A computational method for solving a problem with parameter for linear systems of integro-differential equations

Abstract

This article presents a computational method for solving a problem with parameter for a system of Fredholm integro-differential equations. Some additional parameters are introduced and the problem under consideration is reduced to solving a system of linear algebraic equations. The coefficients and right-hand side of the system are calculated by solving the Cauchy problems for ordinary differential equations. We establish a criterion for the unique solvability of the problem under consideration. A numerical algorithm is offered for solving the problem with parameter. The results are illustrated by numerical examples.

1 Introduction

Control problems, also referred to as boundary value problems with parameters or as parameter identification problems, for ordinary differential and integro-differential equations have been extensively studied by many authors Akhmetov et al. (2002); Alimhan et al. (2015); Dauylbayev and Atakhan (2015); Dauylbaev and Mirzakulova (2017); Kiguradze (1987); Luchka and Nesterenko (2008); Nesterenko (2014); Ronto and Samoilenko (2000). Various methods have been applied to study these problems, such as methods of qualitative theory of differential equations, the calculus of variations and optimization theory, the method of upper and lower solutions, etc. However, there still remain open problems in obtaining effective criteria for the unique solvability of such problems and in developing numerical algorithms to find their optimal solutions.

Consider the following problem with a parameter for a system of Fredholm integro-differential equations with degenerate kernels:

$$\begin{aligned}&\frac{dx}{dt}{=}A(t)x {+} \sum \limits _{k=1}^m \int \limits ^{T}_{0} \varphi _k(t) \psi _k (s) x(s) ds {+} A_0(t) \mu + f(t), \quad x \in R^n, \ \mu \in R^l, \ t\in (0,T),\nonumber \\ \end{aligned}$$

(1)

$$\begin{aligned}&B_0 \mu + B x(0)+ C x(T)=d, \quad d\in R^{n+l}. \end{aligned}$$

(2)

Here the $(n\times n)$ matrices A(t), $\varphi _k(t)$, $\psi _k (\tau )$, $k=\overline{1,m}$, the $(n \times l)$ matrix $A_0(t)$ and the n vector f(t) are continuous on [0, T]; the $((n+l)\times l)$ matrix $B_0$ and the $((n+l)\times n)$ matrices B and C are constant; $\Vert x\Vert =\max \limits _{i=\overline{1,n}}|x_i|$.

By a solution to problem (1), (2) we mean a pair $(x^{*}(t), \mu ^{*})$, where $\mu ^{*} \in R^n$ and $x^{*}(t)$ is a continuous on [0, T] and continuously differentiable on (0, T) vector function satisfying the system of integro-differential Eq. (1) and boundary condition (2) for $\mu = \mu ^{*}$.

The aim of this paper was to establish a criterion for the unique solvability of problem with parameters (1), (2) and propose an algorithm for finding its solutions including its numerical implementation.

For this purpose, we use the parametrization method proposed by Dzhumabayev (1989). This is a constructive method originally developed to investigate and solve boundary value problems for ordinary differential equations. In Dzhumabayev (1989), coefficient criteria were established for the unique solvability of linear boundary value problems. An algorithm for finding their approximate solutions was developed. The method was later extended to boundary value problems, both linear and nonlinear, for various classes of equations. In particular, the parametrization method has been applied to problems for Fredholm integro-differential equations Dzhumabaev (2010, (2013); Dzhumabaev and Bakirova (2013); Dzhumabaev (2015, (2016) and linear boundary value problems with a parameter for ordinary differential equations Minglibayeva (2003); Minglibayeva and Dzhumabaev (2004).

The rest of this paper is organized as follows: Sect. 2 is devoted to the study of the unique solvability of problem (1), (2). We make a partition $\Delta _N$ of the interval [0, T] into N parts and take the values of a solution at the left-end points of the subintervals as additional parameters. We then obtain a special Cauchy problem for a system of integro-differential equations with parameters on the subintervals. The unique solvability of this problem is equivalent to the invertibility of a matrix $I -G(\Delta _N)$ composed of the fundamental matrix of the differential part and the kernel matrices of the integral term. We call a partition $\Delta _N$ regular if the matrix $I-G(\Delta _N)$ has an inverse. Then, assuming $\Delta _N$ to be regular, we construct a system of linear algebraic equations in parameters by using $[I-G(\Delta _N)]^{-1}$, boundary condition (2), and the continuity conditions of a solution at the interior partition points. It is established that the unique solvability of problem (1), (2) is equivalent to the invertibility of the matrix of the constructed system.

In Sect. 3, we develop an algorithm for finding a solution to problem (1), (2). For a chosen partition $\Delta _N$, the matrix $G(\Delta _N)$ is computed. If the matrix $I -G(\Delta _N)$ has an inverse, then we construct the above-mentioned system of linear algebraic equations. The components of $G(\Delta _N)$, the coefficients and right-hand sides of the system are determined by solving the Cauchy problems for ordinary differential equations and by calculating definite integrals of some known functions over the partition subintervals. Solving the system, we find the values of solution at the left-end points of subintervals. Using them and the initial data, we construct a function $F^{*}(t)$. Solving the Cauchy problem for the ordinary differential equations with the right-hand side $F^{*}(t)$, we find the values of the desired solution at the remaining points of [0, T]. Section 3 also provides the numerical implementation of the algorithm. Note that the elements of matrix $G(\Delta _N)$, the coefficients and right-hand side of the system of algebraic equations in parameters can be evaluated by the parallel computing on the partition subintervals. Section 4 presents numerical examples to demonstrate the effectiveness of the proposed method.

2 The unique solvability of the problem with parameter

Let us take a partition $\Delta _N$ of the interval [0, T] by points $t_0 =0< t_1< ... < t_N = T$.

We introduce the following notation:

$C([0,T],R^n)$ is the space of continuous functions $x:[0,T] \rightarrow R^n$ with the norm

$\qquad ||x||_1 = \max \limits _{t\in [0,T]}||x(t)||$;

$C([0,T], \Delta _N, R^{nN})$ is the space of function systems $x[t] = (x_1(t), x_2(t),\ldots , x_N(t))$, where $x_r: [t_{r-1}, t_r) \rightarrow R^n$, $r =\overline{1,N}$, are continuous functions having finite left-sided limits $\lim \limits _{t\rightarrow t_r-0} x_r(t)$, with the norm $||x[\cdot ]||_2 = \max \limits _{r =\overline{1,N}} \sup \limits _{t\in [t_{r-1}, t_r)} ||x_r(t)||.$

Suppose that x(t) is a solution to problem (1), (2) and denote by $x_r(t)$ the restriction of x(t) to the rth subinterval of the partition, i.e. $x_r(t) = x(t)$ for $t\in [t_{r-1}, t_r)$, $r =\overline{1,N}$. We introduce additional parameters $\lambda _r$, $r =\overline{1,N}$, as the values of a solution x(t) at the left endpoints of the partition subintervals: $\lambda _r =x_r(t_{r-1})$. We then compose the vector $\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _N,\lambda _{N+1})$, whose last component is the parameter $\mu$ included in problem (1), (2), i.e. $\lambda _{N+1} =\mu$.

On each partition subinterval, we make the substitution $u_r(t)=x_r(t)-\lambda _r$, $t\in [t_{r-1}, t_r)$, $r =\overline{1,N}$. The problem (1), (2) is then transformed into the multipoint boundary value problem with parameters

$$\begin{aligned} \frac{du_r}{dt}&= A(t)(u_r+\lambda _r)+\sum \limits _{j=1}^{N}\sum \limits _{k=1}^{m}\int \limits ^{t_j}_{t_{j-1}}\varphi _k(t)\psi _k(s)[u_j(s)\nonumber \\&\quad + \lambda _j] ds + A_0(t)\lambda _{N+1} + f(t),\quad t \in [t_{r-1},t_r), \end{aligned}$$

(3)

$$\begin{aligned}&u_r(t_{r-1})= 0, \qquad r= \overline{1,N}, \end{aligned}$$

(4)

$$\begin{aligned}&B_0 \lambda _{N+1} + B \lambda _1 + C\lambda _N + C\lim \limits _{t\rightarrow T-0} u_N(t) = d, \end{aligned}$$

(5)

$$\begin{aligned}&\lambda _{p} + \lim \limits _{t\rightarrow t_p-0} u_{p}(t) - \lambda _{p+1}=0, \qquad p=\overline{1,N-1}, \end{aligned}$$

(6)

where conditions (6) are imposed to ensure the continuity of a solution to (1), (2) at the interior points of the partition $\Delta _N.$ Note that conditions (6) in conjunction with integro-differential equations (3) also ensure the continuity of the derivative of a solution at these points.

A solution to problem (3)–(6) is a pair $(u^{*}[t],\lambda ^{*})$, where $u^{*}[t] =\bigl (u_1^{*}(t),u_2^{*}(t),\ldots , u_N^{*}(t)\bigr )\in C([0,T],\Delta _N,R^{nN})$ with continuously differentiable on $[t_{r-1}, t_r)$ components $u_r^{*}(t)$ and $\lambda ^{*}=(\lambda _1^{*},\lambda _2^{*},\ldots ,\lambda _N^{*}, \lambda _{N+1}^{*})\in R^{nN+l},$ satisfying the system of integro-differential equations (3), initial conditions (4), and relations (5), (6).

