A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters

Gasılov, Nızamı; Kaya, Meral

doi:10.1142/s0219876218501153

Cited by 5 publications

(5 citation statements)

References 14 publications

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“…An interval is a set of real numbers. In similar way, it is natural to consider an interval problem as a set of real (classical) problems ( [5,6,7]). Then, we can define the solution as follows.…”

Section: Preliminariesmentioning

confidence: 99%

Solving a system of linear differential equations with interval coefficients

Gasılov¹

2021

DCDS-B

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In this study, we consider a system of homogeneous linear differential equations, the coefficients and initial values of which are constant intervals. We apply the approach that treats an interval problem as a set of real (classical) problems. In previous studies, a system of linear differential equations with real coefficients, but with interval forcing terms and interval initial values was investigated. It was shown that the value of the solution at each time instant forms a convex polygon in the coordinate plane. The motivating question of the present study is to investigate whether the same statement remains true, when the coefficients are intervals. Numerical experiments show that the answer is negative. Namely, at a fixed time, the region formed by the solution's value is not necessarily a polygon. Moreover, this region can be non-convex. The solution, defined in this study, is compared with the Hukuharadifferentiable solution, and its advantages are exhibited. First, under the proposed concept, the solution always exists and is unique. Second, this solution concept does not require a set-valued, or interval-valued derivative. Third, the concept is successful because it seeks a solution from a wider class of set-valued functions.

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Section: Preliminariesmentioning

confidence: 99%

Solving a system of linear differential equations with interval coefficients

Gasılov¹

2021

DCDS-B

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“…We claim that this interpretation is not the only way. In the present study, we model an interval uncertainty changing with time as a bunch of real functions (see previous studies).…”

Section: Preliminariesmentioning

confidence: 99%

On differential equations with interval coefficients

Gasılov

Amrahov

2019

Math Methods in App Sciences

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In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept.Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case. KEYWORDSbunch of functions, interval differential equations, linear differential equations MSC CLASSIFICATION34A12; 65L05; 03E75; 68W25

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“…If the boundary value problem (17) has a unique solution, the boundary value problem (19) has a unique solution. Comparing with boundary value problems (16) and (19), it can conclude…”

Section: The Analytical Solution Of Nonhomogeneous Boundary Value Promentioning

confidence: 99%

“…According to the assumption, the boundary value problem (19) has the following solutions of each interval:…”

Section: The Analytical Solution Of Nonhomogeneous Boundary Value Promentioning

confidence: 99%

“…They also extended and proved some theorems for nonlinear differential equations by means of differential transformation method. Gasilov and Kaya proposed a numerical solution method for solving the interval boundary value problems of linear differential equation, which is different from the usual method. The complexity of the method is reduced from O ( n 5 ) to O ( n 2 ).…”

Section: Introductionmentioning

confidence: 99%

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Similar constructing method for solving nonhomogeneous mixed boundary value problem of n‐interval composite ode and its application

Dong

Liu

2019

Math Methods in App Sciences

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This paper studies a simple method—Similar Constructing Method (SCM)—for constructing the exact solutions of the nonhomogeneous mixed boundary value problem for sets of n‐interval composite second‐order ordinary differential equation (ODE) with variable coefficient. Then this paper proves the correctness of the solution obtained by SCM. After that, this paper has done simulation experiment. This section uses the SCM to solve the nonhomogeneous boundary value problem of three‐interval composite Bessel equation. Solutions are presented in graphical form for various parameter values, and the influence of parameters on the solution is analyzed. The example shows that using SCM to solve the class of nonhomogeneous mixed boundary value problems of n‐interval composite second‐order linear ODE is easy, convenient, and effective.

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A Method for the Numerical Solution of a Boundary Value Problem for a Linear Differential Equation with Interval Parameters

Cited by 5 publications

References 14 publications

Solving a system of linear differential equations with interval coefficients

Solving a system of linear differential equations with interval coefficients

On differential equations with interval coefficients

Similar constructing method for solving nonhomogeneous mixed boundary value problem of n‐interval composite ode and its application

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