A Problem with Parameter for the Integro-Differential Equations

The article proposes a numerically approximate method for solving a boundary value problem for an integro-differential equation with a parameter and considers its convergence, stability, and accuracy. The integro-differential equation with a parameter is approximated by a loaded differential equation with a parameter. A new general solution to the loaded differential equation with a parameter is introduced and its properties are described. The solvability of the boundary value problem for the loaded differential equation with a parameter is reduced to the solvability of a system of linear algebraic equations with respect to arbitrary vectors of the introduced general solution. The coefficients and the right-hand sides of the system are compiled through solutions of the Cauchy problems for ordinary differential equations. Algorithms are proposed for solving the boundary value problem for the loaded differential equation with a parameter. The relationship between the qualitative properties of the initial and approximate problems is established, and estimates of the differences between their solutions are given.


Introduction
The theory of control problems for a system of ordinary differential equations and for a system of integro-differential equations in partial derivatives, with parameters, is rapidly developing and used in various fields of applied mathematics, biophysics, biomedicine, chemistry, etc. Control problems, also called as boundary value problems with parameters and parameter identification problems for systems of ordinary differential and integro-differential equations with parameters, are intensively studied by many authors [3,4,8,9,17,18,19,20,24,25]. To find solutions to these problems, methods of the qualitative theory of differential equations, variational calculus and optimization theory, the method of upper and lower solutions, etc. were used. Note that the problems of determining effective criteria for unique solvability and constructing numerical algorithms for finding optimal solutions to control problems for systems of ordinary differential and integro-differential equations with parameters are still relevant.
In this article, we extend the methods and results of [14,15] to a boundary value problem for integro-differential equations with parameters. The solvability conditions for boundary value problems for a system of integro-differential equations with parameters are established. A numerically approximate method for solving the investigated boundary value problem is constructed, and its convergence, stability, and accuracy are investigated.
The aim of this paper is to construct approximate and numerical methods for solving the problem with parameter in (1.1)-(1.2) and to ensure their convergence, stability, and accuracy. Section 2 is devoted to a loaded differential equation with a parameter. By the Simpson formula, the integral term of Equation (1.1) is replaced by the integral sum, and problem (1.1)-(1.2) reduces to the problem for a loaded differential equation with parameter. A new approach to the general solution of a loaded differential equation with a parameter is proposed. A new δ m (θ) general solution exists for all loaded linear nonhomogeneous differential equations with parameter and depends on the m + 1 arbitrary vectors λ r ∈ R n . We study the properties of δ m (θ) general solution and obtain solvability criteria for loaded differential equations with a parameter. The condition for the existence of a classical general solution is also established. Numerical and approximate methods for compilation of the δ m (θ) general solution are proposed.
The employment of the δ m (θ) general solution for solving the problem with a parameter for a loaded differential equation is given in Section 3. Substituting the expressions of the δ m (θ) general solution into the boundary condition (1.2) and the continuity conditions for solutions at loaded points, we obtain a system of linear algebraic equations with respect to λ r ∈ R n , r = 1, m + 1. Invertibility of Q * (δ m (θ)), the matrix of the compiled system, is equivalent to the wellposedness of the problem with a parameter for the loaded differential equation. The coefficients and the right-hand sides of this system are constructed through solutions to the Cauchy problems for ordinary differential equations on the subintervals [θ r−1 , θ r ], r = 1, m. Algorithms for solving the problem with a parameter for the loaded differential equation are proposed.
The interrelation between the unique solvability of the initial boundary value problem and the unique solvability of the approximate boundary value problem is established in Section 4. Estimates for the differences between their solutions are given. For an approximate problem with a parameter, a system of linear algebraic equations with respect to arbitrary vectors of a new general solution is composed. It is shown that the conditionality number of this system increases linearly relative to 2N if the approximate problem with the parameter is wellposed. This property of the system guarantees its stable solution. Cauchy problems for ordinary differential equations on subintervals are the main auxiliary problems in the proposed methods. If we choose an approximate method for solving these problems, we obtain an approximate method for solving the problem with parameter (1.1)-(1.2).
Numerical methods for solving Cauchy problems give numerical methods for solving the problem with parameter (1.1)-(1.2).

