Explicit Solutions for Linear Partial Differential Equations Using Bezier Functions

Venkataraman, P.

doi:10.1115/detc2006-99227

Cited by 3 publications

(8 citation statements)

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“…The use of implicit Bézier functions to obtain approximate solutions of BVP (or IVP) is not a new idea, albeit it is quite recent (2004). 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].…”

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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“…Venkataraman solved approximately the linear partial differential equation by using the Bézier method as an extension of the solution of the ordinary differential equation. The method was applied for the solution of several engineering problems, such as the Poison equation, one-dimensional heat equation, and the slender two dimensional cantilever beam . Manikandan and Kamanat solved three dimensional Navier Stokes equations near the rotating disk by using the Bézier curve method, and the results were compared with the numerical solution given by the conventional boundary value problem solver .…”

Section: Introductionmentioning

confidence: 99%

“…According to Venkataraman, the Bézier curve is advantageous when solving differential equations for four main reasons: its approximations are accurate, its formulation is simple, the differential equations can be handled in their original forms, and standard optimization techniques can be applied. 22 …”

Section: Introductionmentioning

confidence: 99%

Bézier Curve Method to Compute Various Meniscus Shapes

Lewis

Matsuura

2023

ACS Omega

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This paper is an extension of our earlier paper in which it was shown that the meniscus shape in a cylindrical capillary could be computed by solving the Young−Laplace equation via optimization of a Beźier curve. This work extends the previous work by demonstrating that the method is applicable to predict the meniscus shape not only in a cylindrical capillary but also in other cases, such as at a tilted plate, between two plates, and for a sessile drop. Numerous works have attempted previously to solve the Young−Laplace equation, and their results all agree with this paper's validating its method. All the preceding approaches, however, used special techniques to solve the differential equation, while the Beźier curve method proposed in this work is more simple, which allows it to maintain greater computational simplicity. Moreover, the Beźier curve method can be applied to solve many other different differential equations in the same way as shown in this work. The effect of the Beźier curve degree on the precision of prediction was also thoroughly investigated. It was found that the 4th degree Beźier curve was required to predict the meniscus shape precisely in a cylindrical capillary, against a tilted plate, and between two plates, while the 5th degree was required for the shape of the sessile drop.

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“…This work uses the Bézier curve method, which approximates the solution to any differential equation by changing the parameters of the Bézier curve to minimize the difference between the two sides of the differential equation. According to Venkataraman, the Bézier curve is advantageous when solving differential equations for four main reasons: its approximations are accurate, its formulation is simple, the differential equations can be handled in their original forms, and standard optimization techniques can be applied …”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Calculation of the Meniscus Shape Formed under Gravitational Force by Solving the Young–Laplace Differential Equation Using the Bézier Curve Method

Lewis

Matsuura

2022

ACS Omega

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This work presents a method to calculate the meniscus shape by solving the differential equation based on the Young−Laplace equation. More specifically, the differential equation is solved by applying the cubic Beźier curve. A complicated nonlinear differential equation is solved using the Beźier control points and the least-squares method while maintaining computational simplicity. The results show all of the expected features of the meniscus under the gravitational force. A brief discussion is also made on the effect of the errors on the results. The method is further validated by its agreement with the numerical solutions reported in the existing literature.

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Explicit Solutions for Linear Partial Differential Equations Using Bezier Functions

Cited by 3 publications

References 0 publications

Least-Squares Solutions of Nonlinear Differential Equations

Least-Squares Solutions of Nonlinear Differential Equations

Bézier Curve Method to Compute Various Meniscus Shapes

Calculation of the Meniscus Shape Formed under Gravitational Force by Solving the Young–Laplace Differential Equation Using the Bézier Curve Method

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