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A numerical solution of problem for essentially loaded differential equations with an integro-multipoint condition

  • Zhazira M. Kadirbayeva EMAIL logo and Symbat S. Kabdrakhova EMAIL logo
Published/Copyright: October 12, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

We study a linear boundary value problem for systems of essentially loaded differential equations with an integro-multipoint condition. We make use of the numerical implementation of the Dzhumabaev parametrization method to obtain the desired result, which is well supported by two numerical examples.

Keywords: loaded differential equation; integro-multipoint condition; parametrization method; numerical solution
MSC 2010: 34B10; 45J05; 65L06

1 Introduction and problem statement

In the last few decades, loaded differential equations with boundary conditions have been studied by many researchers [1,2,3, 4,5,6, 7,8,9, 10,11]. This is because, loaded equations describe problems in optimal control, regulation of the layer of soil water and ground moisture, and underground fluid and gas dynamics. Monographs [12,13] contain various applications of loaded equations as a method for studying problems in mathematical biology, mathematical physics, the theory of mathematical modeling of nonlocal processes and phenomena, and the theory of elastic shells.

Various problems for loaded differential equations with integral conditions and methods for finding their solutions are considered in [14,15,16, 17,18,19, 20,21]. Boundary value problems with integro-multipoint boundary conditions have been studied by many researchers, for example, see [22,23].

In a recent article [24], the author discussed the numerical method for solving the boundary value problem for essentially loaded differential equations based on the Dzhumabaev parametrization method [25,26]. This is a constructive method originally developed to investigate and solve boundary value problems for ordinary differential equations. The Dzhumabaev parametrization method is based on dividing the interval [ 0 , T ] into N parts and introducing the additional parameters. In [25], coefficient criteria were established for the unique solvability of linear boundary value problems. An algorithm for finding their approximate solutions was developed. The Dzhumabaev parametrization method was later extended to boundary value problems, both linear and nonlinear, for various classes of equations. In [24], the term essentially loaded differential equation means that the right side of the differential equation depends on the value of the desired solution and its derivatives at given points, where the order of the derivatives is not less than the order of the differential part of the equation. In [24], by assuming the invertibility of the matrix compiled through the coefficients at the values of the derivative of the desired function at load points, we reduce the considering problem to a two-point boundary value problem for loaded differential equations.

Motivated by the research going on in this direction, in this article, we study the finding a numerical solution to the linear boundary value problem for systems of essentially loaded differential equations with an integro-multipoint condition of the form:

(1) d x d t = A 0 ( t ) x + ∑ i = 1 m A i ( t ) x ˙ ( θ i ) + f ( t ) , t ∈ ( 0 , T ) ,

(2) ∑ j = 0 m + 1 B j x ( θ j ) + ∫ θ 0 θ m + 1 C ( t ) x ( t ) d t = d , d ∈ R n , x ∈ R n .

Here, ( n × n ) -matrices A i ( t ) , ( i = 0 , m ¯ ) , C ( t ) , and n -vector-function f ( t ) are continuous on [ 0 , T ] , B j ( j = 0 , m + 1 ¯ ) are constant ( n × n ) -matrices, and 0 = θ 0 < θ 1 < … < θ m < θ m + 1 = T ; ‖ x ‖ = max i = 1 , n ¯ ∣ x i ∣ .

Let C ( [ 0 , T ] , R n ) denote the space of continuous functions x : [ 0 , T ] → R n with the norm ‖ x ‖ 1 = max t ∈ [ 0 , T ] ‖ x ( t ) ‖ .

A solution to problem (1), (2) is a continuously differentiable on ( 0 , T ) function x ( t ) ∈ C ( [ 0 , T ] , R n ) satisfying the system of essentially loaded differential equations (1) and the integro-multipoint condition (2).

The aim of this article is to propose a numerical implementation of the Dzhumabaev parametrization method for solving the boundary value problem for systems of essentially loaded differential equations with integro-multipoint condition (1), (2).

1.1 The scheme of the Dzhumabaev parametrization method

Definition 1

Problem (1), (2) is called uniquely solvable, if for any function f ( t ) ∈ C ( [ 0 , T ] , R n ) and vector d ∈ R n , it has a unique solution.

The interval [ 0 , T ] is partitioned by the loading points: [ 0 , T ) = ⋃ r = 1 m + 1 [ θ r − 1 , θ r ) .

Let C ( [ 0 , T ] , θ , R n ( m + 1 ) ) denote the space of function systems x [ t ] = ( x 1 ( t ) , x 2 ( t ) , … , x m + 1 ( t ) ) , where x r : [ θ r − 1 , θ r ) → R n are continuous on [ θ r − 1 , θ r ) and have finite left-sided limits lim t → θ r − 0 x r ( t ) for all r = 1 , m + 1 ¯ , with the norm ‖ x [ ⋅ ] ‖ 2 = max r = 1 , m + 1 ¯ sup t ∈ [ θ r − 1 , θ r ) ‖ x r ( t ) ‖ .

The restriction of the function x ( t ) to the r th subinterval [ θ r − 1 , θ r ) is denoted by x r ( t ) , i.e., x r ( t ) = x ( t ) for t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ , and the restriction of the function x ˙ ( t ) to the r th subinterval [ θ r − 1 , θ r ) is denoted by x ˙ r ( t ) , i.e., x ˙ r ( t ) = x ˙ ( t ) for t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ . The problem (1), (2) is then transformed into the equivalent problem:

(3) d x r d t = A 0 ( t ) x r + ∑ i = 1 m A i ( t ) x ˙ i + 1 ( θ i ) + f ( t ) , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ ,

(4) ∑ j = 0 m B j x j + 1 ( θ j ) + B m + 1 lim t → θ m + 1 − 0 x m + 1 ( t ) + ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) x r ( t ) d t = d ,

(5) lim t → θ i − 0 x i ( t ) = x i + 1 ( θ i ) , i = 1 , m ¯ ,

where (5) are conditions for matching the solution at the interior partition points. From conditions (5) and the assumption of the continuity of the coefficients A i ( t ) , ( i = 0 , m ¯ ) , it follows that the derivatives of the solution will also be continuous at the partition points.

A solution of problem (3)–(5) is a system of functions x ∗ [ t ] = ( x 1 ∗ ( t ) , x 2 ∗ ( t ) , … , x m + 1 ∗ ( t ) ) ∈ C ( [ 0 , T ] , θ , R n ( m + 1 ) ) , where functions x r ∗ ( t ) , r = 1 , m + 1 ¯ , are continuously differentiable on [ θ r − 1 , θ r ) , which satisfies system (3) and conditions (4) and (5).

