Solving Inverse ODE Using Bezier Functions

2009 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Discrete Data ◽  
Exact Form ◽  
Data Points ◽  
High Degree ◽  
Efficient Approximation ◽  
The Given

The simplest inverse boundary value problem is to identify the differential equation and the boundary conditions from a given set of discrete data points. For an ordinary differential equation, it would involve finding a function, which when expressed through some function of itself and its derivatives, and integrated using particular boundary conditions would generate the given data. Parametric Bezier functions are excellent candidates for these functions. They allow efficient approximation of data and its derivative content. The Bezier function is smooth and continuous to a high degree. In this paper the best Bezier function to fit the data represents this function which is being sought. This Bezier approximation also determines the boundary conditions. Next, a generic form of the differential equation is assumed. The Bezier function and its derivatives are then used in this generic form to establish the exponents and coefficients of the various terms in the actual differential equation. The paper looks at homogeneous ordinary differential equations and shows it can recover the exact form of both linear and nonlinear differential equations. Two examples are presented. The first example uses data from the Bessel equation, representing a linear equation. The second example uses the data from the Blassius equation which is nonlinear. In both cases the exact form of the equation is identified from the given discrete data.

Inverse Solution to the Coupled Ordinary Differential Equation

2010 ◽  
Author(s):  
P. Venkataraman
Keyword(s):  
Differential Equation ◽  
Genetic Algorithm ◽  
Differential Equations ◽  
Generic Model ◽  
Real Numbers ◽  
Data Points ◽  
Discrete Values ◽  
Linear Differential ◽  
Discrete Programming ◽  
The Given

The identification of the actual form of the constant coefficient coupled differential equations and their boundary conditions, from two sets of discrete data points, is possible through a unique two-step approach developed in this paper. In the first step, the best Bezier function is fitted to the data. This allows an effective approximation of the data and the required number of derivatives for the entire range of the independent variable. In the second step, the known derivatives are introduced in a generic model of the coupled differential equation. This generic form includes two types of unknowns, real numbers and integers. The real numbers are the coefficients of the various terms in the differential equations, while the integers are exponents of the derivatives. The unknown exponents and coefficients are identified using an error formulation. Two examples are solved. The given data is exact, smooth and they represent solutions to coupled linear differential equations. The solution is obtained through discrete programming. Three methods are presented. The first is limited enumeration, which is useful if the coefficients belong to a limited set of discrete values. The second is global search using the genetic algorithm for a larger choice of coefficient values. The third uses a state space integrator driven by the genetic algorithm, to minimize the error between known data and that obtained from numerical integration.

scholarly journals Stability analysis for first-order nonlinear differential equations with three-point boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
pp. 40-52
Author(s):  
Kamala E. Ismayilova
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
First Order ◽  
Fixed Point Principle ◽  
The Stability ◽  
The Given ◽  
Stability Results

AbstractIn the present paper, we study a system of nonlinear differential equations with three-point boundary conditions. The given original problem is reduced to the equivalent integral equations using Green function. Several theorems are proved concerning the existence and uniqueness of solutions to the boundary value problems for the first order nonlinear system of ordinary differential equations with three-point boundary conditions. The uniqueness theorem is proved by Banach fixed point principle, and the existence theorem is based on Schafer’s theorem. Then, we describe different types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability. We discuss the stability results providing suitable example.

scholarly journals Existence and Uniqueness of Solutions for the System of First-order Nonlinear Differential Equations with Three-point and Integral Boundary Conditions

2019 ◽  
Vol 12 (3) ◽  
pp. 756-770
Author(s):  
Misir Mardanov ◽  
Yagub Sharifov ◽  
Kamala Ismayilova ◽  
Sevinc Zamanova
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Nonlinear Differential Equations ◽  
Integral Boundary Conditions ◽  
Uniqueness Of Solutions ◽  
Integral Boundary ◽  
First Order ◽  
The Green Function ◽  
The Given

In the paper, the existence and uniqueness of the solutions for the system of the nonlinear first-order ordinary differential equations with three-point and integral boundary conditions are studied. The Green function is constructed and the considered problem is reduced to the equivalent integral equation. The existence and uniqueness of the solutions for the given problem are analyzed by using the Banach contraction principle. The Schaefer’s fixed point theorem is thenused to prove the existence of the solutions. Finally, the examples are given to verify the given theorems.

scholarly journals Nonlinear Differential Equations Modeling the Antarctic Circumpolar Current

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Jifeng Chu ◽  
Kateryna Marynets
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Topological Degree ◽  
Antarctic Circumpolar Current ◽  
Fixed Point Theorems ◽  
Nonlinear Differential Equations ◽  
Existence Results ◽  
Circumpolar Current ◽  
The Antarctic

AbstractThe aim of this paper is to study one class of nonlinear differential equations, which model the Antarctic circumpolar current. We prove the existence results for such equations related to the geophysical relevant boundary conditions. First, based on the weighted eigenvalues and the theory of topological degree, we study the semilinear case. Secondly, the existence results for the sublinear and superlinear cases are proved by fixed point theorems.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

scholarly journals Repeated alternating loading of a elastoplastic three-layer plate in a temperature field

2021 ◽  
Vol 21 (1) ◽  
pp. 60-75
Author(s):  
Eduard I. Starovoitov ◽  
◽  
Denis V. Leonenko ◽  
Keyword(s):  
Temperature Field ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Plastic Region ◽  
Local Load ◽  
Natural State ◽  
Elastic Plastic

Axisymmetric deformation of a three-layer circular plate under repeated alternating loading from the plastic region by a local load is considered. To describe kinematics of asymmetrical on the thickness of the plate pack is adopted the hypothesis of a broken line. In a thin elastic-plastic load-bearing layers are used the hypothesis of Kirchhoff. A non-linearly elastic relatively thick filler is incompressible in thickness. It is taken to be a hypothesis of Tymoshenko regarding the straightness and the incompressibility of the deformed normals with linear approximation of the displacements through the thickness layer. The work of the filler in the tangential direction is taken into account. The physical relations of stress-strain relations correspond to the theory of small elastic-plastic deformations. The effect of heat flow is taken into account. The temperature field in the plate was calculated by the formula obtained by averaging the thermophysical parameters over the thickness of the package. The system of differential equations of equilibrium under loading of the plate from the natural state is obtained by the Lagrange variational method. Boundary conditions on the plate contour are formulated. The solution of the corresponding boundary value problem is reduced to finding the three desired functions: deflection, shear and radial displacement of the shear surface of the filler. A non-uniform system of ordinary nonlinear differential equations is written for these functions. Its analytical iterative solution is obtained in Bessel functions by the method of elastic solutions of Ilyushin. In case of repeated alternating loading of the plate, the solution of the boundary value problem is constructed using the theory of variable loading of Moskvitin. In this case, the hypothesis of similarity of plasticity functions at each loading step is used. Their analytical form is taken independent of the point of unloading. However, the material constants included in the approximation formulas will be different. The cyclic hardening of the material of the bearing layers is taken into account. The parametric analysis of the obtained solutions under different boundary conditions in the case of a local load distributed in a circle is carried out. The influence of temperature and nonlinearity of layer materials on the displacements in the plate is numerically investigated.

scholarly journals A New Approach to Approximate Solutions for Nonlinear Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Safia Meftah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Asymptotic Expansions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Approximation Methods ◽  
New Approach

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

Sign in / Sign up

Export Citation Format

Share Document