The problems (1), (2) and (3)–(6) are equivalent. Indeed, if a pair $(x^{*}(t),\mu ^{*})$ is a solution to problem (1), (2), then the pair $(u^{*}[t],\lambda ^{*})$ composed of the components $u_r^{*}(t)=x^{*}(t)-x^{*}(t_{r-1}),$$t\in [t_{r-1},t_r),$$\lambda _r^{*}=x^{*}(t_{r-1}),$$r=\overline{1,N}$, $\lambda _{N+1}^{*} = \mu ^{*}$, is a solution to problem (3)–(6). Conversely, if a pair $(\widetilde{u}[t], \widetilde{\lambda })$, with elements $\widetilde{u}[t]\in C([0,T],\Delta _N,R^{nN})$ and $\widetilde{\lambda }\in R^{nN+l}$, is a solution to problem (3)–(6), then the pair $(\widetilde{x}(t), \widetilde{\mu })$ defined by the equalities $\widetilde{x}(t)=\widetilde{u}_r(t)+ \widetilde{\lambda }_r,$$t\in [t_{r-1},t_r),$$r=\overline{1,N}$, $\widetilde{x}(T)=\lim \limits _{t\rightarrow T-0}\widetilde{u}_N(t) + \widetilde{\lambda }_N$, and $\ \widetilde{\mu } = \widetilde{\lambda }_{N+1}$, is a solution to the original problem (1), (2).

For fixed $\lambda _r$, $r=\overline{1,N+1}$, Eqs. (3) and (4) form a special Cauchy problem for the system of Fredholm integro-differential equations.

Consider the system of differential equations:

$$\begin{aligned} \frac{du_r}{dt} = A(t)u_r + g(t), \end{aligned}$$

(7)

subject to the condition

$$\begin{aligned} u_r(t_{r-1}) = u^0_r, \end{aligned}$$

(8)

where g(t) is a continuous on $[t_{r-1},t_r]$ function and $u^0_r$ is a constant vector.

By a fundamental matrix $X_r(t)$ of

$$\begin{aligned} \frac{du_r}{dt} = A(t)u_r \end{aligned}$$

(9)

or

$$\begin{aligned} \frac{dX_r}{dt} = A(t)X_r \end{aligned}$$

(10)

is meant a solution of (10) such that $det X_r(t) \ne 0$.

If $X_r(t)$ is a solution of (10) and c is a constant vector, the principle of superposition states that

$$\begin{aligned} u_r(t) = X_r(t)c \end{aligned}$$

(11)

is a solution of (9). Furthermore, if $X_r(t)$ is a fundamental solution of (10), then every solution of (9) subject to (8) is of the form (11) with $c = X^{-1}_r(t_{r-1})u_r(t_{r-1})$ (see Hartman (1964, p.47)), that is,

$$\begin{aligned} u_r(t) = X_r(t) X^{-1}_r(t_{r-1})u_r(t_{r-1}). \end{aligned}$$

(12)

By Corollary 2.1 Hartman (1964, p.48), the solution to the initial-value problem (7), (8) can be represented by the Cauchy formula

$$\begin{aligned} u_r(t) = X_r(t) c + X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau )g(\tau )d \tau . \end{aligned}$$

Taking into account (12), we get that the initial-value problem (7), (8) is equivalent to the system of integral equations

$$\begin{aligned} u_r(t) = X_r(t) X^{-1}_r(t_{r-1})u^0_r + X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau )g(\tau )d \tau . \end{aligned}$$

Setting $g(t) =A(t)\lambda _r +\sum \limits _{j=1}^{N}\sum \limits _{k=1}^{m}\int \limits ^{t_j}_{t_{j-1}}\varphi _k(t)\psi _k(s)[u_j(s) + \lambda _j] ds + A_0(t)\lambda _{N+1} + f(t)$ and $u^0_r=0$, we get that the special Cauchy problem (3), (4) is reduced to the equivalent system of integral equations

$$\begin{aligned} u_r(t)= & {} X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau )A(\tau )d \tau \lambda _r \nonumber \\&+ X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau )\sum \limits _{j=1}^{N} \sum \limits _{k=1}^{m} \int \limits _{t_{j-1}}^{t_j} \varphi _{k}(\tau )\psi _k(s)[u_j(s)+\lambda _j]ds d\tau +\nonumber \\&+ X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau )A_0(\tau ) d\tau \lambda _{N+1}\nonumber \\&+ X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau )f(\tau ) d\tau , \quad t \in [t_{r-1},t_r), \quad r=\overline{1,N}. \end{aligned}$$

(13)

Let us set $\xi _k=\sum \limits _{j=1}^{N} \int \limits _{t_{j-1}}^{t_j}\psi _k(s)u_j(s)ds$, $k=\overline{1,m}$, and rewrite system (13) in the following form:

$$\begin{aligned} u_r(t)= & {} \sum \limits _{k=1}^{m}X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau ) \varphi _{k}(\tau )d\tau \xi _k + X_r(t)\int \limits _{t_{r-1}}^{t}X_r^{-1}(\tau ) \Big [ A(\tau ) \lambda _r +\nonumber \\&+ \sum \limits _{k=1}^{m} \varphi _{k}(\tau )\sum \limits _{j=1}^{N} \int \limits _{t_{j-1}}^{t_j} \psi _k(s)ds\lambda _j {+} A_0(\tau ) \lambda _{N+1} {+} f(\tau )\Big ]d\tau , \quad t \in [t_{r-1},t_r), \quad r{=}\overline{1,N}.\nonumber \\ \end{aligned}$$

(14)

Multiplying both sides of (14) by $\psi _p(t)$, integrating them over the interval $[t_{r-1},t_r]$, and summing up with respect to r, we obtain the following system of linear algebraic equations in $\xi =(\xi _1,\ldots ,\xi _m)\in R^{nm}$:

$$\begin{aligned} \xi _p=\sum \limits _{k=1}^m G_{p,k}(\Delta _N)\xi _k+\sum \limits ^{N+1}_{r=1}V_{p,r}(\Delta _N)\lambda _r + g_p(f,\Delta _N), \quad p=\overline{1,m}, \end{aligned}$$

(15)

with the $(n\times n)$ matrices

$$\begin{aligned} G_{p,k}(\Delta _N)= & {} \sum \limits _{r=1}^{N}\int \limits _{t_{r-1}}^{t_r} \psi _p(\tau ) X_r(\tau )\int \limits _{t_{r-1}}^{\tau } X_r^{-1}(s)\varphi _k(s) ds d\tau , \quad k=\overline{1,m},\end{aligned}$$

(16)

$$\begin{aligned} V_{p,r}(\Delta _N)= & {} \int \limits _{t_{r-1}}^{t_r} \psi _p(\tau ) X_r(\tau )\int \limits _{t_{r-1}}^{\tau } X^{-1}_r(s)A(s) ds d\tau + \nonumber \\&+\sum \limits _{j=1}^{N}\sum \limits _{k=1}^{m}\int \limits _{t_{j-1}}^{t_j} \psi _p(\tau ) X_j(\tau )\int \limits _{t_{j-1}}^{\tau } X_j^{-1}(\tau _1)\varphi _k(\tau _1) d\tau _1 d\tau \int \limits _{t_{r-1}}^{t_r}\psi _k(s)ds, \quad r=\overline{1,N},\nonumber \\ \end{aligned}$$

(17)

the $(n\times l)$ matrices

$$\begin{aligned} V_{p,N+1}(\Delta _N)= \sum \limits _{r=1}^{N}\int \limits _{t_{r-1}}^{t_r} \psi _p(\tau ) X_r(\tau )\int \limits _{t_{r-1}}^{\tau }X_r^{-1}(s) A_0(s) ds d\tau , \end{aligned}$$

(18)

and the vectors of dimension n

$$\begin{aligned} g_{p}(f,\Delta _N)= \sum \limits _{r=1}^{N}\int \limits _{t_{r-1}}^{t_r} \psi _p(\tau ) X_r(\tau )\int \limits _{t_{r-1}}^{\tau } X_r^{-1}(s)f(s)ds d\tau , \qquad p=\overline{1,m}. \end{aligned}$$

(19)

Using the matrices $G_{p,k}(\Delta _N)$ and $V_{p,r}(\Delta _N)$, we construct the matrices $G(\Delta _N)=(G_{p,k}(\Delta _N)),$$p,k=\overline{1,m},$ and $V(\Delta _N)=(V_{p,r}(\Delta _N)),$$p=\overline{1,m},$$r=\overline{1,N+1}$. Then the system (15) becomes

$$\begin{aligned} {[}I-G(\Delta _N)]\xi = V(\Delta _N)\lambda + g(f,\Delta _N), \end{aligned}$$

(20)

where I is the identity matrix of order nm and $g(f,\Delta _N)=(g_1(f,\Delta _N),\ldots , g_m(f,\Delta _N))\in R^{nm}.$

Definition 2.1

A partition $\Delta _N$ is called regular if the matrix $I-G(\Delta _N)$ is invertible.

