ds:

|u_i|< P₁,|v_i|< P₂, i= 0,1, x, s∈[a, b]

, J = [a^∗;a], I = [a, b], I1= [a, x1], I2= [x1, x2], . . . , Ik= [xk−1, xk], Ik+1 = [xk, b],

B J∪I=ⁿy(x) :y(x)∈C(J∪I)∩ C¹(J)∪C¹(I)∩^k+1S

j=1

C²(I_j),

|y(x)| ≤P1, |y⁰(x)| ≤P2

o.

A function y = y(x) from the space B(J∪I) is called a solution of the problem (1)-(2) if it satisfies the equation (1) on [a;b] (with the possible ex- ception of the set E) and boundary conditions (2).

From the definition of the space B(J∪I) we conclude that the solution of the problem (1)-(2) is continuously differentiable for each x ∈ [a, b] where y⁰(a) is the right derivative.

Let us introduce a norm in the space B(J∪I):

kyk_B= max

8

(b−a)² max

x∈J∪I |y(x)|,_b−a² maxⁿmax

x∈J

y⁰(x),max

x∈I

y⁰(x)

o . The spaceB(J∪I) with this norm is a Banach space.

The boundary value problem (1)-(2) is equivalent to the following integral equation (Grim and Schmitt 1968; Kamensky and Myshkis 1972)

y(x) = Z b

a^∗

"

f s, y(s), y(s−τ₀(s)), y⁰(s), y⁰(s−τ₁(s))+ (3)

+ Z b

a

g s, ξ, y(ξ), y(ξ−τ0(ξ)), y⁰(ξ), y⁰(ξ−τ1(ξ))dξ

#

G¯(x, s)ds +l(x), x∈J∪I,

where G¯(x, s) =

(G(x, s), x, s∈I,

0, otherwise, l(x) =

(ϕ(x), x∈J,

γ−ϕ(a)

b−a (x−a) +ϕ(a), x∈I, and G(x, s) is the Green’s function of the following boundary value problem

y⁰⁰(x) = 0, x∈I, y(a) =y(b) = 0.

We define an operatorT in the spaceB(J∪I) in the following way (T y) (x) =

Z b a^∗

"

f s, y(s), y(s−τ0(s)), y⁰(s), y⁰(s−τ1(s))+ +

Z b a

g s, ξ, y(ξ), y(ξ−τ₀(ξ)), y⁰(ξ), y⁰(ξ−τ₁(ξ))dξ

#

G¯(x, s)ds +l(x), x∈J∪I.

Hence,

(T y)⁰(x) = Z b

a^∗

"

f s, y(s), y(s−τ₀(s)), y⁰(s), y⁰(s−τ₁(s))+ (4)

+ Zb

a

g s, ξ, y(ξ), y(ξ−τ₀(ξ)), y⁰(ξ), y⁰(ξ−τ₁(ξ))dξ

#

G¯⁰_x(x, s)ds +^γ−ϕ(a)_b−a , x∈J∪I.

Theorem 1. Let the following conditions hold:

1) max

maxx∈J |ϕ(x)|, ^(b−a)

2

8 P+ max (|ϕ(a)|,|γ|)

≤P1, 2) max

maxx∈J |ϕ⁰(x)|, ^b−a₂ P+^γ−ϕ(a)_b−a

≤P2,

3) the functions f(x, u₀, u₁, v₀, v₁), g(x, s, u₀, u₁, v₀, v₁) satisfy the Lip- schitz condition for variablesui, vi, i= 0,1 with constants L¹_j, L²_j, j= 1,4 in G and Q,

4) ^(b−a)₈ ²

2

P

j=1

L¹_j + (b−a)L²_j+^b−a₂

4

P

j=3

L¹_j+ (b−a)L²_j<1.

Then there exists a unique solution of the problem (1)-(2) in B(J∪I).

Proof. Based on Green’s function (Hartman 2002)

G(t, s) =







(s−a)(t−b)

b−a , a≤s≤t≤b,

(t−a)(s−b)

b−a , a≤t≤s≤b, we obtain the following estimates

(5)

Z b a

|G(t, s)|ds≤ ^(b−a)₈ ², Z b

a

G⁰_t(t, s)ds≤ ^b−a₂ .

When the conditions 1)-2) and the inequalities (5) are true the operatorT maps the space B(J∪I) on itself.