Problem (1), (2), and (3)–(5) are equivalent. If a system of functions x ˜ [ t ] = ( x ˜ 1 ( t ) , x ˜ 2 ( t ) , … , x ˜ m + 1 ( t ) ) ∈ C ( [ 0 , T ] , θ , R n ( m + 1 ) ) is a solution of problem (3)–(5), then the function x ˜ ( t ) defined by the equalities x ˜ ( t ) = x ˜ r ( t ) , x ˜ ˙ ( t ) = x ˜ ˙ r ( t ) , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ , and x ˜ ( θ m + 1 ) = lim t → θ m + 1 − 0 x ˜ m + 1 ( t ) is a solution of the original problem (1), (2). Conversely, if x ( t ) is a solution of problem (1), (2), then the system of functions x [ t ] = ( x 1 ( t ) , x 2 ( t ) , … , x m + 1 ( t ) ) , where x r ( t ) = x ( t ) , x ˙ r ( t ) = x ˙ ( t ) , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ , and lim t → θ m + 1 − 0 x m + 1 ( t ) = x ( θ m + 1 ) is a solution of problem (3)–(5).

Introducing the additional parameters λ r = lim t → θ r − 1 + 0 x r ( t ) , μ r = lim t → θ r − 1 + 0 x ˙ r ( t ) , r = 1 , m + 1 ¯ , and performing a replacement of the function u r ( t ) = x r ( t ) − λ r on the each interval [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ , we obtain the boundary value problem with parameters λ r , r = 1 , m + 1 ¯ :

(6) d u r d t = A 0 ( t ) ( u r + λ r ) + ∑ i = 1 m A i ( t ) μ i + 1 + f ( t ) , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ ,

(7) u r ( θ r − 1 ) = 0 , r = 1 , m + 1 ¯ ,

(8) ∑ j = 0 m B j λ j + 1 + B m + 1 λ m + 1 + B m + 1 lim t → θ m + 1 − 0 u m + 1 ( t ) + ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) ( u r ( t ) + λ r ) d t = d ,

(9) λ i + lim t → θ i − 0 u i ( t ) = λ i + 1 , i = 1 , m ¯ ,

(10) lim t → θ r − 1 + 0 u ˙ r ( t ) = μ r , r = 1 , m + 1 ¯ .

A solution to problem (6)–(10) is a triple ( λ ∗ , μ ∗ , u ∗ [ t ] ) , with elements λ ∗ = ( λ 1 ∗ , λ 2 ∗ , … , λ m + 1 ∗ ) ∈ R n ( m + 1 ) , μ ∗ = ( μ 1 ∗ , μ 2 ∗ , … , μ m + 1 ∗ ) ∈ R n ( m + 1 ) , and u ∗ [ t ] = ( u 1 ∗ ( t ) , u 2 ∗ ( t ) , … , u m + 1 ∗ ( t ) ) ∈ C ( [ 0 , T ] , θ , R n ( m + 1 ) ) , where u r ∗ ( t ) are continuously differentiable on [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ , and satisfying the system of ordinary differential equations (6) and conditions (7)–(10) at the λ r = λ r ∗ , μ r = μ r ∗ , j = 1 , m + 1 ¯ .

Problem (1), (2) are equivalent to problem (6)–(10). If the function x ∗ ( t ) is a solution to problem (1), (2), then the triple ( λ ∗ , μ ∗ , u ∗ [ t ] ) , where λ ∗ = ( x ∗ ( θ 0 ) , x ∗ ( θ 1 ) , … , x ∗ ( θ m ) ) , μ ∗ = ( x ˙ ∗ ( θ 0 ) , x ˙ ∗ ( θ 1 ) , … , x ˙ ∗ ( θ m ) ) , and u ∗ [ t ] = ( x ∗ ( t ) − x ∗ ( θ 0 ) , x ∗ ( t ) − x ∗ ( θ 1 ) , … , x ∗ ( t ) − x ∗ ( θ m ) ) is a solution to problem (6)–(10). Conversely, if the triple ( λ ˜ , μ ˜ , u ˜ [ t ] ) , with elements λ ˜ = ( λ ˜ 1 , λ ˜ 2 , … , λ ˜ m + 1 ) ∈ R n ( m + 1 ) , μ ˜ = ( μ ˜ 1 , μ ˜ 2 , … , μ ˜ m + 1 ) ∈ R n ( m + 1 ) , u ˜ [ t ] = ( u ˜ 1 ( t ) , u ˜ 2 ( t ) , … , u ˜ m + 1 ( t ) ) ∈ C ( [ 0 , T ] , θ , R n ( m + 1 ) ) , is a solution to problem (6)–(10), then the function x ˜ ( t ) defined by the equalities x ˜ ( t ) = u ˜ r ( t ) + λ ˜ r , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ , and x ˜ ( T ) = λ ˜ m + 1 + lim t → T − 0 u ˜ m + 1 ( t ) will be the solution of the original problem (1), (2).

First, find the values of μ j , j = 1 , m + 1 ¯ . By passing on the right-hand side of (6) to the limit as t → θ r − 1 + 0 and r = 1 , m + 1 ¯ , and substituting the expression into (10), we obtain the system of equations for the unknown parameters μ j , j = 1 , m + 1 ¯ :

(11) μ r − ∑ i = 1 m A i ( θ r − 1 ) μ i + 1 = A 0 ( θ r − 1 ) λ r + f ( θ r − 1 ) , r = 1 , m + 1 ¯ ,

that is, we can rewrite equation (11) in the following form:

(12) G ( θ ) μ = H ( θ , λ ) , μ ∈ R n ( m + 1 ) .

Here,

G ( θ ) = I − A 1 ( θ 0 ) − A 2 ( θ 0 ) … − A m − 1 ( θ 0 ) − A m ( θ 0 ) O I − A 1 ( θ 1 ) − A 2 ( θ 1 ) … − A m − 1 ( θ 1 ) − A m ( θ 1 ) O − A 1 ( θ 2 ) I − A 2 ( θ 2 ) … − A m − 1 ( θ 2 ) − A m ( θ 2 ) … … … … … O − A 1 ( θ m − 1 ) − A 2 ( θ m − 1 ) … I − A m − 1 ( θ m − 1 ) − A m ( θ m − 1 ) O − A 1 ( θ m ) − A 2 ( θ m ) … − A m − 1 ( θ m ) I − A m ( θ m ) ,

where I is the identity matrix of dimension n and O is the zero matrix of dimension n ,

H ( θ , λ ) = ( H 1 ( θ , λ ) , H 2 ( θ , λ ) , … , H m + 1 ( θ , λ ) ) ′ , H r ( θ , λ ) = A 0 ( θ r − 1 ) λ r + f ( θ r − 1 ) , r = 1 , m + 1 ¯ .