Definition 2.2

The special Cauchy problem (3), (4) is called uniquely solvable if it has a unique solution for any $\lambda \in R^{nN+l}$ and $f(t)\in C([0,T],R^n)$.

Thus, the special Cauchy problem (3), (4) is equivalent to the system of integral Eq. (13). This system, due to the kernel degeneracy, is equivalent to the system of algebraic Eq. (15) in $\xi = (\xi _1,\ldots , \xi _m)\in R^{nm}.$ Therefore, the special Cauchy problem is uniquely solvable if and only if the partition $\Delta _N,$ generating this problem, is regular.

Let $\sigma (m,[0,T])$ denote the set of regular partitions $\Delta _N$ of [0, T] for the Eq. (1).

Since the special Cauchy problem is uniquely solvable for a partition with a sufficiently small step size $h>0$ (see Dzhumabaev (2010), p.1152), the set $\sigma (m,[0,T])$ is not empty.

Take a partition $\Delta _N\in \sigma (m,[0,T])$ and represent the matrix $[I-G(\Delta _N)]^{-1}$ in the form

$$\begin{aligned} {[}I-G(\Delta _N)]^{-1}=\Big (M_{k,p}(\Delta _N)\Big ), \qquad k, p=\overline{1,m}, \end{aligned}$$

where $M_{k,p}(\Delta _N)$ are square matrices of order n. Then, in view of (20), the elements of the vector $\xi \in R^{nm}$ can be determined by the equalities

$$\begin{aligned} \xi _k=\sum \limits _{j=1}^{N+1}\Big (\sum \limits _{p=1}^{m} M_{k,p}(\Delta _N)V_{p,j}(\Delta _N)\Big ) \lambda _j + \sum \limits ^m_{p=1}M_{k,p}(\Delta _N)g_p(f,\Delta _N), \quad k=\overline{1,m}. \end{aligned}$$

(21)

By replacing $\xi _k$ in (14) with the right-hand side of (21), we get the following representation of the functions: $u_r(t)$ via $\lambda _j,$$j=\overline{1,N+1}:$

$$\begin{aligned} u_r(t)= & {} \sum \limits _{j=1}^{N} \biggl \{ \sum \limits _{k=1}^m X_r(t)\int \limits _{t_{r-1}}^{t} X_r^{-1}(\tau )\varphi _k(\tau ) d\tau \biggl [\sum \limits _{p=1}^{m} M_{k,p}(\Delta _N)V_{p,j}(\Delta _N)+\int \limits _{t_{j-1}}^{t_j} \psi _k(s)ds\biggr ]\biggr \} \lambda _j+\nonumber \\&+ X_r(t)\int \limits _{t_{r-1}}^{t} X_r^{-1}(\tau )A(\tau ) d\tau \lambda _r +\nonumber \\&+ X_r(t)\int \limits _{t_{r-1}}^{t} X_r^{-1}(\tau ) \biggl [\sum \limits _{k=1}^m \varphi _k(\tau ) \sum \limits _{p=1}^{m} M_{k,p}(\Delta _N)V_{p,N+1}(\Delta _N) + A_0(\tau )\biggr ] d\tau \lambda _{N+1} +\nonumber \\&+ X_r(t)\int \limits _{t_{r-1}}^{t} X_r^{-1}(\tau )\Big [\sum \limits _{k=1}^m\varphi _k(\tau )\sum \limits _{p=1}^m M_{k,p}(\Delta _N)g_{p}(f,\Delta _N)\nonumber \\&+f(\tau )\Big ]d\tau ,\quad t \in [t_{r-1},t_r), \quad r=\overline{1,N}. \end{aligned}$$

(22)

We introduce the following notation:

$$\begin{aligned} D_{r,j}(\Delta _N)= & {} \sum \limits _{k=1}^m X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau )\varphi _k(\tau )d\tau \biggl [\sum \limits _{p=1}^m M_{k,p}(\Delta _N)V_{p,j}(\Delta _N)+\int \limits _{t_{j-1}}^{t_j}\psi _k(s)ds\biggr ],\nonumber \\&j\ne r, \qquad r,j=\overline{1,N}, \end{aligned}$$

(23)

$$\begin{aligned} D_{r,r}(\Delta _N)= & {} \sum \limits _{k=1}^m X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau ) \varphi _k(\tau )d\tau \biggl [\sum \limits _{p=1}^m M_{k,p}(\Delta _N)V_{p,r}(\Delta _N)+\int \limits _{t_{r-1}}^{t_r}\psi _k(s)ds\biggr ]+\nonumber \\&+ \sum \limits _{k=1}^m X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau )A(\tau ) d\tau , \end{aligned}$$

(24)

$$\begin{aligned} D_{r,N+1}(\Delta _N)= & {} \sum \limits _{k=1}^m X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau ) \varphi _k(\tau ) d\tau \sum \limits _{p=1}^{m} M_{k,p}(\Delta _N)V_{p,N+1}(\Delta _N) +\nonumber \\&+ X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau ) A_0(\tau ) d\tau , \end{aligned}$$

(25)

$$\begin{aligned} F_{r}(\Delta _N)= & {} \sum \limits _{k=1}^m X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau ) \varphi _k(\tau ) d\tau \sum \limits _{p=1}^m M_{k,p}(\Delta _N)g_{p}(f,\Delta _N)+\nonumber \\&+\sum \limits _{k=1}^m X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau )f(\tau ) d\tau , \quad r=\overline{1,N}. \end{aligned}$$

(26)

Then from (22) we get

$$\begin{aligned} \lim \limits _{t\rightarrow t_r-0}u_r(t)=\sum \limits _{j=1}^{N+1} D_{r,j}(\Delta _N)\lambda _j+F_r(\Delta _N). \end{aligned}$$

(27)

If we substitute the right-hand side of (27) into the boundary condition (5) and the continuity condition (6), we obtain the following system of linear algebraic equations in parameters $\lambda _r,$$r=\overline{1,N+1}:$

$$\begin{aligned}&{[}B+CD_{N,1}(\Delta _N)]\lambda _1+\sum \limits _{j=2}^{N-1}CD_{N,j}(\Delta _N)\lambda _j+C[I+D_{N,N}(\Delta _N)]\lambda _N +\nonumber \\&\quad + [B_0 +C D_{N,N+1}(\Delta _N)]\lambda _{N+1} = d-CF_N(\Delta _N), \end{aligned}$$

(28)

$$\begin{aligned}&{[}I+D_{p,p}(\Delta _N)]\lambda _p-[I-D_{p,p+1}(\Delta _N)]\lambda _{p+1}\nonumber \\&\quad +\sum \limits _{{}^{\,\,\,\,\,\,\,\,\,\,j=1}_{j \ne p, \, j \ne p+1}}^{N+1}D_{p,j}(\Delta _N)\lambda _j=-F_p(\Delta _N),\quad p=\overline{1,N-1}. \end{aligned}$$

(29)

Let $Q_{*}(\Delta _N)$ denote the matrix corresponding to the left-hand side of this system. Then we can represent (28), (29) as follows:

$$\begin{aligned} Q_{*}(\Delta _N)\lambda =- F_{*}(\Delta _N), \quad \lambda \in R^{nN+l}, \end{aligned}$$

(30)

where $F_{*}(\Delta _N)=\Big (-d+CF_{N}(\Delta _N), F_{1}(\Delta _N), \ldots , F_{N-1}(\Delta _N)\Big )\in R^{nN+l}.$

Lemma 2.1

For $\Delta _N\in \sigma (m,[0,T])$ the following assertions hold:

(i) the vector $\lambda ^{*}=(\lambda _1^{*},\ldots ,\lambda _{N+1}^{*})\in R^{nN+l},$ composed of the values of a solution $(x^{*}(t), \mu ^{*})$ to problem (1), (2) at the partition points $\lambda _r^{*} = x^{*}(t_{r-1}),$$r=\overline{1,N}$, and $\lambda _{N+1}^{*} = \mu ^{*}$, satisfies the system (30);

(ii) if $\widetilde{\lambda }=(\widetilde{\lambda }_1,\ldots , \widetilde{\lambda }_{N+1})\in R^{nN+l}$ is a solution to system (30) and the function system $\widetilde{u}[t]=(\widetilde{u}_{1}(t),\ldots ,\widetilde{u}_{N}(t))$ is a solution to the special Cauchy problem (3), (4) with $\lambda _r=\widetilde{\lambda }_r,$$r=\overline{1,N+1},$ then the pair $(\widetilde{x}(t), \widetilde{\mu })$, where the function $\widetilde{x}(t)$ and the parameter $\widetilde{\mu }$ are defined by the equalities:

$$\begin{aligned} \widetilde{x}(t)=\widetilde{\lambda }_r+\widetilde{u}_r(t),\quad t\in [t_{r-1},t_r),\quad r=\overline{1,N},\qquad \widetilde{x}(T){=}\widetilde{\lambda }_N+\lim \limits _{t\rightarrow T-0}\widetilde{u}_N(t),\qquad \widetilde{\mu } {=} \widetilde{\lambda }_{N+1}, \end{aligned}$$

is a solution to problem (1), (2).

