Problems For Ordinary Differential Equations Research Articles

One–Step Hybrid Block Scheme for the Numerical Approximation for Solution of Third Order Initial Value Problems

In this research, we have proposed the simulation of linear block algorithm for modeling third order highly stiff problem without reduction to a system of first order ordinary differential equation, to address the weaknesses in reduction method. The method is derived using the linear block method through interpolation and collocation. The basic properties of the block method were recovered and was found to be consistent, convergent and zero-stability. The new block method is been applied to model third order initial value problems of ordinary differential equations without reducing the equations to their equivalent systems of first order ordinary differential equations. The result obtained on the process on some sampled modeled third order linear problems give better approximation than the existing methods which we compared our result with.

Transformasi Fourier Multiplikatif Dan Aplikasinya Pada Persamaan Diferensial Multiplikatif

This research is a development of multiplicative calculus. This study is about the Fourier multiplicative transformation and its application to the multiplicative differential equation. This study aims to determine the Fourier multiplicative transformation as well as the multiplicative differential equation. This study contains numerical simulations to solve the problem of ordinary multiplicative differential equations of the first order. The methods used in this research are descriptive research methods through the study of literature. The results of this study are the application of multiplicative Fourier transformations to multiplicative differential equations and numerical solutions of ordinary multiplicative differential equations with the Adam Bashforth-Moulton multiplicative method. Â Keywords: Multiplicative Calculus, Fourier Multiplicative Transformation, Multiplicative Differential Equations, Adams Bashforth Moulton Multiplicative Method

Correlations in superradiance of a two-dimensional system

The process of cooperative spontaneous emission in a system of two-level emitters interacting via electromagnetic field is analyzed using the Bogolyubov reduced description method, which provides the possibility of studying the state of the generated field. Field macrostates are described with electric and magnetic field amplitudes and binary correlation functions of their components. Since the previously obtained set of differential equations describing the Dicke system evolution with considering field correlations is very cumbersome, we must find some way of simplifying the model to make it accessible for numerical modeling. The general picture of correlation development can be elucidated with using one- and two-dimensional models. The paper presents the set of evolution equations for Dicke system with fixed orientation of dipole moments of emitters forming a two-dimensional structure. The material equations and equilibrium correlations are discussed. The transition to the Cauchy problem for ordinary differential equations is proposed.

Нелінійні коливання сандвіч-пластини зі стільниковим наповнювачем, виго-товленим методом 3d-друку

A three-layer sandwich plate with a FDM-printed honeycomb core made of polycarbonate is considered. The upper and lower faces of the sandwich are made of a carbon fiber-reinforced composite. To study the response of the sandwich plate, the honeycomb core is replaced with a homogeneous layer with appropriate mechanical properties. To verify the honeycomb core model, a finite-element simulation of the representative volume of the core was performed using the ANSYS software package. A modification of the high-order shear theory is used to describe the structure dynamics. The assumed-mode method is used to simulate nonlinear forced oscillations of the plate. The Rayleigh–Ritz method is used to calculate the eigenfrequencies and eigenmodes of the plate, in which the displacement of the plate points during nonlinear oscillations are expanded. This technique allows one to obtain a finite-degree-of-freedom nonlinear dynamic system, which describes the oscillations of the plate. The frequency response of the system is calculated using the continuation approach applied to a two-point boundary value problem for nonlinear ordinary differential equations and the Floquet multiplier method, which allows one to determine the stability and bifurcations of periodic solutions. The resonance behavior of the system is analyzed using its frequency response. The proposed technique is used to analyze the forced oscillations of a square three-layer plate clamped along the contour. The results of the analysis of the free oscillations of the plate are compared with those of ANSYS finite-element simulation, and the convergence of the results with increasing number of basis functions is analyzed. The comparison shows that the results are in close agreement. The analysis of the forced oscillations shows that the plate executes essentially nonlinear oscillations with two saddle-node bifurcations in the frequency response curve, in which the periodic motion stability of the system changes. The nonlinear oscillations of the plate near the first fundamental resonance are mostly monoharmonic. They may be calculated using the describing function method.