Assume that the matrix G ( θ ) is invertible. Denote by S ( θ ) the inverse matrix G ( θ ) , i.e., S ( θ ) = [ G ( θ ) ] − 1 , where S ( θ ) = s p , k ( θ ) , p , k = 1 , m + 1 ¯ . Then from (12), we can uniquely determine μ :

μ = [ G ( θ ) ] − 1 H ( θ , λ ) = S ( θ ) H ( θ , λ ) ,

i.e.,

(13) μ r = ∑ k = 1 m + 1 s r , k ( θ ) { A 0 ( θ k − 1 ) λ k + f ( θ k − 1 ) } , r = 1 , m + 1 ¯ .

In (6), substituting the right-hand side of (13) instead of μ i + 1 , i = 1 , m + 1 ¯ , we obtain

(14) d u r d t = A 0 ( t ) ( u r + λ r ) + ∑ j = 1 m + 1 D j ( t ) λ j + F ( t ) , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ ,

where

D j ( t ) = ∑ i = 1 m A i ( t ) s i + 1 , j ( θ ) A 0 ( θ j − 1 ) , j = 1 , m + 1 ¯ , F ( t ) = ∑ i = 1 m ∑ k = 1 m + 1 A i ( t ) s i + 1 , k ( θ ) f ( θ k − 1 ) + f ( t ) .

Definition 2

The Cauchy problem (14) and (7) is called uniquely solvable, if for any λ ∈ R n ( m + 1 ) , f ( t ) ∈ C ( [ 0 , T ] , R n ) it has a unique solution.

Let Φ r ( t ) be a fundamental matrix of the differential equation d x d t = A 0 ( t ) x on [ θ r − 1 , θ r ] , r = 1 , m + 1 ¯ .

Then the unique solution to the Cauchy problem for the system of ordinary differential equations (14) and (7) with the fixed values

(15) u r ( t ) = Φ r ( t ) ∫ θ r − 1 t Φ r − 1 ( τ ) A 0 ( τ ) d τ λ r + Φ r ( t ) ∫ θ r − 1 t Φ r − 1 ( τ ) ∑ j = 1 m + 1 D j ( τ ) d τ λ j + Φ r ( t ) ∫ θ r − 1 t Φ r − 1 ( τ ) F ( τ ) d τ , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ .

As a rule, construction of fundamental matrices for the systems of ordinary differential equations with variable coefficients fails. Therefore, later, we propose a numerical method for solving problem (1), (2). For this purpose, we consider the Cauchy problems for ordinary differential equations on subintervals

(16) d z d t = A ( t ) z + P ( t ) , z ( θ r − 1 ) = 0 , t ∈ [ θ r − 1 , θ r ] , r = 1 , m + 1 ¯ ,

where P ( t ) is either ( n × n ) matrix, or n vector, both continuous on [ θ r − 1 , θ r ] , r = 1 , m + 1 ¯ . Consequently, the solution to problem (16) is a square matrix or a vector of dimension n . Denote by a r ( P , t ) a solution to the Cauchy problem (16). Obviously,

(17) a r ( P , t ) = Φ r ( t ) ∫ θ r − 1 t Φ r − 1 ( τ ) P ( τ ) d τ , t ∈ [ θ r − 1 , θ r ] , r = 1 , m + 1 ¯ ,

where Φ r ( t ) is a fundamental matrix of the differential equation (16) on the r th subinterval.

Now, taking into account (17), we can rewrite (15) in the following form:

(18) u r ( t ) = a r ( A 0 , t ) λ r + ∑ j = 1 m + 1 a r ( D j , t ) λ j + a r ( F , t ) , t ∈ [ θ r − 1 , θ r ) , r = 1 , m + 1 ¯ .

Substituting the right-hand side of (18) into conditions (8) and (9), at the corresponding limit values, we obtain the following system of linear algebraic equations with respect to parameters λ r , r = 1 , m + 1 ¯ :

(19) ∑ j = 0 m B j λ j + 1 + B m + 1 λ m + 1 + B m + 1 { a m + 1 ( A 0 , θ m + 1 ) λ m + 1 + ∑ j = 1 m + 1 a m + 1 ( D j , θ m + 1 ) λ j } + ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) λ r d t + ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) { a r ( A 0 , t ) λ r + ∑ j = 1 m + 1 a r ( D j , t ) λ j } d t = d − B m + 1 a m + 1 ( F , θ m + 1 ) − ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) a r ( F , t ) d t ,

(20) λ i + a i ( A 0 , θ i ) λ i + ∑ j = 1 m + 1 a i ( D j , θ i ) λ j − λ i + 1 = − a i ( F , θ i ) , i = 1 , m ¯ .

We denote the matrix corresponding to the left-hand side of the system of equations (19) and (20) by Q ∗ ( θ ) and write the system in the following form:

(21) Q ∗ ( θ ) λ = F ∗ ( θ ) , λ ∈ R n ( m + 1 ) ,

where F ∗ ( θ ) = ( d − B m + 1 a m + 1 ( F , θ m + 1 ) − ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) a r ( F , t ) d t , − a 1 ( F , θ 1 ) , … , − a m ( F , θ m ) ) ′ ,