Approximate loaded linear differential equation with parameter: new general solution and its properties
Replacing the integral term of Equation (1.1) by a suitable integral sum, we obtain the loaded differential equation with parameter is continuously differentiable on (0, T ), the parameter µ * ∈ R l satisfies the equation for all t ∈ (0, T ).
Equation (2.1) is also referred to as a differential boundary equation. Loaded differential equations frequently are used in the applied mathematics. Particularly, in [21]- [22], they are used to describe the problems of long-term forecasting and control of groundwater level in the soil moisture. Various problems for these equations and methods for solving them are studied in [1,2,5,6,18,23]. The Fredholm integro-differential equation with a parameter has several features that should be considered when methods for research and solving problems with Equation (1.1) are created. The existence of an unsolvable Fredholm integro-differential equation with a parameter is one of such features. Another feature is that the Cauchy problem for Equation (1.1) can be unsolvable, although the problem with parameters for this equation has a unique solution. These features are also the characteristics of loaded differential equations with a parameter.
Suppose (x(t, c), µ(c)) is a solution to Equation (2.1) for all c ∈ R n , and each solution ( x(t), µ(c)) of Equation (2.1) coincides with the function (x(t, c), µ(c)) for a unique c = c. Then the pair (x(t, c), µ(c)) is said to be the classical general solution of the loaded linear differential equation with parameter (2.1) if it exists for all f (t) ∈ C([0, T ], R n ). General solution is one of the main tools for research and solving problems for differential and integro-differential equations. Since there are unsolvable loaded linear differential equations, the classical general solution also exists not for all equations (2.1). Therefore, we propose a new concept for the general solution of a loaded differential equation with a parameter.
Let C([0, T ], θ, R nm ) denote the space of function systems x r (t) .
) is a function system consisting of restrictions of the function x(t) to the sub-intervals [θ r−1 , θ r ), r = 1, m, then the following equations hold: (2.14) These equations are called continuity conditions for solution of Equation (2.1).
It is clear that the boundary value problem might be considered only for solvable loaded differential equations with a parameter. Thus, we apply the δ m (θ) general solution to establish the solvability criteria for Equation (2.1). Substituting the appropriate expressions from (2.6) and (2.7) into the continuity conditions (2.13) and (2.14), we obtain the system of mn linear algebraic equations with respect to the m + 2 unknown vectors λ j ∈ R n , j = 1, m + 1, Let D * (δ m (θ)) denote the nm × (n(m + 1) + l) matrix corresponding to the left-hand side of (2.16). We rewrite (2.16) in the form: Theorems 1, 2 and well-known statements of linear algebra lead to the following assertion. is valid for ∀η ∈ Ker(D * (δ m (θ))) , where (·, ·) is the inner product in R nm .
Equation (2.17) and properties of the general solution allow us to obtain the following statement on the existence of a classical general solution to Equation (2.1).
Theorem 4. A loaded linear differential equation with parameter (2.1) has a classical general solution if rank D * (δ m (θ)) = mn.
Formulas (2.9)-(2.12), which determine the coefficients and the right-hand sides of the δ m (θ) general solution, contain the fundamental matrices X r (t), r = 1, m. As a rule, the construction of fundamental matrices for systems of ordinary differential equations with variable coefficients fails. Therefore, below we propose numerical and approximate methods for constructing a δ m (θ) general solution.
To do this, we consider the Cauchy problems for ordinary differential equations on sub-intervals: Now, taking into account (2.21), we can determine the coefficients and the right-hand sides of the δ m (θ) general solution from the equations: In the following numerical method for constructing the δ m (θ) general solution, we use the fourth-order Runge-Kutta method to solve the Cauchy problems for ordinary differential equations on sub-intervals. Divide each interval [θ r−1 , θ r ] into N r parts with step h r = (θ r − θ r−1 )/N r , r = 1, m. Suppose that the variablet takes only discrete values:t = θ r−1 ,t = θ r−1 + h r , ...,t = θ r−1 + (N r − 1)h r ,t = θ r , r = 1, m, and denote by {θ r−1 , θ r } the set of such values of t.
Using the Lagrange polynomial [9] with functions ω r,i (t) = ω 1 r,i (t)/ω 2 where t r,i = θ r−1 +ih r , r = 1, m, i = 0, 1, . . . , N r , we get an approximate δ m (θ) general solution of Equation (2.1). We determine the approximate coefficients and right-hand sides by the equalities: a hr r (f, t r−1,i )ω r,i (t), Then, the approximate δ m (θ) general solution of Equation (2.1) has the form: Similarly, solving the Cauchy problems for ordinary differential equations (2.27)-(2.29) by an approximate method, we obtain an approximate δ m (θ) general solution of Equation (2.1). The accuracy of the coefficients and the righthand sides of the numerical and approximate δ m (θ) general solutions depends on the accuracy of the method used to solve Cauchy problems for ordinary differential equations (2.20).
Using the properties of the δ m (θ) general solution, it is easy to prove that the solvability of the problem with parameter (2.1), (1.2) is equivalent to the solvability of system (3.3). From the well-known statements of linear algebra the following two statements follow.
As was noted in Section 2, for Equation (2.1) with the variable matrix A(t), in most cases, the construction of the δ m (θ) general solution in the explicit form fails. Therefore, we propose another way to solve problem (2.1), (1.2). This method is based on compilation of a system of linear algebraic equations (3.1)-(3.2) and does not require the construction of the δ m (θ) general solution. Using relations (2.22)-(2.25), we solve problem (2.1), (1.2) using the following Algorithm B: Step 1. Solve the Cauchy problems for ordinary differential equations and find a r (A + K r , θ r ), a r (K j , θ r ), a r (A 0 , θ r ) and a r (f, θ r ).
In Algorithm B, Cauchy problems for ordinary differential equations on subintervals are the main auxiliary problems. Using approximate methods for solving these problems leads to approximate methods for solving problem (2.1), (1.2). Numerical methods used for solving Cauchy problems for ordinary differential equations lead to numerical methods for solving the problem (2.1), (1.2). Consider the following problem with parameter for the loaded differential equation: A 0 (t) .
Substituting the corresponding expressions of z(δ 2N (h), t, λ) and υ(δ 2N (h), λ) into the boundary condition (5.2) and the continuity conditions for solution at the points t s = sh, s = 1, 2, ..., 2N , then multiplying the boundary condition by h > 0, we obtain the system of linear algebraic equations:
In [4] we also propose a numerically approximate method for solving Cauchy problems for ordinary differential equations on subintervals, which is illustrated by numerical examples. The authors can present numerical results obtained using MathCad15.