Splines and anti-periodic boundary-value problems

Ahmadinia, M.; Loghamni, G. B.

doi:10.1080/00207160701336466

Cited by 2 publications

(5 citation statements)

References 4 publications

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“…In this section, the convergence of the presented method is investigated. Note that the convergence analysis is based on Radjabalipour, Loghmani and Alavizadeh, Ahmadinia and Loghamni, and Loghmani . By substituting in the OCPs to and assuming [ t 0 , t 1 ] = [0,1] the following special case, which is a calculus of variation problem, is obtained:

\min_{x} ({}, I ((), x)),

where,

I ((), x) \approx I ((), x_{n}^{(1)} ((), s)) = \int_{0}^{1} l ((),,, s, x_{n}^{(1)} ((), s), {\dot{x}}_{n}^{(1)} ((), s)) italicds + ψ ((), x_{n}^{(1)} ((), 1)),

in which

l ((),,, s, x ((), s), \overset{x}{true} ((), s)) = L ((),,, s, x ((), s), ϕ ((),,, s, x ((), s), \overset{x}{true} ((), s)))

and,

J ((), x, u) = J ((),, x, ϕ ((),,, s, x ((), s), \overset{x}{true} ((), s)))

J ((), x, u) = J ((),, x, ϕ ((),,, s, x ((), s), \overset{x}{true} ((), s))) = \int_{0}^{1} l ((),,, s, x ((), s), \overset{x}{true} ((), s)) italicds + ψ ((), x ((), 1)) = I ((), x),

…”

Section: Convergence Analysismentioning

confidence: 99%

“…In this section, the convergence of the presented method is investigated. Note that the convergence analysis is based on Radjabalipour, 48 Loghmani and Alavizadeh, 49 Ahmadinia and Loghamni, 50 and Loghmani. 51 By substituting (5) in the OCPs (1) to (3) and assuming [t 0 , t 1 ] = [0,1] the following special case, which is a calculus of variation problem, is obtained:…”

Section: Convergence Analysismentioning

confidence: 99%

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Using B‐spline functions (BSFs) of various degrees to obtain a powerful method for numerical solution for a special class of optimal control problems (OCPs)

Kafash¹,

Alavizadeh

2019

Int J Numerical Modelling

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In this paper, the optimal control problems (OCPs) are converted into a constrained optimization problem based on state parameterization via the Bspline functions (BSFs). In fact, the state variable can be considered as a series of the BSFs with unknown coefficients, and the OCPs are transformed into a constrained optimization problem. With the proposed method, the control and state variables also the performance index can be obtained approximately. Also, the convergence theorem of the presented approach is proved in details, some illustrative examples are reported. Also, an example, which has analytic noncontinuous state and control variable, is presented to show the efficiency and reliability of the purposed method, compared with other existing methods.

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\min_{x} ({}, I ((), x)),

where,

I ((), x) \approx I ((), x_{n}^{(1)} ((), s)) = \int_{0}^{1} l ((),,, s, x_{n}^{(1)} ((), s), {\dot{x}}_{n}^{(1)} ((), s)) italicds + ψ ((), x_{n}^{(1)} ((), 1)),

in which

l ((),,, s, x ((), s), \overset{x}{true} ((), s)) = L ((),,, s, x ((), s), ϕ ((),,, s, x ((), s), \overset{x}{true} ((), s)))

and,

J ((), x, u) = J ((),, x, ϕ ((),,, s, x ((), s), \overset{x}{true} ((), s)))

J ((), x, u) = J ((),, x, ϕ ((),,, s, x ((), s), \overset{x}{true} ((), s))) = \int_{0}^{1} l ((),,, s, x ((), s), \overset{x}{true} ((), s)) italicds + ψ ((), x ((), 1)) = I ((), x),

…”

Section: Convergence Analysismentioning

confidence: 99%

Section: Convergence Analysismentioning

confidence: 99%

Using B‐spline functions (BSFs) of various degrees to obtain a powerful method for numerical solution for a special class of optimal control problems (OCPs)

Kafash¹,

Alavizadeh

2019

Int J Numerical Modelling

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“…Thus, we conclude that the obstacle problem (18) is equivalent to solving the variational inequality problem (22). This equivalence has been used to study the existence of a unique solution of (18), [1,4,5].…”

Section: Applicationsmentioning

confidence: 99%

“…This equivalence has been used to study the existence of a unique solution of (18), [1,4,5]. Now using the idea of Lewy and Stampacchia [8], problem (22) can be written as…”

Section: Applicationsmentioning

confidence: 99%

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Numerical Solution of Obstacle Problems by B-Spline Functions

Loghmani¹,

Mahdifar²,

Alavizadeh³

2011

AJCM

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In this study, we use B-spline functions to solve the linear and nonlinear special systems of differential equations associated with the category of obstacle, unilateral, and contact problems. The problem can easily convert to an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. The numerical examples and computational results illustrate and guarantee a higher accuracy for this technique.

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Splines and anti-periodic boundary-value problems

Cited by 2 publications

References 4 publications

Using B‐spline functions (BSFs) of various degrees to obtain a powerful method for numerical solution for a special class of optimal control problems (OCPs)

Using B‐spline functions (BSFs) of various degrees to obtain a powerful method for numerical solution for a special class of optimal control problems (OCPs)

Numerical Solution of Obstacle Problems by B-Spline Functions

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