The proof of Lemma 2.1 is similar to that of Lemma 1 in Dzhumabaev (2010, p. 1155).

Let us introduce the following notation:

$$\begin{aligned} \alpha= & {} \max \limits _{t\in [0,T]}\Vert A(t)\Vert , \quad \alpha _0 =\max \limits _{t\in [0,T]}\Vert A_0(t)\Vert ,\quad \overline{\omega }=\max \limits _{r=\overline{1,N}}(t_r-t_{r-1}),\\ \overline{\varphi }(m)= & {} \max \limits _{r=\overline{1,N}}\int \limits _{t_{r-1}}^{t_r} \sum \limits _{k=1}^{m} \Vert \varphi _k(t)\Vert dt, \quad \overline{\psi }(T)= \max \limits _{p=\overline{1,m}}\int \limits _{0}^{T} \Vert \psi _p(t)\Vert dt. \end{aligned}$$

Theorem 2.1

Let $\Delta _N\in \sigma (m,[0,T])$ and the matrix ${Q}_{*}(\Delta _N):R^{nN+l}\rightarrow R^{nN+l}$ be invertible. Then problem (1), (2) has a unique solution $(x^{*}(t), \mu ^{*})$ for any $f(t)\in C([0,T],R^{n}),$$d\in R^{n+l},$ and the estimate

$$\begin{aligned} \max (\Vert x^{*}\Vert _1, ||\mu ^{*}||) \le \mathcal {N}(m,\Delta _N)\max (\Vert d\Vert ,\Vert f\Vert _1), \end{aligned}$$

(31)

holds, where

$$\begin{aligned} \mathcal {N}(m,\Delta _N)= & {} e^{\alpha \overline{\omega }}\Big \{\overline{\varphi }(m) \Big [\Vert [I-G(\Delta _N)]^{-1}\Vert \cdot \overline{\psi }(T) \Big (e^{\alpha \overline{\omega }}-1+ e^{\alpha \overline{\omega }} \cdot \overline{\varphi }(m)\cdot \overline{\psi }(T)\Big )+\nonumber \\&+ \overline{\psi }(T)+ \alpha _0 \Big ]+ 1 + \alpha _0 \Big \} \gamma _{*}(\Delta _N) (1+\Vert C\Vert )\max \Big \{1,\overline{\omega }e^{\alpha \overline{\omega }}\Big [1+ e^{\alpha \overline{\omega }}\cdot \overline{\varphi }(m)\cdot \nonumber \\&\Vert [I-G(\Delta _N)]^{-1}\Vert \cdot \overline{\psi }(T)\Big \}+ e^{\alpha \overline{\omega }}\overline{\omega }\Big [\overline{\varphi }(m)\cdot \Vert [I-G(\Delta _N)]^{-1}\Vert \cdot \overline{\psi }(T)\cdot e^{\alpha \overline{\omega }}+1\Big ].\nonumber \\ \end{aligned}$$

(32)

Definition 2.3

Problem (1), (2) is said to be well-posed if it has a unique solution $(x(t),\mu )$ for any pair (f(t), d),  with $f(t)\in C([0,T],R^n)$ and $d\in R^{n+l},$ and the following inequality holds:

$$\begin{aligned} \max (\Vert x\Vert _1, ||\mu ||) \le K\max (\Vert f\Vert _1,\Vert d\Vert ), \end{aligned}$$

where K is a constant, independent of f(t) and d.

Theorem 2.2

Problem (1), (2) is well-posed if and only if for any $\Delta _N \in \sigma (m,[0,T])$ the matrix $Q_{*}(\Delta _N):$$R^{nN+l}\rightarrow R^{nN+l}$ is invertible.

The proofs of Theorems 2.1 and 2.2 repeat with minor changes in the proofs of Theorems 2.1 and 2.2 in Dzhumabaev (2016, pp. 347–349).

3 An algorithm for solving problem (1), (2) and its numerical realization

An essential part of the proposed algorithm is solving auxiliary Cauchy problems for ordinary differential equations on the partition subintervals:

$$\begin{aligned} \frac{dx}{dt} = A(t)x + P(t), \qquad x(t_{r-1})=0, \quad t\in [t_{r-1},t_r],\quad r=\overline{1,N}. \end{aligned}$$

(33)

Here P(t) is either a square matrix of order n or a vector of dimension n, both continuous on $[t_{r-1},t_r],$$r=\overline{1,N}$. Hence a solution to problem (33) is either a square matrix or a vector. Let $E_{*,r}(A(\cdot ),P(\cdot ),t)$ denote such a solution. We then have

$$\begin{aligned} E_{*,r}(A(\cdot ),P(\cdot ),t)=X_r(t)\int \limits _{t_{r-1}}^{t} X^{-1}(\tau )P(\tau )d\tau , \quad t\in [t_{r-1},t_r], \end{aligned}$$

(34)

where $X_r(t)$ is a fundamental matrix of a homogeneous differential equation corresponding to (33) on the r-th subinterval.

An appropriate choice of a regular partition is another important part of the algorithm. We can start with $\Delta _1,$ when the interval[0; T] is not partitioned.

Let us now formulate the Algorithm for solving problem (1), (2).

I. Choose a partition $\Delta _N, N=1,2,\ldots$.

II. Solve the $N\cdot m$ auxiliary Cauchy problems for matrix ordinary differential equations

$$\begin{aligned} \frac{dx}{dt} = A(t)x + \varphi _k(t), \qquad x(t_{r-1})=0, \quad t\in [t_{r-1},t_r], \end{aligned}$$

(35)

to get the matrix functions

$$\begin{aligned} E_{*,r}(A(\cdot ),\varphi _k(\cdot ),t), \quad t\in [t_{r-1},t_r], \quad r=\overline{1,N}, \quad k=\overline{1,m}. \end{aligned}$$

(36)

III. Multiply each $(n\times n)$ matrix (36) by the $(n\times n)$ matrix $\psi _p(t),$$p=\overline{1,m}$, and integrate the product over $[t_{r-1},t_r]:$

$$\begin{aligned} \widehat{\psi }_{p,r}(\varphi _k)=\int \limits _{t_{r-1}}^{t_r}\psi _p(t) E_{*,r}(A(\cdot ),\varphi _k(\cdot ),t)dt. \end{aligned}$$

(37)

Summing up (37) with respect to r,  we obtain the $(n\times n)$ matrices

$$\begin{aligned} G_{p,k}(\Delta _N)=\sum \limits _{r=1}^N\widehat{\psi }_{p,r}(\varphi _k), \quad p,k=\overline{1,m}, \end{aligned}$$

which follows from (18) and (34).

Compose the $(nm\times nm)$ matrix $G(\Delta _N)=(G_{p,k}(\Delta _N)),$$p,k=\overline{1,m}$, and check whether the matrix $[I-G(\Delta _N)]: R^{nm}\rightarrow R^{nm}$ is invertible.

If so, find its inverse and represent it in the form $[I-G(\Delta _N)]^{- 1}=(M_{p,k}(\Delta _N)),$ where $M_{p,k}(\Delta _N))$ are square matrices of order n, $p,k=\overline{1,N}$. Then move on to the next step of Algorithm.

If there is no inverse of $[I-G(\Delta _N)]$, i.e. the partition $\Delta _N$ is not regular, then take a new partition of interval [0, T], and the algorithm starts over. A simple way for selecting a new partition is to choose the partition $\Delta _{2N},$ where each interval of the partition $\Delta _N$ is divided into two parts.

IV. By solving again the auxiliary Cauchy problem for ordinary differential equations

$$\begin{aligned} \frac{dx}{dt}= & {} A(t)x + A(t), \quad x(t_{r-1})=0, \quad t\in [t_{r-1},t_r],\\ \frac{dx}{dt}= & {} A(t)x + A_0(t), \quad x(t_{r-1})=0, \quad t\in [t_{r-1},t_r],\\ \frac{dx}{dt}= & {} A(t)x +f(t), \quad x(t_{r-1})=0, \quad t\in [t_{r-1},t_r],\quad r=\overline{1,N}, \end{aligned}$$

find their respective solutions $E_{*,r}(A(\cdot ),A(\cdot ),t)$, $E_{*,r}(A(\cdot ),A_0(\cdot ),t),$ and $E_{*,r}(A(\cdot ),f(\cdot ),t),$$r=\overline{1,N}.$

V. Evaluate the integrals

$$\begin{aligned} \widehat{\psi }_{p,r}= & {} \int \limits _{t_{r-1}}^{t_r}\psi _p(t)dt, \quad \widehat{\psi }_{p,r}(A)=\int \limits _{t_{r-1}}^{t_r}\psi _p(t)E_{*,r}(A(\cdot ),A(\cdot ),t)dt,\\ \widehat{\psi }_{p,r}(A_0)= & {} \int \limits _{t_{r-1}}^{t_r}\psi _p(t)E_{*,r}(A(\cdot ),A_0(\cdot ),t)dt, \quad \widehat{\psi }_{p,r}(f)=\int \limits _{t_{r-1}}^{t_r}\psi _p(t)E_{*,r}(A(\cdot ),f(\cdot ),t)dt. \end{aligned}$$