Q ∗ ( θ ) = ( Q p , k ( θ ) ) , p , k = 1 , m + 1 ¯ , i.e. , Q 1 , 1 ( θ ) = B 0 + B m + 1 a m + 1 ( D 1 , θ m + 1 ) + ∫ θ 0 θ 1 C ( t ) d t + ∫ θ 0 θ 1 C ( t ) a 1 ( A 0 , t ) d t + ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) a r ( D 1 , t ) d t , Q 1 , k ( θ ) = B k − 1 + B m + 1 a m + 1 ( D k , θ m + 1 ) + ∫ θ k − 1 θ k C ( t ) d t + ∫ θ k − 1 θ k C ( t ) a k ( A 0 , t ) d t + ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) a r ( D k , t ) d t , k = 2 , m ¯ , Q 1 , m + 1 ( θ ) = B m + B m + 1 + B m + 1 a m + 1 ( A 0 , θ m + 1 ) + B m + 1 a m + 1 ( D m + 1 , θ m + 1 ) + ∫ θ m θ m + 1 C ( t ) d t + ∫ θ m θ m + 1 C ( t ) a m + 1 ( A 0 , t ) d t + ∑ r = 1 m + 1 ∫ θ r − 1 θ r C ( t ) a r ( D m + 1 , t ) d t , Q p , p − 1 ( θ ) = I + a p − 1 ( A 0 , θ p − 1 ) + a p − 1 ( D p − 1 , θ p − 1 ) , p = 2 , m + 1 ¯ , Q p , p ( θ ) = a p − 1 ( D p , θ p − 1 ) − I , p = 2 , m + 1 ¯ , Q p , k ( θ ) = a p − 1 ( D k , θ p − 1 ) , k ≠ p , k ≠ p − 1 , k = 1 , m + 1 ¯ , p = 2 , m + 1 ¯ .

1.2 Numerical implementation of the method

We offer the following numerical implementation of the Dzhumabaev parametrization method for solving linear boundary value problem for essentially loaded differential equations with integro-multipoint condition based on the Runge-Kutta method of fourth-order and Simpson’s method.

  1. Suppose we have a partition: 0 = θ 0 < θ 1 < θ 2 < ⋯ < θ m − 1 < θ m < θ m + 1 = T . Divide each r th interval [ θ r − 1 , θ r ] , r = 1 , m + 1 ¯ , into N r parts with step size h r = ( θ r − θ r − 1 ) ∕ N r . Assume that on each interval [ θ r − 1 , θ r ] , r = 1 , m + 1 ¯ , the variable θ ^ takes its discrete values: θ ^ = θ r − 1 , θ ^ = θ r − 1 + h r , … , θ ^ = θ r − 1 + ( N r − 1 ) h r , θ ^ = θ r , and denote by { θ r − 1 , θ r } , r = 1 , m + 1 ¯ , the set of such points.

  2. We find the values of ( n × n ) matrices a r ( A 0 , θ ^ ) , a r ( D i , θ ^ ) , i = 1 , m + 1 ¯ , and n vector a r ( F , θ ^ ) on { θ r − 1 , θ r } , r = 1 , m + 1 ¯ .

  3. Applying Simpson’s method on the set { θ r − 1 , θ r } , r = 1 , m + 1 ¯ , we evaluate the definite integrals

    W r ( C ) = ∫ θ r − 1 θ r C ( t ) d t , W r ( F ) = ∫ θ r − 1 θ r C ( t ) a r ( F , t ) d t , r = 1 , m + 1 ¯ , W r ( A 0 ) = ∫ θ r − 1 θ r C ( t ) a r ( A 0 , t ) d t , W r ( D j ) = ∫ θ r − 1 θ r C ( t ) a r ( D j , t ) d t , j = 1 , m + 1 ¯ .

  4. Construct the system of linear algebraic equations in parameters

    (22) Q ∗ h ˜ ( θ ) λ = F ∗ h ˜ ( θ ) , λ ∈ R n ( m + 1 ) ,

    and find its solution λ h ˜ . As noted earlier, the elements of λ h ˜ = ( λ 1 h ˜ , λ 2 h ˜ , … , λ m h ˜ , λ m + 1 h ˜ ) are the values of an approximate solution to problem (1), (2) at the left end-points of the subintervals: x h ˜ r ( θ r − 1 ) = λ r h ˜ , r = 1 , m + 1 ¯ .

  5. To define the values of an approximate solution at the remaining points of set { θ r − 1 , θ r } , r = 1 , m + 1 ¯ , we solve the Cauchy problems:

    d x d t = A 0 ( t ) x + ∑ j = 1 m + 1 D j ( t ) λ j h ˜ + F ( t ) , x ( θ r − 1 ) = λ r h ˜ , t ∈ [ θ r − 1 , θ r ] , r = 1 , m + 1 ¯ .

    We found the solutions to Cauchy problems by using the fourth-order Runge-Kutta method. Thus, the algorithm allows us to find the numerical solution to the problem (1), (2), when the matrices G ( θ ) , Q ∗ ( θ ) are invertible.

1.3 Solvability of the problem

In this section, we establish conditions for the unique solvability of problem (1), (2). To prove the main assertion, we need the following lemma.

Lemma 1

If G ( θ ) is invertible, then the following assertions hold:

  1. If λ ˜ ∈ R n ( m + 1 ) and μ ˜ ∈ R n ( m + 1 ) are solutions of systems (21) and (12), respectively, and the function system u ˜ [ t ] ∈ C ( [ 0 , T ] , θ , R n ( m + 1 ) ) is a solution to the Cauchy problem (6), (7) with λ r = λ ˜ r , μ r = μ ˜ r , r = 1 , m + 1 ¯ , then the function x ˜ ( t ) , defined by the equalities:

    x ˜ ( t ) = u ˜ r ( t ) + λ ˜ r , t ∈ [ θ r − 1 , θ r ) , lim t → θ r − 1 + 0 x ˜ ˙ ( t ) = μ ˜ r , r = 1 , m + 1 ¯ , x ˜ ( T ) = λ ˜ m + 1 + lim t → T − 0 u ˜ m + 1 ( t ) ,

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

  2. The vectors λ ∗ ∈ R n ( m + 1 ) and μ ∗ ∈ R n ( m + 1 ) , composed of the values of the solution x ∗ ( t ) and its derivative x ˙ ∗ ( t ) to problem (1), (2) at the partition points λ r ∗ = lim t → θ r − 1 + 0 x ∗ ( t ) , and μ r ∗ = lim t → θ r − 1 + 0 x ˙ ∗ ( t ) , r = 1 , m + 1 ¯ , satisfy the systems (21) and (12), respectively.

The evidence is similar in concept to the proof of Lemma 1 with slight variations [27].

We present the main theorem on the existence of a unique solution to problem (1), (2) in terms of matrices G ( θ ) and Q ∗ ( θ ) .

Theorem 1

Let the matrices G ( θ ) : R n ( m + 1 ) → R n ( m + 1 ) and Q ∗ ( θ ) : R n ( m + 1 ) → R n ( m + 1 ) be invertible. Then boundary value problem (1), (2) has a unique solution x ∗ ( t ) for any f ( t ) ∈ C ( [ 0 , T ] , R n ) , d ∈ R n .