Let y1, y2 ∈B(J∪I). Considering the condition 3) and the estimates (5), we get:

|(T y₁) (t)−(T y2) (t)| ≤

≤ Z b

a^∗

L¹₁+L¹₂ max

t∈ J∪I |y₁(t)−y₂(t)|

+L¹₃+L¹₄max

maxt∈ I

y⁰₁(t)−y₂⁰ (t), max

t∈ J

y⁰₁(t)−y₂⁰ (t)

+ (b−a)L²₁+L²₂ max

t∈ J∪I |y₁(t)−y₂(t)|

+ (b−a)L²₃+L²₄max

max

t ∈I

y⁰₁(t)−y⁰₂(t),max

t∈ J

y⁰₁(t)−y⁰₂(t)

G¯(t, s)ds

≤ ^(b−a)₈ ² (b−a)²

8

L¹₁+L¹₂+ (b−a)L²₁+L²₂ +^b−a₂ L¹₃+L¹₄+ (b−a)L²₃+L²₄

ky₁−y₂k_B,

(T y1)⁰(t)−(T y2)⁰(t)≤

≤ ^b−a₂ (b−a)²

8

L¹₁+L¹₂+ (b−a)L²₁+L²₂ +^b−a₂ L¹₃+L¹₄+ (b−a)L²₃+L²₄

ky₁−y2k_B.

Based on the obtained estimates and on the definition of the norm in the space B(J∪I) we have:

k(T y₁) (t)−(T y2) (t)k_B≤ (6)

≤

"

(b−a)² 8

2

X

i=1

L¹_i + (b−a)L²_i+^b−a₂

4

X

i=3

L¹_i + (b−a)L²_i

#

ky₁−y2k_B. The inequality (6) and the condition 4) imply that the operator T is a contraction inB(J∪I) and it has a single fixed point in this space, therefore the boundary value problem (1)-(2) has a unique solution y(t) ∈ B(J∪I).

The proof is complete.

3. CUBIC SPLINES WITH DEFECT TWO

Let us consider an irregular grid ∆ ={a=x₀ < x₁ < . . . < x_n=b}on the segment [a;b], E ⊂ ∆. We denote by S(y, x) an interpolating cubic spline with defect two on ∆ which belongs to the spaceB(J∪I).

We can obtain a formula ofS(y, x) (Nikolova and Bainov 1981; Nastasyeva and Cherevko 1999; Dorosh and Cherevko 2014):

S(y, x) =M_j−1^{+ (xj−x)}

3

6hj +M_j^{−(x−xj−1)}

3

6hj +

yj−1−^M

+ j−1h²_j

6

xj−x hj

+

y_j−^M

− j h²_j

6

x−xj−1

hj , x∈[xj−1;x_j], h_j =x_j −xj−1, j= 1, n, (7)

where M_j⁺ = S⁰⁰(y, x_j + 0), j = 0, n−1, M_j⁻ = S⁰⁰(y, x_j−0), j = 1, n satisfy the following system of equations

h_j+1yj−1−(h_j+h_j+1)y_j+h_jy_j+1=

= ^hjh₆^j+1hjM_j−1⁺ + 2hjM_j⁻+ 2hj+1M_j⁺+hj+1M_j+1⁻ , j= 1, n−1, (8)

y₀ =ϕ(a), yn=γ.

We shall present the equations (8) in a matrix form Ay =BM+d,

(9) where

A=









−(h₁+h₂) h₁ 0 0 · · · 0 h3 −(h2+h3) h2 0 · · · 0

· · · · 0 0 · · · 0 hn −(hn−1+hn)









is an (n−1)×(n−1) matrix, d = (−h₂y0, 0, . . . , 0, −h_n−1yn)^T, B is a right side of the relations (8) coefficient matrix with dimensions(n−1)×2n,

M =M₀⁺, M₁⁻, M₁⁺, M₂⁻, M₂⁺, . . . , M_n−1⁻ , M_n−1⁺ , M_n^−T.

Lemma. The following correlations are true:

1) det (A) = (−1)ⁿ⁻¹h2h3. . . hn−1(b−a), (10)

2)kA⁻¹k ≤ _8h^K2₃ (b−a), (11)

3) max

1≤i<n−2 n−1

X

j=1

a⁻¹_i+1,j−a⁻¹_i,j≤ ^K2_2h^(b−a)₂ , (12)

4)kBk ≤H³, (13)

where h= min

i h_i, H= max

i h_i, K = ^H_h and a⁻¹_ij are elements of a matrixA⁻¹. The proof of the lemma statements is easy to obtain by applying the prin- ciple of mathematical induction and using the structure of the matrices A, B.