By equalities (19), (20), and (34), determine the $(n\times n)$ matrices

$$\begin{aligned} V_{p,r}(\Delta _N)=\widehat{\psi }_{p,r}(A)+\sum \limits _{j=1}^N\sum \limits _{k=1}^m \widehat{\psi }_{p,j}(\varphi _k)\cdot \widehat{\psi }_{k,r}, \quad r=\overline{1,N}, \end{aligned}$$

the $(n\times l)$ matrices

$$\begin{aligned} V_{p,N+1}(\Delta _N)= \sum \limits _{r=1}^N \widehat{\psi }_{p,r}(A_0), \quad p=\overline{1,m}, \end{aligned}$$

and the n vectors

$$\begin{aligned} g_{p}(f,\Delta _N)=\sum \limits _{r=1}^N\widehat{\psi }_{p,r}(\Delta _N), \quad p=\overline{1,m},\quad r=\overline{1,N}. \end{aligned}$$

VI. Form the system of linear algebraic equations in parameters

$$\begin{aligned} Q_{*}(\Delta _N)\lambda =-F_{*}(\Delta _N), \quad \lambda \in R^{nN+l}. \end{aligned}$$

(38)

The elements of the matrix $Q_{*}(\Delta _N):$$R^{nN+l}\rightarrow R^{nN+l}$ and the vector $F_{*}(\Delta _N)=(-d+CF_N(\Delta _N),F_1(\Delta _N),\ldots ,F_{N-1}(\Delta _N))\in R^{nN+l}$ are defined by the equalities (23), (24), (25), and (26), where, in view of (34), we replace

$$\begin{aligned} X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau )\varphi _k(\tau )d\tau \quad \text{ and } \quad X_r(t_r)\int \limits _{t_{r-1}}^{t_r} X_r^{-1}(\tau )f(\tau )d\tau \end{aligned}$$

with $E_{*,r}(A(\cdot ),\varphi _k(\cdot ),t_r) \quad \text{ and } \quad E_{*,r}(A(\cdot ),f(\cdot ),t_r)$, respectively.

As it follows from Theorem 2.2, the invertibility of matrix $Q_{*}(\Delta _N)$ is equivalent to the well-posedness of problem (1), (2). By solving the system (38), find $\lambda ^{*}=(\lambda _1^{*},\ldots ,\lambda _{N+1}^{*})\in R^{nN+l}.$

VII. Determine the components of $\xi ^{*}=(\xi ^{*}_1,\ldots ,\xi ^{*}_m)\in R^{nm}$ by the equalities

$$\begin{aligned} \xi ^{*}_k=\sum \limits _{j=1}^{N+1}\Big (\sum \limits _{p=1}^m M_{k,p}(\Delta _N)V_{p,j}(\Delta _N) \Big )\lambda ^{*}_j + \sum \limits _{p=1}^m M_{k,p}(\Delta _N)g_{p}(f,\Delta _N) \end{aligned}$$

(39)

and construct the function

$$\begin{aligned} \mathcal {F}^{*}(t)=\sum \limits _{k=1}^m \varphi _k(t)\Big [\xi ^{*}_k+\sum \limits _{r=1}^N \int \limits _{t_{r-1}}^{t_r}\psi _k(s)ds\lambda ^{*}_r\Big ]+ A_0(t)\lambda ^{*}_{N+1} + f(t). \end{aligned}$$

(40)

Recall that $\lambda _r^{*}=x^{*}(t_{r-1}),$$r=\overline{1,N}$, $\lambda _{N+1}^{*} = \mu ^{*}$, where $(x^{*}(t)$, $\mu ^{*})$ is a solution to the problem with parameter (1), (2). Therefore, the solution of system (38) provides us with the values of the function $x^{*}(t)$ at the left-end points of the partition subintervals and the parameter $\mu ^{*}$.

The values of $x^{*}(t)$ at the remaining points of the subinterval $[t_{r-1},t_r)$ determine by solving the following Cauchy problem for the ordinary differential equation:

$$\begin{aligned} \frac{dx}{dt} = A(t)x + \mathcal {F}^{*}(t), \quad x(t_{r-1})=\lambda _r^{*}, \quad t\in [t_{r-1},t_r), \quad r=\overline{1,N}. \end{aligned}$$

Thus, the offered algorithm consists of seven interconnected steps.

If the fundamental matrices $X_r(t),$$r=\overline{1,N}$, are known, then equalities (23), (24), (25), and (26) allow us to construct the system (38). Using the solution $\lambda ^{*}$ to (38), by (39) and (40), we construct the function $\mathcal {F}^{*}(t)$. Therefore, the solution to the problem with parameter (1), (2) is defined by the equalities

$$\begin{aligned} x^{*}(t)= & {} X_r(t)X_r^{-1}(t_{r-1})\lambda _r^{*}+X_r(t)\int \limits _{t_{r-1}}^{t} X_r^{-1}(\tau )\mathcal {F}^{*}(\tau )d\tau , \quad t\in [t_{r-1},t_r), \ r=\overline{1,N},\nonumber \\ \end{aligned}$$

(41)

$$\begin{aligned} x^{*}(T)= & {} X_N(T)X_N^{-1}(t_{N-1})\lambda _N^{*}+X_N(T)\int \limits _{t_{N-1}}^{T} X_N^{-1}(\tau )\mathcal {F}^{*}(\tau )d\tau , \end{aligned}$$

(42)

$$\begin{aligned} \mu ^{*}= & {} \lambda _{N+1}^{*}. \end{aligned}$$

(43)

However, it is not always possible to construct a fundamental matrix for a system of ordinary differential equations with variable coefficients. We, therefore, offer the numerical implementation of Algorithm that involves numerical solution of auxiliary Cauchy problems and numerical integration.

The numerical algorithm for solving problem (1), (2) performs as follows:

I. Take a partition $\Delta _N: t_0=0<t_1<\ldots<t_{N-1}<t_N=T$. Divide each subinterval $[t_{r-1},t_r],$$r=\overline{1,N}$, into $N_r$ parts with step size $h_r=(t_r-t_{r-1})/N_r.$

Let $\widehat{t}$ be a variable taking on the discrete values $\widehat{t}=t_{r-1}, t_{r-1}+h_r,\ldots ,t_{r-1}+(N_r-1)h_r, t_r$ on the subinterval $[t_{r-1},t_r]$. We denote the set of such points by $\{t_{r-1},t_r\}$.

II. Find the numerical solutions to Cauchy problems (33) by using one of numerical methods for solving initial value problems for ordinary differential equations. Determine the values of the $(n\times n)$ matrices $E_{*,r}^{h_r}(A(\cdot ),\varphi _k(\cdot ),\widehat{t})$ on the set $\{t_{r-1},t_r\}$, $r=\overline{1,N}$, $k=\overline{1,m}.$

III. Using the values of $(n\times n)$ matrices $\psi _k(s)$ and $E_{*,r}^{h_r}\Big (A(\cdot ),\varphi (\cdot ),\widehat{t}\Big )$ on $\{t_{r-1},t_r\}$, and applying a numerical quadrature rule, calculate the $(n\times n)$ matrices

$$\begin{aligned} \widehat{\psi }_{p,r}^{h_r}(\varphi _k)=\int \limits _{t_{r-1}}^{t_r}\psi _p(\tau )E_{*,r}^{h_r}(A(\cdot ),\varphi _k(\cdot ),\tau )d\tau , \quad p,k=\overline{1,m}, \quad r=\overline{1,N}. \end{aligned}$$

Summing up the matrices $\widehat{\varphi }_{p,r}^{h_r}(\psi _k)$ with respect to r, determine the $(n\times n)$ matrices $G_{p,k}^{\widetilde{h}}(\Delta _N)=\sum \limits _{r=1}^N \widehat{\varphi }_{p,r}^{h_r}(\psi _k)$, where $\widetilde{h}=(h_1,h_2,\ldots ,h_N)\in R^n$. Using them, compose the $nm\times nm$ matrix $G^{\widetilde{h}}(\Delta _N)=(G_{p,k}^{\widetilde{h}}(\Delta _N)),$$p,k=\overline{1,m}$.

Check whether the matrix $I-G^{\widetilde{h}}(\Delta _N)$ is invertible. If so, calculate its inverse $[I-G^{\widetilde{h}}(\Delta _N)]^{-1}=(M_{p,k}^{\widetilde{h}}(\Delta _N)),$$p,k=\overline{1,m},$ and move on to the next step.

If the matrix is not invertible, take a new partition. In particular, each subinterval can be divided into two parts. Then go back to Step I.