On the inverse Cauchy problem for linear ordinary differential equations

AbstractThe Cauchy problem characterizes the solutions of a linear ordinary differential equation that satisfies initial conditions. In this paper, we investigate the converse problem, namely, given a function that is known to satisfy a linear ordinary differential equation of a fixed order, determine the coefficients of the ordinary differential equation and the initial conditions. The techniques used to investigate the inverse Cauchy problem come from the algebraic estimation problem introduced by Fliess and Sira‐Ramírez. From the perfect observation of the solution, i.e., without external perturbation and noise corrupting it, the initial value problem can be explicitly reconstructed using only iterative indefinite integrals of the solution.

A New Modified Algorithm Based Upon Exponential and Polynomial Functions for Solving Initial Value Problems

The main focus of this paper is to propose a fourth-order convergent modified numerical algorithm by using an exponential and polynomial type function for getting the approximate solution of initial value problems (IVPs) in ordinary differential equations (ODEs). The proposed algorithm has been tested and analyzed for the convergence, stability, consistency, of several numerical experiments that proved the superiority of the proposed algorithm against algorithms including the fourth-order arithmetic mean (AM), the fourth-order harmonic mean (HM), and the fourth-order Contra harmonic mean( ). Keywords: Modified numerical algorithm, Ordinary differential equation, Order of convergence, Initial value problems, Stability, and Consistency. DOI: 10.7176/MTM/11-6-01 Publication date: December 31 st 2021

Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

An adaptive one-point second-derivative Lobatto-type hybrid method for solving efficiently differential systems

This manuscript presents a one-step method incorporating a second-derivative applied to obtain approximate solutions of first-order initial-value problems of ordinary and time-dependent partial differential equations. The new scheme is derived through interpolation and collocation approaches, and the characteristics of the obtained method are analyzed. An embedding-like technique is considered and executed in adaptive mode to get better performance of the proposed method. The new scheme is used for solving some real-life application problems to determine its efficiency and accuracy. The numerical solutions obtained show that the proposed error estimate strategy is suitable in a variable step-size mode, robust and competitive compared with the results provided by other methods in the available literature.

TWO-POINT SECOND BOUNDARY VALUE PROBLEM FOR A QUADRATIC SIMPLE SECOND-ORDER DIFFERENTIAL EQUATION SOLVED BY THE BERNOULLI EQUATION

This article examines the uniqueness of the solution of the boundary problem for the second regular ordinary differential equation, which is solved in the Bernoulli equation. The uniqueness of the issue is proved by the principle of extremes.

On self-similar solutions of time and space fractional sub-diffusion equations

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.

A thorough description of one-dimensional steady open channel flows using the notion of viscosity solution

Determining water surface profiles of steady open channel flows in a one-dimensional bounded domain is one of the well-trodden topics in conventional hydraulic engineering. However, it involves Dirichlet problems of scalar first-order quasilinear ordinary differential equations, which are of mathematical interest. We show that the notion of viscosity solution is useful in thoroughly describing the characteristics of possibly non-smooth and discontinuous solutions to such problems, achieving the conservation of momentum and the entropy condition. Those viscosity solutions are the generalized solutions in the space of bounded measurable functions. Generalized solutions to some Dirichlet problems are not always unique, and a necessary condition for the non-uniqueness is derived. A concrete example illustrates the non-uniqueness of discontinuous viscosity solutions in a channel of a particular cross-sectional shape.

Composite stocky box girders of variable thickness in high-support expressways: Constrained torsional analysis

The constrained torsional problem of composite stocky box girders (CSBG) of variable thickness in high-support expressways is still an intractable challenge in nowadays structural analysis. To surmount this challenge, a kind of new theory for CSBGs is formulated by employing the mindset of mathematical approximation theory in the study. Thusly, the new CSBG’s theory or the real shapes of a CSBG after torsional deformation is expressed as a set of families of curved surfaces by using nodal line method. By the grounds of the new CSBG’s theory and of the energy principle in elasticity, the mathematical model for the constrained torsion problem of a CSBG is transformed into a system of ordinary differential equations (ODEs) containing a finite number of unknown univariate basis functions. This system of ODEs together with their boundary conditions constitute a boundary value problem of ODEs, which can be solved by ODE’s solver. The numerical solutions of the mathematical model of examples reveal some valuable unprecedented occurrences about the behaviours of the CSBG due to pure constrained torsion, and indirectly verify that the CSBG’s theory is reasonable, and that the numerical analysis method illustrated in the study is effective.