4. COMPUTATIONAL SCHEME

A) Choose a cubic splineSy⁽⁰⁾, xrandomly so that the boundary condi- tions (2) are enforced, for instance,Sy⁽⁰⁾, x= ^γ−ϕ(a)_b−a (x−a)+ϕ(a).

B) Using the original equation (1) and the spline Sy^(k), x, find M_j^+(k+1)=f

x_j, Sy^(k), x_j+ 0, Sy^(k), x_j −τ₀(x_j) + 0, (14)

S⁰y^(k), xj + 0, S⁰y^(k), xj−τ₁(xj) + 0

+ +

Z b a

g

x_j, s, Sy^(k), s, Sy^(k), s−τ₀(s), S⁰y^(k), s, S⁰y^(k), s−τ1(s)

ds, j= 0, n−1,

M_j^−(k+1)=f

xj, Sy^(k), xj−0, Sy^(k), xj −τ0(xj)−0, (15)

S⁰y^(k), x_j −0, S⁰y^(k), x_j−τ₁(x_j)−0

+ +

Z b a

g

x_j, s, Sy^(k), s, Sy^(k), s−τ₀(s), S⁰y^(k), s, S⁰y^(k), s−τ₁(s)

ds, j= 1, n.

In (14), (15) putS^(p)y^(k), t=ϕ^(p)(t), p= 0,1 ift < a.

C) Computeⁿy_j^k+1o, j= 0, n from the equations (8).

D) Build a cubic spline Sy^(k+1), x according to (7), using the values of ny^k+1_j ^o, M_j^+(k+1), M_j^−(k+1). This spline will be the next approxima- tion.

Let us denote

λ1=L¹₁+L¹₂+ (b−a)L²₁+L²₂, λ2=L¹₃+L¹₄+ (b−a)L²₃+L²₄, u= ^K₈⁵(b−a)²+^H₈², v= ^K₂⁵ (b−a) +^2H₃ .

(16)

Theorem2. Assume that the conditions of Theorem1hold. If the following inequality is true

(17) θ=uλ₁+vλ₂<1,

then there exists H^∗ > 0 such that for each 0 < H < H^∗ the sequence of splines ⁿSy^(k), x^o, k= 0,1. . . converges uniformly on [a;b].

Proof. The equation (10) implies that it is possible to construct an iterative spline sequence Sy^(k), x, k = 0,1, . . . using the scheme A)-D). We shall demonstrate that the series

S^(p)y⁽⁰⁾, x+

∞

X

i=1

hS^(p)y⁽ⁱ⁾, x−S^(p)y⁽ⁱ⁻¹⁾, xⁱ, p= 0,1

are uniformly convergent on [a;b] and thus the sequences S^(p)y^(k), x, k = 0,1, . . . , p= 0,1 are also uniformly convergent.

Let us define scalar functionsy(x), M(x) on [a;b] and denote the following vectors

y¯=y(x₁), . . . , y(xn−1)^T,

M¯ =M(x₀+ 0), M(x₁−0), M(x₁+ 0), . . . , M(xn−1−0), M(xn−1+ 0), M(x_n−0)

T

.

We shall write the iterative algorithm A)-D) in a matrix form y¯^(k+1)=A⁻¹BM¯^k+1+A⁻¹d,

(18)

where the vector ¯M components are defined according to (14)-(15) and the constant vectorddepends only on the boundary conditions (2).

From (18) we obtain the estimate

y^(k+1)−y^(k)=A⁻¹BM^k+1−A⁻¹BM^k (19)

≤A⁻¹kBkM¯^k+1−M¯^k.

From (14)-(15) and the properties of the functions f and g we obtain the following inequalities

M_j^+(k+1)−M_j^+(k)≤λ1max

x∈[a;b]

Sy^(k), x−Sy^(k−1), x (20)

+λ₂max

x∈[a;b]

S⁰y^(k), x−S⁰y^(k−1), x, j = 0,1, . . . , n−1,

M_j^−(k+1)−M_j^−(k)≤λ₁max

x∈[a;b]

Sy^(k), x−Sy^(k−1), x +λ₂max

x∈[a;b]

S⁰y^(k), x−S⁰y^(k−1), x, j = 1,2, . . . , n.