IV. Solve numerically the Cauchy problem (33), (35) and find the values of the $(n\times n)$ matrix $E_{*,r}(A(\cdot ),A(\cdot ),\widehat{t})$, the $(n\times l)$ matrix $E_{*,r}(A(\cdot ),A_0(\cdot ),\widehat{t})$, and the n vector $E_{*,r}(A(\cdot ),f(\cdot ),\widehat{t})$ on the grid $\{t_{r-1},t_r\},$$r=\overline{1,N}.$

V. On the set $\{t_{r-1},t_r\}$, evaluate the definite integrals

$$\begin{aligned}&\widehat{\psi }_{p,r}^{h_r}=\int \limits _{t_{r-1}}^{t_r}\psi _p(s)ds, \quad \widehat{\psi }_{p,r}^{h_r}(A)=\int \limits _{t_{r-1}}^{t_r}\psi _p(\tau )E_{*,r}^{h_r} (A(\cdot ),A(\cdot ),\tau ) d\tau ,\\&\widehat{\psi }_{p,r}^{h_r}(A_0)=\int \limits _{t_{r-1}}^{t_r}\psi _p(\tau )E_{*,r}^{h_r} (A(\cdot ),A_0(\cdot ),\tau ) d\tau ,\\&\widehat{\psi }_{p,r}^{h_r}(f)=\int \limits _{t_{r-1}}^{t_r}\psi _p(\tau )E_{*,r}^{h_r} (A(\cdot ),f(\cdot ),\tau )d\tau , \quad r=\overline{1,N}, \quad p=\overline{1,m}. \end{aligned}$$

Determine the $(n\times n)$ matrices $V_{p,r}^{\widetilde{h}}(\Delta _N)$, the $(n\times l)$ matrices $V_{p,N+1}^{\widetilde{h}}(\Delta _N)$, and the n vectors $g_{p}^{\widetilde{h}}(f,\Delta _N)$ by the respective equalities

$$\begin{aligned}&V_{p,r}^{\widetilde{h}}(\Delta _N)=\widehat{\psi }_{p,r}^{h_r}(A)+ \sum \limits _{j=1}^{N}\sum \limits _{k=1}^{m}\widehat{\psi }_{p,j}^{h_j}(\varphi _k) \cdot \widehat{\psi }_{k,r}^{h_r}, \qquad r=\overline{1,N},\\&V_{p,N+1}^{\widetilde{h}}(\Delta _N) = \sum \limits _{r=1}^{N} \widehat{\psi }_{p,r}^{h_r}(A_0), \qquad g_{p}^{\widetilde{h}}(f,\Delta _N)=\sum \limits _{r=1}^{N}\widehat{\psi }_{p,r}^{h_r}(f), \qquad p=\overline{1,m}. \end{aligned}$$

VI. Construct the system of linear algebraic equations in parameters

$$\begin{aligned} Q_{*}^{\widetilde{h}}(\Delta _N)\lambda =-F_{*}^{\widetilde{h}}(\Delta _N), \quad \lambda \in R^{nN+l}, \end{aligned}$$

(44)

where the elements of the matrix $Q_{*}^{\widetilde{h}}(\Delta _N)$ and the vector

$$\begin{aligned} F_{*}^{\widetilde{h}}(\Delta _N)=(-d+CF_{N}^{\widetilde{h}}(\Delta _N),F_{1}^{\widetilde{h}}(\Delta _N), \ldots ,F_{N-1}^{\widetilde{h}}(\Delta _N)) \end{aligned}$$

are defined by the equalities

$$\begin{aligned}&D_{r,j}^{\widetilde{h}}(\Delta _N)=\sum \limits _{k=1}^{m}E_{*,r}^{h_r}(A(\cdot ),\varphi _k(\cdot ),t_r) \Big [\sum \limits _{p=1}^{m}M_{k,p}^{\widetilde{h}}(\Delta _N)V_{p,j}^{\widetilde{h}}(\Delta _N)\\&\quad +\widehat{\psi }_{k,j}^{h_j}\Big ], \quad j\ne r, \quad r,j=\overline{1,N},\\&D_{r,r}^{\widetilde{h}}(\Delta _N)=\sum \limits _{k=1}^{m}E_{*,r}^{h_r}(A(\cdot ),\varphi _k(\cdot ),t_r) \Big [\sum \limits _{p=1}^{m}M_{k,p}^{\widetilde{h}}(\Delta _N)V_{p,r}^{\widetilde{h}}(\Delta _N) +\widehat{\psi }_{k,r}^{h_r}\Big ]\\&\quad +E_{*,r}^{h_r}(A(\cdot ),A(\cdot ),t_r),\quad r=\overline{1,N},\\&D_{r,N+1}^{\widetilde{h}}(\Delta _N)=\sum \limits _{k=1}^{m}E_{*,r}^{h_r}(A(\cdot ),\varphi _k(\cdot ),t_r) \sum \limits _{p=1}^{m}M_{k,p}^{\widetilde{h}}(\Delta _N)V_{p,N+1}^{\widetilde{h}}(\Delta _N)\\&\quad + E_{*,r}^{h_r}(A(\cdot ),A_0(\cdot ),t_r),\quad r=\overline{1,N},\\&F_{r}^{\widetilde{h}}(\Delta _N)=\sum \limits _{k=1}^{m}E_{*,r}^{h_r}(A(\cdot ),\varphi _k(\cdot ),t_r) \sum \limits _{p=1}^{m}M_{k,p}^{\widetilde{h}}(\Delta _N)g_{p}^{\widetilde{h}}(\Delta _N)\\&\quad +E_{*,r}^{h_r}(A(\cdot ),f(\cdot ),t_r), \quad r=\overline{1,N}. \end{aligned}$$

Using the constructed matrix $Q_{*}^{\widetilde{h}}(\Delta _N),$ we can establish the well-posedness of problem (1), (2). Suppose that the matrix $Q_{*}^{\widetilde{h}}(\Delta _N)$ is invertible and the estimate $||Q_{*}(\Delta _N)-Q_{*}^{\widetilde{h}}(\Delta _N)||\le \varepsilon (\widetilde{h})$ holds. By Theorem 4 Dzhumabaev (2015, p.212), if the inequality $||[Q_{*}^{\widetilde{h}}(\Delta _N)]^{-1}||\cdot \varepsilon (\widetilde{h}) < 1$ holds, then $Q_{*}(\Delta _N)$ is invertible. Thus it follows from Theorem 2.2 that the problem (1), (2) is well-posed.

By solving system (44) find $\lambda ^{\widetilde{h}}\in R^{nN+l}.$ As noted above, the elements of $\lambda ^{\widetilde{h}}=(\lambda ^{\widetilde{h}}_1, \ldots , \lambda ^{\widetilde{h}}_{N+1})$ are the values of approximate solution to problem (1), (2), i.e. the approximate values of x(t) at the left endpoints of the subintervals: $x^{\widetilde{h}_r}(t_{r-1})=\lambda ^{\widetilde{h}}_r,$$r=\overline{1,N},$ and the approximate value of the parameter $\mu$: $\mu ^{\widetilde{h}_r} =\lambda ^{\widetilde{h}}_{N+1}$.

VII. To define the values of an approximate solution at the remaining points of the set $\{t_{r-1},t_r\}$, we first find

$$\begin{aligned} \xi _k^{\widetilde{h}}=\sum \limits _{j=1}^{N+1}\Big (\sum \limits _{p=1}^{m}M_{k,p}^{\widetilde{h}}(\Delta _N)V_{p,j} ^{\widetilde{h}}(\Delta _N)\Big )\lambda _j^{h}+ \sum \limits _{p=1}^{m}M_{k,p}^{\widetilde{h}}(\Delta _N)g_{p} ^{\widetilde{h}}(f,\Delta _N), \quad k=\overline{1,m}, \end{aligned}$$

and then numerically solve the Cauchy problems

$$\begin{aligned} \frac{dx}{dt} = A(t)x + \mathcal {F}^{\widetilde{h}}(t), \quad x(t_{r-1})=\lambda _r^{\widetilde{h}}, \quad t\in [t_{r-1},t_r], \quad r=\overline{1,N}. \end{aligned}$$

Here

$$\begin{aligned} \mathcal {F}^{\widetilde{h}}(t)=\sum \limits _{k=1}^{m}\varphi _k(t)\Big (\xi _k^{\widetilde{h}}+ \sum \limits _{j=1}^{N}\widehat{\psi } _{k,j}^{{h}_j}\lambda _j^{h}\Big )+ A_0(t)\lambda _{N+1}^{h} + f(t). \end{aligned}$$

Thus the algorithm offered provides us with the numerical solution to the problem (1), (2).