Proof

Let f ( t ) ∈ C ( [ 0 , T ] , R n ) and d ∈ R n . We find an unique solution to the systems (12) and (21), using the invertibility of matrices G ( θ ) and Q ∗ ( θ ) : μ ∗ = [ G ( θ ) ] − 1 P ( θ , λ ∗ ) and λ ∗ = − [ Q ∗ ( θ ) ] − 1 F ∗ ( θ ) . Solving the Cauchy problem (6), (7) with λ = λ ∗ and μ = μ ∗ , define the function system u ∗ [ t ] . The invertibility of the matrices G ( θ ) and Q ∗ ( θ ) lead to the existence of unique function system u ∗ [ t ] with the elements u r ∗ [ t ] , defined by the right-hand side of (15) at λ = λ ∗ . Then, according to Lemma 1, the function x ∗ ( t ) , defined by the equalities: x ∗ ( t ) = u r ∗ ( t ) + λ r ∗ , t ∈ [ θ r − 1 , θ r ) , lim t → θ r − 1 + 0 x ˙ ∗ ( t ) = μ r ∗ , r = 1 , m + 1 ¯ , x ∗ ( T ) = λ m + 1 ∗ + lim t → T − 0 u m + 1 ∗ ( t ) , is a solution to problem (1), (2). Uniqueness of the solution is proved by contradiction. Theorem 1 is proved.□

To illustrate the fulfillment of the theorem’s conditions and the proposed approach of the numerical solving of the boundary value problem for systems of essentially loaded differential equations with integro-multipoint condition (1), (2) based on the Dzhumabaev parametrization method, let us consider the following examples.

1.4 Examples

Example 1

We consider a linear boundary value problem for essentially loaded differential equations with an integro-multipoint condition:

(23) d x d t = A 0 ( t ) x + A 1 ( t ) x ˙ ( θ 1 ) + A 2 ( t ) x ˙ ( θ 2 ) + A 3 ( t ) x ˙ ( θ 3 ) + f ( t ) , t ∈ ( 0 , 1 ) ,

(24) ∑ j = 0 4 B j x ( θ j ) + ∫ θ 0 θ 4 C ( t ) x ( t ) d t = d , d ∈ R 3 , x ∈ R 3 .

Here,

θ 0 = 0 , θ 1 = 1 4 , θ 2 = 1 2 , θ 3 = 3 4 , θ 4 = T = 1 , f ( t ) = 36 t − 108 t 3 − 132 t 2 − 48 t 4 − 396 48 t 2 − 90 t 3 − 12 t 4 − 252 t − 696 − 360 t 4 − 228 t 3 − 108 t 2 − 408 t − 54 ,

A 0 ( t ) = t 2 t 2 t − 1 t 3 0 3 t 2 5 6 t , A 1 ( t ) = 1 t t 2 4 t 6 0 3 t 4 t 2 , A 2 ( t ) = t 0 6 t 2 4 − 5 0 t − 5 7 t , A 3 ( t ) = t 2 3 t 0 t + 2 6 7 0 t 3 1 , B 0 = 1 4 5 6 3 − 2 8 0 − 1 , B 1 = − 3 0 5 3 2 1 − 5 3 5 , B 2 = 1 6 8 3 0 7 6 − 4 2 , B 3 = 5 0 6 2 1 − 6 4 6 8 , B 4 = 2 6 − 7 6 0 5 − 5 11 3 , C ( t ) = − t 8 0 − 4 2 1 0 3 t − 5 , d = 368 869 454 .

We use the numerical implementation of algorithm. The accuracy of the solution depends on the accuracy of solving the Cauchy problem on subintervals and evaluating definite integrals. We provide the results of the numerical implementation of algorithm by partitioning the subintervals [ 0 , 0.25 ] , [ 0.25 , 0.5 ] , [ 0.5 , 0.75 ] , [ 0.75 , 1 ] with step h = 0.025 . Solving the system of equations (22), we obtain the numerical values of the parameters

λ 1 h ˜ = 12.00002405 − 6.00001267 0.0000129 , λ 2 h ˜ = 12.00002727 − 3.00001292 6.00000624 , λ 3 h ˜ = 24.00002692 − 0.00001382 18.00000584 , λ 4 h ˜ = 48.00001942 2.99998431 36.00000785 .

Exact solution of problem (23) and (24) is the following: x ∗ ( t ) = 96 t 2 − 24 t + 12 12 t − 6 48 t 2 + 12 t .

The differences between the exact and approximate solutions to problem (23) and (24)  ε i = ∣ x ( i ) ∗ ( t k ) − x ˜ ( i ) ( t k ) ∣ , i = 1 , 3 ¯ , are provided in Table 1. Table 2 provides the difference between numerical and exact solutions of problem (23) and (24), where we solve Cauchy problems by using the Runge-Kutta Fehlberg method.

Table 1

Error analysis in Example 1 (using the fourth-order Runge-Kutta method)