Therefore, taking into account the above mentioned lemma, (19) can be writ- ten in the following way

y^(k+1)−y^(k)≤ ^K₈⁵ (b−a)

λ₁Sy^(k), x−Sy^(k−1), x (21)

+λ2

S⁰y^(k), x−S⁰y^(k−1), x

. Letx∈[xj−1;xj]. Considering (7), we have

Sy^(k+1), x−Sy^(k), x≤^x_6h^j−x

j

(xj−x)²−h²_j+ + ^x−x_6h^j−1

j

(x−xj−1)²−h²_j M¯^k+1−M¯^k+ (22)

+y^k+1_j−1 −y_j^kxj_h^−x

j

+y_j^k+1−y_j^kx−x_h^j−1

j

. It is easy to show that

max

x∈[xj−1;xj]

xj−x 6hj

h²_j −(x_j−x)²+^x−x_6h^j−1

j

h²_j −(x−xj−1)²≤ ^H₈². (23)

Using (20), (21), (23), from (22) we obtain

Sy^(k+1), x−Sy^(k), x≤ ^H₈² M¯^k+1−M¯^k+y^(k+1)−y^(k) (24)

≤^K₈⁵(b−a)²+^H₈²

λ₁Sy^(k), x−Sy^(k−1), x+ +λ2

S⁰y^(k), x−S⁰y^(k−1), x

.

According to the spline (7), we get

S⁰y^(k+1), x−S⁰y^(k), x≤ (25)

≤

hj

6 −^(xj_2h^−x)2

j

M_j−1^+(k+1)−M_j−1^+(k) +

(x−xj−1)² 2hj −^h₆^j

M_j^−(k+1)−M_j^−(k)+_h¹

j

y^k+1_j −y_j−1^k+1−y^k_j −y^k_j−1. One can show that

max

x∈[xj−1;xj]

hj

6 − ^(xj_2h^−x)2

j

+

(x−xj−1)² 2hj −^h₆^j

≤ ^2H₃ , (26)

max

1<j<n

y^k+1_j −y_j−1^k+1−y^k_j −y_j−1^k ≤ ^K₂⁴ (b−a)HM¯^k+1−M¯^k. (27)

Due to (26)-(27), the inequality (25) implies that

S⁰y^(k+1), x−S⁰y^(k), x≤

≤^K₂⁵(b−a) +²₃H

λ₁Sy^(k), x−Sy^(k−1), x (28)

+λ2

S⁰y^(k), x−S⁰y^(k−1), x

.

After iterating (24), (28) and considering the notations (16)-(17) we obtain

Sy^(k+1), x−Sy^(k), x≤ (29)

≤uθ^k−1λ1

Sy⁽¹⁾, x−Sy⁽⁰⁾, x+λ2

S⁰y⁽¹⁾, x−S⁰y⁽⁰⁾, x,

S⁰y^(k+1), x−S⁰y^(k), x≤

≤vθ^k−1λ₁Sy⁽¹⁾, x−Sy⁽⁰⁾, x+λ₂S⁰y⁽¹⁾, x−S⁰y⁽⁰⁾, x. The correlations (29) with the condition (17) ensure the convergence of the sequencesⁿS^(p)y^(k), x^o, k= 0,1, . . . , p= 0,1. Theorem 2 is proved.

Let us denote lim

k→∞S^(p)y^(k), x = S^(p)(˜y, x), p = 0,1, . . . Note that the parametersM^f_j⁺,M^f_j⁻of the splineS(˜y, x) satisfy the system (8) and equations (14)-(15).

Let S(y, x) be a cubic spline with defect 2 which interpolates the solution y(x) of the boundary value problem (1)-(2). Thus,

S^(p)(˜y, x)−y^(p)(x)≤S^(p)(˜y, x)−S^(p)(y, x)

+S^(p)(y, x)−y^(p)(x), p= 0,1.

(30)

For the second term on the right side of the inequality (30) it is true (Alberg, Nilson and Walsh 1967) that

S^(p)(y, x)−y^(p)(x)≤K_pH^2−pω y⁰⁰(x), H, (31)

p= 0,1,2, K₀ = ⁵₂, K₁=K₂= 5, where ω(y⁰⁰(x), H) = max

1≤r≤k+1ωr(y⁰⁰(x), H), ωr(y⁰⁰(x), H) is a modulus of continuity fory⁰⁰(x) on I_r= [xr−1;x_r].