4 Numerical examples

Example 1

Consider the following problem with parameter for the system of integro-differential equations:

$$\begin{aligned}&\frac{dx}{dt}=A(t)x+\varphi (t)\int \limits _0^T\psi (\tau )x(\tau )d\tau + A_0(t)\mu +f(t),\qquad x\in R^2, \quad \mu \in R^1, \end{aligned}$$

(45)

$$\begin{aligned}&B_0\mu +B x(0) +Cx(T) =d, \qquad d\in R^3, \end{aligned}$$

(46)

where

$$\begin{aligned} T= & {} 1, \qquad A(t)=\begin{pmatrix} t&{} t^2 \\ 0&{} t-4 \end{pmatrix}, \qquad A_0(t)=\begin{pmatrix} t+5\\ t^3-2 \end{pmatrix},\\ \varphi (t)= & {} \begin{pmatrix} 3t&{} t^3\\ 4&{} t-2 \end{pmatrix}, \quad \psi (t)=\begin{pmatrix} t&{} t^2-2\\ t+4&{} 3 \end{pmatrix}, \quad f(t)= \begin{pmatrix} 5t^4-t^5-t^6+\frac{13t^3}{7}-\frac{318t}{7}-45 \\ 10t^2-5t^3-t^4-\frac{267t}{7}+\frac{85}{21} \\ \end{pmatrix},\\ B_0= & {} \begin{pmatrix} 2\\ 7\\ -5 \end{pmatrix},\quad B =\begin{pmatrix} 2&{} -5\\ 0&{}6\\ -4&{} 11 \end{pmatrix}, \quad C =\begin{pmatrix} 3&{} 0\\ -12&{}5\\ 9&{}17 \end{pmatrix}, \quad d=\begin{pmatrix} 36\\ 9\\ -153 \end{pmatrix}. \end{aligned}$$

To implement the numerical algorithm for solving problem (45),(46), we use Simpson’s rule for estimation of definite integrals and the fourth-order Runge-Kutta method. To do this, we divide each interval [0, 0.5] and [0.5, 1] into $N = 10$ subintervals with the step $h = 0.05$.

We compute the matrix $I-G^{\widetilde{h}}(\Delta _2)=\begin{pmatrix} 1.6665474 &{} -0.4153697\\ -3.6533113 &{} 1.5407901 \end{pmatrix}$, where I is the second-order identity matrix. The invertibility of this matrix implies the regularity of $\Delta _2$.

We then construct the system of linear algebraic equations with respect to parameters

$$\begin{aligned} Q_{*}^{\widetilde{h}}(\Delta _2)\lambda =-F_{*}^{\widetilde{h}}(\Delta _2), \quad \lambda \in R^{5}, \end{aligned}$$

(47)

where

$$\begin{aligned}&Q_{*}^{\widetilde{h}}(\Delta _2)=\begin{pmatrix} 9.5083951 &{}-7.0229286 &{}16.5414502 &{}-0.0873712 &{}41.3179432\\ -30.5783061 &{}12.889927 &{}-66.2339239 &{}0.2548226&{} -151.3556546\\ 16.6731188&{} 0.8451376 &{}49.3927316&{} -0.5839651&{} 109.2686315\\ 1.6033622&{} -0.1482411&{} -0.2174947 &{}-0.0605072 &{}4.7172016\\ -0.5501459&{} -0.0171558 &{}-0.6994877 &{}-1.2253692&{} -1.908402 \end{pmatrix},\\&F_{*}^{\widetilde{h}}(\Delta _2)=\Big (-508.2954741, 1863.211432, -1321.586001, -51.2392, 19.6852962\Big )^{'}. \end{aligned}$$

By solving (47) we find $\lambda ^{\widetilde{h}}=(\lambda ^{\widetilde{h}}_1, \lambda ^{\widetilde{h}}_2,\lambda ^{\widetilde{h}}_{3})\in R^5$ with

$$\begin{aligned} \lambda _1^{\widetilde{h}}= \begin{pmatrix} 6.0000271\\ 0.0000055 \end{pmatrix}, \quad \lambda _2^{\widetilde{h}}= \begin{pmatrix} 4.7812568\\ -3.3750023 \end{pmatrix}, \quad \lambda _3^{\widetilde{h}}=8.9999902. \end{aligned}$$

To define the values of an approximate solution at the remaining points of set $\{t_{r-1},t_r\}$, $r=\overline{1,2},$ we first find $\xi ^{\widetilde{h}}= \begin{pmatrix} 1.8759368\\ -8.9001442 \end{pmatrix},$ and then solve the Cauchy problems:

$$\begin{aligned}&\frac{d\widetilde{x}}{dt}=A(t)\widetilde{x} +\varphi (t)\xi ^{\widetilde{h}}+\varphi (t)\int \limits _0^{0.5}\psi (\tau )d\tau \lambda _1^{\widetilde{h}} +\varphi (t)\int \limits _{0.5}^{1}\psi (\tau )d\tau \lambda _2^{\widetilde{h}}+ A_0(t)\lambda ^{\widetilde{h}}_{3} + f(t),\\&\widetilde{x}(t_{r-1})=\lambda ^{\widetilde{h}}_{r}, \quad t\in [t_{r-1},t_r], \quad r=\overline{1,2}. \end{aligned}$$

The exact solution to problem with parameter (45), (46) is the pair $(x^{*}(t), \mu ^{*})$ with

$$\begin{aligned} x^{*}(t)=\begin{pmatrix} t^5-5t^2+6\\ t^3-7t\end{pmatrix}, \mu ^{*}=9. \end{aligned}$$

Table 1 provides the values of the exact solution and the numerical solution $(\widetilde{x}(t), \widetilde{\mu })$. The calculations were carried out in the MathCad software package.

Table 1 Comparison of exact and numerical solutions to problem (45), (46)

The error estimates obtained by using the Runge–Kutta method are as follows:

$$\begin{aligned} \Vert \mu ^{*}-\widetilde{\mu }\Vert<0.00001, \qquad \max \limits _{j=\overline{0,20}}\Vert x^{*}(t_j)-\widetilde{x}(t_j)\Vert <0.00003. \end{aligned}$$

Example 2

Consider the problem with parameter for system of integro-differential equations

$$\begin{aligned}&\frac{dx}{dt}=A(t)x+\varphi (t)\int \limits _0^1\psi (\tau )x(\tau )d\tau + A_0(t)\mu +f(t),\qquad x\in R^2, \quad \mu \in R^2, \end{aligned}$$

(48)

$$\begin{aligned}&B_0\mu +B x(0) +Cx(T) =d, \quad d\in R^4, \end{aligned}$$

(49)

where $A(t)=\begin{pmatrix} e^t&{} 1 \\ t^3&{} cos(t) \end{pmatrix}, \quad A_0(t)=\begin{pmatrix} 4 &{} t^2\\ sin(t) &{} 0 \end{pmatrix}, \quad \varphi (t)=\begin{pmatrix} t&{} t^2\\ 0&{} t+3 \end{pmatrix}, \quad \psi (t)=\begin{pmatrix} t^3&{} t-2\\ 0&{} e^t \end{pmatrix},$

$$\begin{aligned} f(t)= & {} \begin{pmatrix} \frac{43t}{56}+t^2+4t^3-28-t^2e+e^t(4-6t^3-t^4)\\ 5t-3e-7sin(t)-t^2cos(t)-te+4t^3-6t^6-t^7+tcos(t)+8 \\ \end{pmatrix},\\ B_0= & {} \begin{pmatrix} 4 &{}5\\ 2 &{}3\\ 0 &{} -3\\ 1 &{}2 \end{pmatrix},\quad B =\begin{pmatrix} 1&{} 7\\ 3&{}0\\ -5&{} 2\\ 9 &{}3 \end{pmatrix}, \quad C =\begin{pmatrix} 1&{} 9\\ 0&{}-8\\ 19&{}4\\ 9 &{} 7 \end{pmatrix}, \quad d=\begin{pmatrix} 122\\ 59\\ 20\\ 36 \end{pmatrix}. \end{aligned}$$

We use the numerical implementation of algorithm. The accuracy of the solution depends on that of solving the Cauchy problems on the subintervals. We provide the results of the numerical implementation of algorithm based on the Bulirsch–Stoer method Atkinson et al. (2009), Butcher (2000), Stoer and Bulirsch (2002) by partitioning the subintervals [0, 0.5], [0.5, 1] with step size h = 0.05.

The exact solution to problem with parameter (48), (49) is the pair $(x^{*}(t), \mu ^{*})$ with

$$\begin{aligned} x^{*}(t)=\begin{pmatrix} t^4+6t^3-4\\ t^2-t\end{pmatrix}, \mu ^{*}=\begin{pmatrix} 7\\ 19\end{pmatrix}. \end{aligned}$$

In Table 2, the values of the exact solution and numerical solution $(x^{*}(t_k), \mu ^{*})$ and $(\widetilde{x}(t_k), \widetilde{\mu })$, $k=\overline{0,20}$, are shown.