k t k ε 1 ε 2 ε 3 k t k ε 1 ε 2 ε 3
0 0 2.4 × 1 0 − 5 1.27 × 1 0 − 5 1.29 × 1 0 − 5 20 0.5 2.69 × 1 0 − 5 1.38 × 1 0 − 5 0.58 × 1 0 − 5
1 0.025 2.45 × 1 0 − 5 1.27 × 1 0 − 5 1.19 × 1 0 − 5 21 0.525 2.66 × 1 0 − 5 1.39 × 1 0 − 5 0.61 × 1 0 − 5
2 0.05 2.49 × 1 0 − 5 1.27 × 1 0 − 5 1.09 × 1 0 − 5 22 0.55 2.62 × 1 0 − 5 1.41 × 1 0 − 5 0.63 × 1 0 − 5
3 0.075 2.52 × 1 0 − 5 1.27 × 1 0 − 5 1.01 × 1 0 − 5 23 0.575 2.57 × 1 0 − 5 1.42 × 1 0 − 5 0.66 × 1 0 − 5
4 0.1 2.56 × 1 0 − 5 1.27 × 1 0 − 5 0.93 × 1 0 − 5 24 0.6 2.51 × 1 0 − 5 1.44 × 1 0 − 5 0.69 × 1 0 − 5
5 0.125 2.59 × 1 0 − 5 1.27 × 1 0 − 5 0.86 × 1 0 − 5 25 0.625 2.44 × 1 0 − 5 1.46 × 1 0 − 5 0.72 × 1 0 − 5
6 0.15 2.63 × 1 0 − 5 1.27 × 1 0 − 5 0.8 × 1 0 − 5 26 0.65 2.36 × 1 0 − 5 1.48 × 1 0 − 5 0.74 × 1 0 − 5
7 0.175 2.66 × 1 0 − 5 1.28 × 1 0 − 5 0.75 × 1 0 − 5 27 0.675 2.28 × 1 0 − 5 1.5 × 1 0 − 5 0.77 × 1 0 − 5
8 0.2 2.68 × 1 0 − 5 1.28 × 1 0 − 5 0.7 × 1 0 − 5 28 0.7 2.18 × 1 0 − 5 1.52 × 1 0 − 5 0.78 × 1 0 − 5
9 0.225 2.71 × 1 0 − 5 1.29 × 1 0 − 5 0.66 × 1 0 − 5 29 0.725 2.07 × 1 0 − 5 1.54 × 1 0 − 5 0.79 × 1 0 − 5
10 0.25 2.73 × 1 0 − 5 1.29 × 1 0 − 5 0.62 × 1 0 − 5 30 0.75 1.94 × 1 0 − 5 1.57 × 1 0 − 5 0.79 × 1 0 − 5
11 0.275 2.74 × 1 0 − 5 0.13 × 1 0 − 5 0.59 × 1 0 − 5 31 0.775 1.8 × 1 0 − 5 1.6 × 1 0 − 5 0.76 × 1 0 − 5
12 0.3 2.76 × 1 0 − 5 1.31 × 1 0 − 5 0.57 × 1 0 − 5 32 0.8 1.65 × 1 0 − 5 1.63 × 1 0 − 5 0.71 × 1 0 − 5
13 0.325 2.77 × 1 0 − 5 1.31 × 1 0 − 5 0.55 × 1 0 − 5 33 0.825 1.47 × 1 0 − 5 1.67 × 1 0 − 5 0.63 × 1 0 − 5
14 0.35 2.77 × 1 0 − 5 1.32 × 1 0 − 5 0.54 × 1 0 − 5 34 0.85 1.28 × 1 0 − 5 1.71 × 1 0 − 5 0.51 × 1 0 − 5
15 0.375 2.77 × 1 0 − 5 1.33 × 1 0 − 5 0.54 × 1 0 − 5 35 0.875 1.06 × 1 0 − 5 1.75 × 1 0 − 5 0.34 × 1 0 − 5
16 0.4 2.77 × 1 0 − 5 1.34 × 1 0 − 5 0.54 × 1 0 − 5 36 0.9 0.82 × 1 0 − 5 1.8 × 1 0 − 5 0.1 × 1 0 − 5
17 0.425 2.76 × 1 0 − 5 1.35 × 1 0 − 5 0.54 × 1 0 − 5 37 0.925 0.54 × 1 0 − 5 1.85 × 1 0 − 5 0.22 × 1 0 − 5
18 0.45 2.74 × 1 0 − 5 1.36 × 1 0 − 5 0.55 × 1 0 − 5 38 0.95 0.23 × 1 0 − 5 1.91 × 1 0 − 5 0.66 × 1 0 − 5
19 0.475 2.72 × 1 0 − 5 1.37 × 1 0 − 5 0.57 × 1 0 − 5 39 0.975 0.12 × 1 0 − 5 1.98 × 1 0 − 5 1.23 × 1 0 − 5
20 0.5 2.69 × 1 0 − 5 1.38 × 1 0 − 5 0.58 × 1 0 − 5 40 1 0.53 × 1 0 − 5 2.06 × 1 0 − 5 1.97 × 1 0 − 5
Table 2

Error analysis in Example 1 (using the Runge-Kutta Fehlberg method)

k t k ε 1 ε 2 ε 3 k t k ε 1 ε 2 ε 3
0 0 1.11 × 1 0 − 9 8.88 × 1 0 − 10 1.46 × 1 0 − 9 20 0.5 1.35 × 1 0 − 9 1.02 × 1 0 − 9 8.27 × 1 0 − 10
1 0.025 1.14 × 1 0 − 9 8.92 × 1 0 − 10 1.39 × 1 0 − 9 21 0.525 1.32 × 1 0 − 9 1.03 × 1 0 − 9 8.19 × 1 0 − 10
2 0.05 1.17 × 1 0 − 9 8.97 × 1 0 − 10 1.32 × 1 0 − 9 22 0.55 1.30 × 1 0 − 9 1.04 × 1 0 − 9 8.22 × 1 0 − 10
3 0.075 1.19 × 1 0 − 9 9.02 × 1 0 − 10 1.26 × 1 0 − 9 23 0.575 1.27 × 1 0 − 9 1.05 × 1 0 − 9 8.27 × 1 0 − 10
4 0.1 1.22 × 1 0 − 9 9.07 × 1 0 − 10 1.21 × 1 0 − 9 24 0.6 1.24 × 1 0 − 9 1.06 × 1 0 − 9 8.35 × 1 0 − 10
5 0.125 1.24 × 1 0 − 9 9.13 × 1 0 − 10 1.16 × 1 0 − 9 25 0.625 1.20 × 1 0 − 9 1.07 × 1 0 − 9 8.45 × 1 0 − 10
6 0.15 1.27 × 1 0 − 9 9.19 × 1 0 − 10 1.11 × 1 0 − 9 26 0.65 1.15 × 1 0 − 9 1.09 × 1 0 − 9 8.57 × 1 0 − 10
7 0.175 1.29 × 1 0 − 9 9.25 × 1 0 − 10 1.07 × 1 0 − 9 27 0.675 1.10 × 1 0 − 9 1.10 × 1 0 − 9 8.72 × 1 0 − 10
8 0.2 1.31 × 1 0 − 9 9.32 × 1 0 − 10 1.03 × 1 0 − 9 28 0.7 1.04 × 1 0 − 9 1.11 × 1 0 − 9 8.88 × 1 0 − 10
9 0.225 1.33 × 1 0 − 9 9.38 × 1 0 − 10 9.97 × 1 0 − 10 29 0.725 9.80 × 1 0 − 10 1.13 × 1 0 − 9 9.05 × 1 0 − 10
10 0.25 1.34 × 1 0 − 9 9.45 × 1 0 − 10 9.65 × 1 0 − 10 30 0.75 9.07 × 1 0 − 10 1.14 × 1 0 − 9 9.23 × 1 0 − 10
11 0.275 1.36 × 1 0 − 9 9.52 × 1 0 − 10 9.50 × 1 0 − 10 31 0.775 8.00 × 1 0 − 10 1.16 × 1 0 − 9 7.98 × 1 0 − 10
12 0.3 1.37 × 1 0 − 9 9.59 × 1 0 − 10 9.25 × 1 0 − 10 32 0.8 7.08 × 1 0 − 10 1.18 × 1 0 − 9 7.96 × 1 0 − 10
13 0.325 1.38 × 1 0 − 9 9.67 × 1 0 − 10 9.03 × 1 0 − 10 33 0.825 6.08 × 1 0 − 10 1.20 × 1 0 − 9 7.89 × 1 0 − 10
14 0.35 1.38 × 1 0 − 9 9.75 × 1 0 − 10 8.83 × 1 0 − 10 34 0.85 4.96 × 1 0 − 10 1.22 × 1 0 − 9 7.74 × 1 0 − 10
15 0.375 1.38 × 1 0 − 9 9.83 × 1 0 − 10 8.67 × 1 0 − 10 35 0.875 3.74 × 1 0 − 10 1.25 × 1 0 − 9 7.49 × 1 0 − 10
16 0.4 1.38 × 1 0 − 9 9.91 × 1 0 − 10 8.53 × 1 0 − 10 36 0.9 2.38 × 1 0 − 10 1.28 × 1 0 − 9 7.09 × 1 0 − 10
17 0.425 1.38 × 1 0 − 9 9.99 × 1 0 − 10 8.42 × 1 0 − 10 37 0.925 8.99 × 1 0 − 11 1.31 × 1 0 − 9 6.51 × 1 0 − 10
18 0.45 1.37 × 1 0 − 9 1.01 × 1 0 − 9 8.34 × 1 0 − 10 38 0.95 7.39 × 1 0 − 11 1.35 × 1 0 − 9 5.69 × 1 0 − 10
19 0.475 1.36 × 1 0 − 9 1.02 × 1 0 − 9 8.29 × 1 0 − 10 39 0.975 2.55 × 1 0 − 10 1.39 × 1 0 − 9 4.56 × 1 0 − 10
20 0.5 1.35 × 1 0 − 9 1.02 × 1 0 − 9 8.27 × 1 0 − 10 40 1 4.55 × 1 0 − 10 1.43 × 1 0 − 9 3.02 × 1 0 − 10