We shall denote max

x∈[a;b]

S^(p)(˜y, x)−S^(p)(y, x)=α_p, p= 0,1.

According to the properties of the functionsf,gand estimates (31), we obtain

M_j⁺−fx_j, y(x_j), y(x_j−τ₀(x_j)), y⁰(x_j), y⁰(x_j−τ₁(x_j))

− Z b

a

gx_j, s, y(s), y(s−τ₀(s)), y⁰(s), y⁰(s−τ₁(s))ds (32)

≤51 +¹₂λ1H²+λ2Hω y⁰⁰(x), H,

M_j⁻−fx_j, y(x_j), y(x_j−τ₀(x_j)), y⁰(x_j), y⁰(x_j−τ₁(x_j))

− Z b

a

gx_j, s, y(s), y(s−τ₀(s)), y⁰(s), y⁰(s−τ₁(s))ds (33)

≤51 +¹₂λ1H²+λ2Hω y⁰⁰(x), H.

Using the formulas of S(˜y, x), S(y, x) and the inequalities (32)-(33), one can get the following system of inequalities

α0 ≤uα0λ1+α1λ2+ 51 +¹₂λ1H²+λ2Hω y⁰⁰(x), H, α1 ≤vα0λ1+α1λ2+ 51 +¹₂λ1H²+λ2Hω y⁰⁰(x), H. (34)

Solving the system (34), we find estimates for the first terms on the right side of (30):

α₀ ≤ ⁵(1+¹₂λ1H²+λ2H)u

1−θ ω y⁰⁰(x), H, α1≤ ⁵(1+¹₂λ1H²+λ2H)v

1−θ ω y⁰⁰(x), H. Now the inequalities (30) can be written in the following form

S^(p)(˜y, x)−y^(p)(x)≤Kpω y⁰⁰(x), H, p= 0,1, (35)

whereK₀ = sup

H≤H^∗

_uµ

1−θ+ ^5H₂², K₁ = sup

H≤H^∗

_vµ

1−θ + 5H.

We can summarize the aforementioned arguments concerning accuracy of approximating the solution of the boundary value problem (1)-(2) based on the spline sequence as the following theorem.

Theorem3. Let the solution of the boundary value problem(1)-(2)exist, be unique and belong to the space B[a^∗;b]. If the condition(17) holds, then there exists H^∗ > 0 such that for any H < H^∗ the spline sequence ⁿSy^(k), x^o is approximating the solution of the boundary value problem (1)-(2) and the correlations (35) are true.

5. EXAMPLE

Let us consider the usage of this calculation scheme for finding an approxi- mate solution of the following boundary value problem

y⁰⁰(x) =−αy⁰ x−^π₂+ Z ^π

2 0

y(t−^π₂)dt+ cosx, 0≤x≤ ^π₂, y(x) = sin(x) + 1, −^π₂ ≤x <0,

y(0) = 1, y(^π₂) = 2 +α.

In this example L¹₁ =L¹₂ = L¹₃ = 0, L¹₄ = α, L²₁ = L²₃ = L²₄ = 0, L²₂ = 1, so λ1 = ^π₂, λ2 = α, h = H = ₄₀^π, K = 1, u = ^π₃₂² + ^H₈², v = ^π₄ + ²₃H, θ = ^π₃₂² +^H₈^2π₂ +^π₄ +²₃Hα. If we put α = ¹₄, then θ ≈ 0.695 < 1 and therefore the conditions of the Theorems 1 and 2 are satisfied. The precise solutionyp(x) of this boundary value problem, which was found using the step method, is

yp(x) =αsinx−cosx+ ^π₂ −1^x₂² +^π₄ 1−^π₂x+ 2.

x ya(x) yp(x) ∆ δ

0 1 1 0 0%

π

8 1.03971 1.03976 0.00005 0.01%

π

4 1.2935 1.2936 0.00010 0.01%

3π

8 1.7162 1.7163 0.00010 0.01%

π

2 2.25 2.25 0 0%

Table 1. Precise and approximate solutions.

The results of the calculation are given in Table 1, wherey_p(x) is the pre- cise solution, ya(x) is the approximate solution obtained with h= ₄₀^π after 2 iterations, ∆ is the absolute error and δ is the relative error.

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Received by the editors: June 17, 2015.