Table 2 Comparison of exact and numerical solutions to problem (48), (49)

Example 3

Consider the following problem with parameter for the system of integro-differential equations:

$$\begin{aligned}&\frac{dx}{dt}=A(t)x+\sum \limits _{k=1}^2\int \limits _0^1\varphi _k(t)\psi _k(\tau )x(\tau )d\tau + A_0(t)\mu +f(t),\qquad x\in R^2, \quad \mu \in R^3,\nonumber \\ \end{aligned}$$

(50)

$$\begin{aligned}&B_0\mu +B x(0) +Cx(T) =d, \qquad d\in R^5, \end{aligned}$$

(51)

where

$$\begin{aligned} A(t)= & {} \begin{pmatrix} sin(t)&{} 1 \\ t^2&{} 0 \end{pmatrix}, \qquad A_0(t)=\begin{pmatrix} t&{} 1&{} t+2\\ t^2-7&{}2t &{} 8 \end{pmatrix}, \qquad f(t)= \begin{pmatrix} f_1(t) \\ f_2(t) \\ \end{pmatrix},\\ \varphi _1(t)= & {} \begin{pmatrix} t&{} 0\\ 2t^3&{} t-3 \end{pmatrix}, \quad \varphi _2(t)=\begin{pmatrix} 1&{} t\\ t^3&{}t+5\end{pmatrix}, \quad \psi _1(t)=\begin{pmatrix} t&{} e^t\\ t^2&{} 4t \end{pmatrix}, \quad \psi _2(t)=\begin{pmatrix} 2&{} t^2\\ t&{}e^t\end{pmatrix},\\ f_1(t)= & {} -\frac{1}{6\pi } (163 \pi -6\pi ^2 \cos (\pi t)+6\pi \sin (t) t+6\pi \sin (t) \sin (\pi t)\\&+\, 6\pi t^3+30\pi t^2-56\pi t+12t+144\pi e t +24),\\ f_2(t)= & {} -\frac{1}{60\pi ^3} (240\pi ^3 t^2-397\pi ^3 t -1150 \pi ^3 t^3+ 60\pi ^3 t^2\sin (\pi t)\\&+ 360\pi ^2 t^3 + 1440 \pi ^3 e t^3+\\&+ 120\pi ^2 t- 240t-6749\pi ^3+ 120\pi ^2+720+720\pi ^3 t e+3600\pi ^3 e),\\ B_0= & {} \begin{pmatrix} 1&{} 2&{} 5\\ 3&{}0 &{} 1\\ 4&{} 2&{} 1\\ 12&{} 1&{} 4\\ 2&{} 0&{} -5\\ \end{pmatrix},\quad B =\begin{pmatrix} 4&{} 5\\ 1&{}2\\ 5&{} 6\\ 2&{} -5\\ 0&{} 5 \end{pmatrix}, \quad C =\begin{pmatrix} -8&{} 9\\ 5&{}2\\ 6&{}5\\ 15&{} 6\\ 1&{} 3 \end{pmatrix}, \quad d=\begin{pmatrix} 216\\ 83\\ 164\\ 176\\ 60 \end{pmatrix}. \end{aligned}$$

The exact solution to problem with parameter (50), (51) is the pair $(x^{*}(t), \mu ^{*})$ with $x^{*}(t)=\begin{pmatrix} t+sin(\pi t)\\ t^3+5t^2+9\end{pmatrix},$$\mu ^{*}=\begin{pmatrix} 7\\ -4 \\ 9\end{pmatrix}.$

Case 1. Let $N=1$. We introduce the additional parameters $\lambda _{1},\lambda _2\in R^2$ setting $\lambda _{1}=x(0)$ and $\lambda _{2}=\mu$. Let $\xi _k=\int \limits _{0}^{1}\psi _k(s)u(s)ds$, $k=1,2,$ where $u(s)=x(s)-\lambda _{1}$. Then for the function u(t) we have the equality

$$\begin{aligned} u(t)= & {} E_{*,1}^{h_1}(A(\cdot ),A(\cdot ),\widehat{t})\lambda _{1}\nonumber \\&+E_{*,1}^{h_1}(A(\cdot ),A_0(\cdot ),\widehat{t})\lambda _{2}+ E_{*,1}^{h_1}(A(\cdot ),f(\cdot ),\widehat{t})+E_{*,1}^{h_1}(A(\cdot ),\varphi _1(\cdot ),\widehat{t})\xi _1+\nonumber \\&+E_{*,1}^{h_1}(A(\cdot ),\varphi _2(\cdot ),\widehat{t})\xi _2 +E_{*,1}^{h_1}(A(\cdot ),\varphi _1(\cdot ),\widehat{t})\int _{0}^{1}\psi _1(s)ds\lambda _{1}\nonumber \\&+ E_{*,1}^{h_1}(A(\cdot ),\varphi _2(\cdot ),\widehat{t})\int _{0}^{1}\psi _2(s)ds\lambda _{1}. \end{aligned}$$

(52)

Table 3 Comparison of numerical solutions to problem (50, 51). Case 1

Fig. 1

Comparison of the exact solutions and numerical solutions to problem (50, 51) ($N=2$)

Multiplying both sides of (52) by $\psi _p(t)$, $p=1,2,$ and then integrating them over the interval [0, 1], we get the system of linear algebraic equations in $\xi =(\xi _1,\xi _2)\in R^{4}$. By solving this system we determine $\xi$ and substitute the corresponding expression into the right-hand side of (52). We then obtain the representation of function u(t) through $\lambda _{1}$ and $\lambda _{2}$.

Table 4 Comparison of numerical solutions to problem (50),(51). Case 2

To implement the numerical algorithm for solving problem (50),(51), we use Simpson’s rule for estimation of definite integrals and the fourth-order Runge–Kutta method, the Adams method, and the Bulirsch–Stoer method to solve auxiliary Cauchy problems for ordinary differential equations. The calculations were carried out in the MathCad software package.

For the chosen step size $h=0.1$, the algorithm was performed three times, by separately using the Adams method, the fourth-order Runge–Kutta method, and the Bulirsch–Stoer method. Table 3 and Fig. 1 provide the comparative results obtained for the numerical solution $(\widetilde{x}(t),\widetilde{\mu })$ to problem (50),(51).

The error estimates obtained by using the three methods are as follows:

Adams method: $\Vert \mu ^{*}-\widetilde{\mu }\Vert <0.0832155$, $\max \limits _{j=\overline{0,10}}\Vert x^{*}(t_j)-\widetilde{x}(t_j)\Vert <0.0482514$;

Runge–Kutta method: $\Vert \mu ^{*}-\widetilde{\mu }\Vert <0.0440641$, $\max \limits _{j=\overline{0,10}}\Vert x^{*}(t_j)-\widetilde{x}(t_j)\Vert <0.0255510$;

Bulirsch–Stoer method: $\Vert \mu ^{*}-\widetilde{\mu }\Vert <0.0414742$, $\max \limits _{j=\overline{0,10}}\Vert x^{*}(t_j)-\widetilde{x}(t_j)\Vert <0.0240500$.

Table 5 Comparison of numerical solutions to problem (50),(51). Case 3

Case 2. Let $N=2$. We perform the numerical algorithm by the partitioning the interval [0, 1] with step size $h=0.5$ and the subintervals [0, 0.5],  [0.5, 1] with step size $h_1=0.05$. The results are presented in Table 4.

Case 3. Let us take $N=4$ and perform the numerical algorithm for solving problem (50),(51). The four partition subintervals [0, 0.25],  [0.25, 0.5], [0.5, 0.75],  [0.75, 1] are in turn divided with step size $h_1=0.025$.

Again, we implement the algorithm three times using different methods for solving auxiliary Cauchy problems.

The results are presented in Table 5 and Fig. 2.

To compare the results, we obtain the following error estimates:

Adams method: $\qquad \Vert \mu ^{*}-\widetilde{\mu }\Vert <0.0309316$, $\qquad \max \limits _{j=\overline{0,40}}\Vert x^{*}(t_j)-\widetilde{x}(t_j)\Vert <0.01793120;$

Runge–Kutta method:$\qquad \Vert \mu ^{*}-\widetilde{\mu }\Vert <0.0001708$, $\qquad \max \limits _{j=\overline{0,40}}\Vert x^{*}(t_j)-\widetilde{x}(t_j)\Vert <0.000099;$

Bulirsch–Stoer method: $\qquad \Vert \mu ^{*}-\widetilde{\mu }\Vert <0.0001604$, $\qquad \max \limits _{j=\overline{0,40}}\Vert x^{*}(t_j)-\widetilde{x}(t_j)\Vert <0.000093.$

5 Conclusion

The proposed computational method for solving problems with parameters for integro-differential equations is based on the parametrization method with the choice of a regular partition. The algorithm includes two auxiliary problems: the Cauchy problems for ordinary differential equations and the evaluation of definite integrals. The numerical solutions to the Cauchy problems were obtained by the Adams method, the fourth-order Runge–Kutta method, and the Bulirsch–Stoer method; the integrals were evaluated by Simpson’s rule. If we use other numerical or approximate methods, we obtain a new numerical or approximate implementation of the algorithm. By choosing various regular partitions, we obtain a family of algorithms.

The proposed method can be extended to problems for impulsive integro-differential equations, integro-differential equations of mixed type, and fractional integro-differential equations. One of the possible options for the further development of the proposed computational method is its combination with computational methods for fractional dynamical systems Burgos et al. (2019), Harrat et al. (2018), Kim et al. (2020), Liu et al. (2018), Manimaran et al. (2019).

Fig. 2

The exact solution (light blue solid line) and the numerical solution values obtained by the Bulirsch–Stoer method (’o’)