From Tables 1 and 2, it can be seen that the Runge-Kutta Fehlberg method is able to produce lower error rates compared with those from the fourth-order Runge-Kutta method. All results were received by using MathCad15.

Example 2

We consider a linear boundary value problem for essentially loaded differential equations with an integro-multipoint condition:

(25) d x d t = t 1 0 2 t 2 x + 1 2 t t 3 0 x ˙ 1 4 + 0 t 4 t + 3 1 x ˙ 1 2 + 3 t 6 t 2 2 x ˙ 3 4 + 64 t 3 − 33 t 4 − 9 t 2 + 1,039 16 t − 3,139 16 1,509 16 t 2 − 131 16 t 3 − 24 t 4 + 349 4 t + 407 4 , t ∈ ( 0 , 1 ) ,

(26) 1 0 3 5 x ( 0 ) + − 3 8 1 3 x 1 4 + 2 − 3 5 7 x 1 2 + 9 0 3 5 x 3 4 + 1 6 7 3 x ( 1 ) + ∫ 0 1 t 2 4 1 7 t x ( t ) d t = 761 30 8,495 96 , x ∈ R 2 .

The exact solution of problem (25) and (26) is the following: x ∗ ( t ) = t 3 − 64 t 2 + 18 12 t 2 + 20 t .

We use the numerical implementation of algorithm. The accuracy of the solution depends on the accuracy of solving the Cauchy problem on subintervals and evaluating definite integrals. We provide the results of the numerical implementation of algorithm by partitioning the subintervals [ 0 , 0.25 ] , [ 0.25 , 0.5 ] , [ 0.5 , 0.75 ] , [ 0.75 , 1 ] with step h 1 = 0.025 . For the difference of the corresponding values of the exact and constructed solutions of the problem, the following estimate is true:

max j = 0 , 40 ¯ ‖ x ∗ ( t j ) − x ˜ ( t j ) ‖ < 0.0000014 .

With step h 2 = 0.0125 :

max j = 0 , 80 ¯ ‖ x ∗ ( t j ) − x ˜ ( t j ) ‖ < 0.000000088 .

With step h 3 = 0.00625 :

max j = 0 , 160 ¯ ‖ x ∗ ( t j ) − x ˜ ( t j ) ‖ < 0.0000000055 .

With step h 4 = 0.003125 :

max j = 0 , 320 ¯ ‖ x ∗ ( t j ) − x ˜ ( t j ) ‖ < 0.00000000034 .

Acknowledgments

The authors thank the reviewers for their constructive remarks on our work.

  1. Funding information: This research was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09058457).

  2. Conflict of interest: The authors state no conflict of interest.

References

[1] A. M. Nakhushev, Loaded Equations and Their Applications, Nauka, Moscow, 2012 (in Russian). Search in Google Scholar

[2] A. M. Krall, The development of general differential and general differential-boundary systems, Rocky Mt. J. Math. 5 (1975), no. 4, 493–542, https://doi.org/10.1216/RMJ-1975-5-4-493. Search in Google Scholar

[3] D. S. Dzhumabaev, E. A. Bakirova, and S. T. Mynbayeva, A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation, Math. Methods Appl. Sci. 43 (2020), no. 4, 1788–1802, DOI: https://doi.org/10.1002/mma.6003. 10.1002/mma.6003Search in Google Scholar

[4] A. M. Nakhushev, A method for approximate solution of boundary value problems for differential equations and its applications to the dynamics of soil moisture and groundwater, Differ. Equ. 18 (1982), 72–81 (in Russian). Search in Google Scholar

[5] A. A. Alikhanov, A. M. Berezkov, and M. Kh. Shkhanukhov-Lafishev, boundary value problems for certain classes of loaded differential equations and solving them by finite difference methods, Comput. Math. Math. Phys. 48 (2005), no. 9, 1581–1590, https://doi.org/10.1134/S096554250809008X. Search in Google Scholar

[6] A. T. Assanova and Zh. M. Kadirbayeva, On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations, Comput. Appl. Math. 37 (2018), 4966–4976, https://doi.org/10.1007/s40314-018-0611-9. Search in Google Scholar

