Bezier Curves Method for Fourth-Order Integrodifferential Equations

Ghomanjani, Fateme; Kamyad, Ali Vahidian; Kılıçman, Adem

doi:10.1155/2013/672058

Cited by 6 publications

(8 citation statements)

References 11 publications

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“…Venkataraman has attacked the problem using optimization techniques [4], [5], [6] while Zheng uses analytical LS approach [7]. Bézier curves have been adopted also to solve specific problems such as singular perturbed BVP [8] as well as integro-DE [9]. Two-point BVP are usually solved by iterative techniques.…”

Section: Least-squares Solution Of Boundary Value Problemsmentioning

confidence: 99%

Least-Squares Solutions of Nonlinear Differential Equations

Mortari

Johnston

Smith

2018

2018 Space Flight Mechanics Meeting

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This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second order differential equations. The proposed method has two steps. The first step consists of writing a constrained expression, introduced in Ref. [1], that has embedded the differential equation constraints. These expressions are given in term of a new unknown function, g(t), and they satisfy the constraints, no matter what g(t) is. The second step consists of expressing g(t) as a linear combination of m independent known basis functions, g(t) = ξ T h(t). Specifically, Chebyshev orthogonal polynomials of the first kind are adopted for the basis functions. This choice requires rewriting the differential equation and the constraints in term of a new independent variable, x ∈ [−1, +1]. The procedure leads to a set of linear equations in terms of the unknown coefficients vector, ξ, that is then computed by least-squares. Numerical examples are provided to quantify the solutions accuracy for initial and boundary values problems as well as for a control-type problem, where the state is defined in one point and the costate in another point.

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Section: Least-squares Solution Of Boundary Value Problemsmentioning

confidence: 99%

Least-Squares Solutions of Nonlinear Differential Equations

Mortari

Johnston

Smith

2018

2018 Space Flight Mechanics Meeting

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“…Since the nonlocal term under the integral sign will cause some mathematical difficulties, the analytical solutions for IDEs are usually not easy to obtain. For the linear fourth-order boundary value problems governed by IDEs like (3), only a few studies have been carried out by using numerical methods; see, e.g., [7][8][9][10][11][12][13] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

Wang,

Yao

2024

Advances in Mathematical Physics

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In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

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“…The better accuracy in terms of error for the approximate solution of higher order ODEs can be obtained by using LSM through the polynomial or piecewise polynomial functions [32]. Meanwhile, many researchers used LSM in solving different types of differential equations approximately [33][34][35][36][37][38][39][40]. The LSM is employed based on the control points of Bézier curves by developing the least square objective function for the discretization of integrals to improve the approximate solutions of higher order ODEs [29].…”

Section: Introductionmentioning

confidence: 99%

“…Lyche and Morken [31] stated that the LSM is an efficient and simple method when employed to solve higher order ODEs approximately. From past reviews, the application of the LSM to find the approximate solution of higher order ODEs based on the Bézier curve's control points, the result is only satisfied, but not on the required level in terms of error [30,36,[39][40]47].…”

Section: Introductionmentioning

confidence: 99%

Solving Ordinary Differential Equations (ODEs) Using Least Square Method Based on Wang Ball Curves

Bhatti¹,

Karim²

2022

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Numerical methods are regularly established for the better approximate solutions of the ordinary differential equations (ODEs). The best approximate solution of ODEs can be obtained by error reduction between the approximate solution and exact solution. To improve the error accuracy, the representations of Wang Ball curves are proposed through the investigation of their control points by using the Least Square Method (LSM). The control points of Wang Ball curves are calculated by minimizing the residual function using LSM. The residual function is minimized by reducing the residual error where it is measured by the sum of the square of the residual function of the Wang Ball curve's control points. The approximate solution of ODEs is obtained by exploring and determining the control points of Wang Ball curves. Two numerical examples of initial value problem (IVP) and boundary value problem (BVP) are illustrated to demonstrate the proposed method in terms of error. The results of the numerical examples by using the proposed method show that the error accuracy is improved compared to the existing study of Bézier curves. Successfully, the convergence analysis is conducted with a two-point boundary value problem for the proposed method.

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Bezier Curves Method for Fourth-Order Integrodifferential Equations

Cited by 6 publications

References 11 publications

Least-Squares Solutions of Nonlinear Differential Equations

Least-Squares Solutions of Nonlinear Differential Equations

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

Solving Ordinary Differential Equations (ODEs) Using Least Square Method Based on Wang Ball Curves

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