Unique solvability of boundary value problem for functional differential equations with involution

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

Richardson-Kalitkin method in abstract description

An abstract description of the RichardsonKalitkin method is given for obtaining a posteriori estimates for the proximity of the exact and found approximate solution of initial problems for ordinary differential equations (ODE). The problem Ρ{{\Rho}} is considered, the solution of which results in a real number uu. To solve this problem, a numerical method is used, that is, the set Hℝ{H\subset \mathbb{R}} and the mapping uh:Hℝ{u_h:H\to\mathbb{R}} are given, the values of which can be calculated constructively. It is assumed that 0 is a limit point of the set HH and uh{u_h} can be expanded in a convergent series in powers of h:uh=u+c1hk+...{h:u_h=u+c_1h^k+...}. In this very general situation, the RichardsonKalitkin method is formulated for obtaining estimates for uu and cc from two values of uh{u_h}. The question of using a larger number of uh{u_h} values to obtain such estimates is considered. Examples are given to illustrate the theory. It is shown that the RichardsonKalitkin approach can be successfully applied to problems that are solved not only by the finite difference method.

Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces.

Methods for solving the initial value problem with a two-sided estimate of the local error

Numerical methods for solving the initial value problem for ordinary differential equations are proposed. Embedded methods of order of accuracy 2(1), 3(2) and 4(3) are constructed. To estimate the local error, two-sided calculation formulas were used, which give estimates of the main terms of the error without additional calculations of the right-hand side of the differential equation, which favorably distinguishes them from traditional two-sided methods of the Runge- Kutta type.

New similarity variable to transform the fluid flow from PDEs into fractional-order ODEs: Numerical study

The recent research aims to study the fractional-order differential equations of the higher-order. The problems consist of momentum and thermal boundary layers. The similarity variables are introduced to convert the partial differential equations (PDEs) into the fractional-order ordinary differential equations (ODEs). The two nonlinear problems of fractional order ODEs are tackled through the (Fractional Differential equation-12) FDE-12 method. The obtained results show that the fractional-order problems are more nonlinear as compared to the integer-order differential equations. The fractional-order increasing the nonlinearity and consequently more efforts required to obtain the outputs. The impact of the physical parameters in the case of the integer-order and fractional-order are obtained and discussed. It is observed that the impact of the physical parameters more compact in the case of fractional order as compared to the integer-order case. The important features of the drag force and heat transfer rate in the case of the fractional-order boundary value problem also have been analyzed and discussed.

On the Solvability of a Periodic Problem for Nonlinear Ordinary Differential Equation of the Second Order

On the Solvability of a Periodic Problem for Nonlinear Ordinary Differential Equation of the Second Order

Spectral properties of a beam equation with eigenvalue parameter occurring linearly in the boundary conditions

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$.

Approximate Solution of Higher Order Fuzzy Initial Value Problems of Ordinary Differential Equations Using Bezier Curve Representation

The Bezier curve is a parametric curve used in the graphics of a computer and related areas. This curve, connected to the polynomials of Bernstein, is named after the design curves of Renault's cars by Pierre Bézier in the 1960s. There has recently been considerable focus on finding reliable and more effective approximate methods for solving different mathematical problems with differential equations. Fuzzy differential equations (known as FDEs) make extensive use of various scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under uncertainty. This article discusses the use of Bezier curves for solving elevated order fuzzy initial value problems (FIVPs) in the form of ordinary differential equation. A Bezier curve approach is analyzed and updated with concepts and properties of the fuzzy set theory for solving fuzzy linear problems. The control points on Bezier curve are obtained by minimizing the residual function based on the least square method. Numerical examples involving the second and third order linear FIVPs are presented and compared with the exact solution to show the capability of the method in the form of tables and two dimensional shapes. Such findings show that the proposed method is exceptionally viable and is straightforward to apply.