[7] T. K. Yuldashev and O. Kh. Abdullaev, Unique solvability of a boundary value problem for a loaded fractional parabolic-hyperbolic equation with nonlinear terms, Lobachevskii J. Math. 42 (2021), no. 5, 1113–1123, DOI: https://doi.org/10.1134/S1995080221050218. 10.1134/S1995080221050218Search in Google Scholar

[8] T. K. Yuldashev, B. I. Islomov, and E. K. Alikulov, Boundary-value problems for loaded third-order parabolic-hyperbolic equations in infinite three-dimensional domains, Lobachevskii J. Math. 41 (2020), no. 5, 926–944, DOI: https://doi.org/10.1134/S1995080220050145. 10.1134/S1995080220050145Search in Google Scholar

[9] U. Baltaeva, Solvability of the analogs of the problem Tricomi for the mixed type loaded equations with parabolic-hyperbolic operators, Bound. Value Probl. 2014 (2014), 211, https://doi.org/10.1186/s13661-014-0211-6. Search in Google Scholar

[10] A. T. Assanova, A. E. Imanchiyev, and Zh. M. Kadirbayeva, Numerical solution of systems of loaded ordinary differential equations with multipoint conditions, Comput. Math. Math. Phys. 58 (2018), 508–516, DOI: https://doi.org/10.1134/S096554251804005X. 10.1134/S096554251804005XSearch in Google Scholar

[11] A. T. Assanova and Zh. M. Kadirbayeva, Periodic problem for an impulsive system of the loaded hyperbolic equations, Electron. J. Differential Equations 2018 (2018), no. 72, 1–8. 10.1063/1.5000620Search in Google Scholar

[12] A. M. Nakhushev, Equations of Mathematical Biology, Vyshaiya shkola, Moscow, 1995 (in Russian). Search in Google Scholar

[13] M. T. Dzhenaliev and M. I. Ramazanov, Loaded Equations as Perturbations of Differential Equations, Gylym, Almaty, 2010 (in Russian). Search in Google Scholar

[14] V. M. Abdullaev and K. R. Aida-zade, Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations, Comput. Math. Math. Phys. 54 (2014), 1096–1109, DOI: https://doi.org/10.1134/S0965542514070021. 10.1134/S0965542514070021Search in Google Scholar

[15] V. M. Abdullaev and K. R. Aida-zade, On the numerical solution to loaded systems of ordinary differential equations with non-separated multipoint and integral conditions, Numer. Anal. Appl. 7 (2014), no. 1, 1–14, DOI: https://doi.org/10.1134/S1995423914010017. 10.1134/S1995423914010017Search in Google Scholar

[16] V. M. Abdullaev and K. R. Aida-zade, Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions, J. Inverse Ill-Posed Probl. 24 (2016), no. 5, 543–558, DOI: https://doi.org/10.1515/jiip-2015-0011. 10.1515/jiip-2015-0011Search in Google Scholar

[17] I. N. Parasidis, Extension method for a class of loaded differential equations with nonlocal integral boundary conditions, Bullet. Karaganda Univ. 96 (2019), no. 4, 58–68. 10.31489/2019M4/58-68Search in Google Scholar

[18] I. N. Parasidis and E. Providas, An exact solution method for a class of nonlinear loaded difference equations with multipoint boundary conditions, J. Differential Equations Appl. 24 (2018), no. 10, 1649–1663, DOI: https://doi.org/10.1080/10236198.2018.1515928. 10.1080/10236198.2018.1515928Search in Google Scholar

[19] I. N. Parasidis, E. Providas, and V. Dafopoulos, Loaded differential and Fredholm integro-differential equations with nonlocal integral boundary conditions, Appl. Math. Control Sci. 3 (2018), 50–68, https://doi.org/10.15593/2499-9873/2018.3.04. Search in Google Scholar

[20] I. N. Parasidis and E. Providas, Closed-form solutions for some classes of loaded difference equations with initial and nonlocal multipoint conditions, in: N. Daras and T. Rassias (eds), Modern Discrete Mathematics and Analysis, Springer Optimization and Its Applications, vol. 131, Springer, Cham, 2018, pp. 363–387, https://doi.org/10.1007/978-3-319-74325-7_19. Search in Google Scholar

[21] E. Providas and I. N. Parasidis, On the solution of boundary value problems for loaded ordinary differential equations, in: I. N. Parasidis, E. Providas, and T. M. Rassias (eds), Mathematical Analysis in Interdisciplinary Research, Springer Optimization and Its Applications, vol. 179, Springer, Cham, 2021, pp. 641–659, https://doi.org/10.1007/978-3-030-84721-0_29. Search in Google Scholar

[22] C. Nuchpong, S. K. Ntouyas, and J. Tariboon, Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions, Open Math. 18 (2020), 1879–1894, https://doi.org/10.1515/math-2020-0122. Search in Google Scholar

[23] A. Alsaedi, B. Ahmad, B. Alghamdi, and S. K. Ntouyas, On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions, Open Math. 19 (2021), 760–772, https://doi.org/10.1515/math-2021-0069. Search in Google Scholar

[24] Zh. M. Kadirbayeva, A numerical method for solving boundary value problem for essentially loaded differential equations, Lobachevskii J. Math 42 (2021), no. 3, 551–559, https://doi.org/10.1134/S1995080221030112. Search in Google Scholar

[25] D. S. Dzhumabaev, Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation, USSR Comput. Math. Math. Phys. 29 (1989), 34–46. 10.1016/0041-5553(89)90038-4Search in Google Scholar

[26] S. M. Temesheva, D. S. Dzhumabaev, and S. S. Kabdrakhova, On one algorithm to find a solution to a linear two-point boundary value problem, Lobachevskii J. Math. 42 (2021), no. 3, 606–612, DOI: https://doi.org/10.1134/S1995080221030173. 10.1134/S1995080221030173Search in Google Scholar

[27] D. S. Dzhumabaev, On one approach to solve the linear boundary value problems for Fredholm integro-differential equations, J. Comput. Appl. Math. 294 (2016), 342–357, https://doi.org/10.1016/j.cam.2015.08.023. Search in Google Scholar
Received: 2022-02-22
Revised: 2022-07-21
Accepted: 2022-09-07
Published Online: 2022-10-12

© 2022 Zhazira M. Kadirbayeva and Symbat S. Kabdrakhova, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0496
Keywords for this article
loaded differential equation; integro-multipoint condition; parametrization method; numerical solution
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