Fourth-order Ordinary Differential Equation Research Articles

On the Boundedness of Solution of the Second Order Ordinary Differential Equation with Involution

In the present paper, the initial value problem for the second order ordinary differential equation with involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with involution is established.

Elzaki transform method for finding solutions to two-dimensional elasticity problems in polar coordinates formulated using Airy stress functions

In this paper, the Elzaki transform method is used for solving two-dimensional (2D) elasticity problems in plane polar coordinates. Airy stress function was used to express the stress compatibility equation as a biharmonic equation. Elzaki transform was applied with respect to the radial coordinate to a modified form of the stress compatibility equation, and the biharmonic equation simplified to a fourth order ordinary differential equation (ODE). The general solution for the Airy stress potential function in the Elzaki transform space was obtained by solving the ODE. By inversion, the general solution for the Airy stress potential function was obtained in the physical domain space variables in terms of four unknown integration constants. Normal stresses and shear stress fields were also determined for the general case of 2D elasticity problems. The Flamant problem was solved as a particular illustration of 2D elasticity problems. The stress boundary conditions and the requirement of equilibrium of the internal stress resultants and the external forces were used simultaneously to determine the four constants of integration. The Airy stress potential function and the normal and shear stress fields were thus completely determined. The principle of superposition was used to obtain the elasticity solutions for the stress fields in the elastic half plane due to strip load of infinite extent, and solutions for horizontal stresses on smooth rigid retaining walls due to strip loads and parallel line load of infinite extent acting on the elastic half plane.

New Efficient Numerical Model for Solving Second, Third and Fourth Order Ordinary Differential Equations Directly

This article presents a two-step hybrid linear multistep block method for solving second, third and fourth order initial value problems of ordinary differential equations directly. The derivation of the method was done using collocation and interpolation techniques while approximated power series was used as an interpolating polynomial. The fourth derivative of the power series is collocated at the entire grid and off grid points while the fifth and sixth derivatives of the polynomial are collocated at the end point only. The basic properties of the developed method, that is, order, error constant, zero stability, region of absolute stability, convergence and consistence of the method are properly investigated. The numerical results demonstrated that the scheme developed handles second, third and fourth order ordinary differential equations efficiently and also better in accuracy when compared with existing methods. The proposed method takes away the burden of developing separate method for the solution of second, third and fourth order initial value problem of ordinary differential equations.

Analog Hawking effect: A master equation

We consider further on the problem of the analogue Hawking radiation. We propose a fourth order ordinary differential equation, which allows to discuss the problem of Hawking radiation in analogue gravity in a unified way, encompassing fluids and dielectric media. In a suitable approximation, involving weak dispersive effects, WKB solutions are obtained far from the horizon (turning point), and furthermore an equation governing the behaviour near the horizon is derived, and a complete set of analytical solutions is obtained also near the horizon. The subluminal case of the original fluid model introduced by Corley and Jacobson, the case of dielectric media are discussed. We show that in this approximation scheme there is a mode which is not directly involved in the pair-creation process. Thermality is verified and a framework for calculating the grey-body factor is provided.

Symmetry solutions and conservation laws of a (3+1)-dimensional generalized KP-Boussinesq equation in fluid mechanics

In this paper we analyse a (3+1)-dimensional generalized Kadomtsev-Petviashvili-Boussinesq equation, which describes the evolution of shallow water waves. This equation was formulated not long ago, by including the term uttto the generalized (3+1)-dimensional Kadomtsev-Petviashvili equation. Invoking the Lie symmetry methods we perform several symmetry reductions and reduce the equation to a fourth-order nonlinear ordinary differential equation. Consequently, this ordinary differential equation is solved and its general solution is derived connected with Weierstrass zeta function. The profiles of solutions are displayed graphically. In addition to those, we construct some travelling waves by engaging Kudryashov’s method. To complete our study we finally derive some conservation laws of the equation under consideration by appealing to the Ibragimov’s conservation theorem.

Symmetry analysis for a fourth-order noise-reduction partial differential equation

We apply the theory of Lie symmetries in order to study a fourth-order 1+2 evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries for the specific 1+2 partial differential equations and we apply the invariant functions to determine similarity solutions. For the static solutions we observe that the reduced fourth-order ordinary differential equations are reduced to second-order ordinary differential equations which are maximally symmetric. Finally, nonstatic closed-form solutions are also determined.

Novel Lagrangian hierarchies, generalized variational ODE’s, and families of regular and embedded solitary waves

Hierarchies of Lagrangians of degree two, each only partly determined by the choice of leading terms and with some coefficients remaining free, are considered. The free coefficients they contain satisfy the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation, and derived by solution of the full inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations (ODEs) respectively. However, our Lagrangians have significantly greater freedom since our existence conditions are for individual coefficients in the Lagrangian. In particular, the classes of Lagrangians derived here have four arbitrary or free functions, including allowing the leading coefficient in the resulting variational ODEs to be arbitrary, and with models based on the earlier general criteria for a variational representation being special cases. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations, some of which recover known physical models. Families of regular and embedded solitary waves are derived for some of these generalized variational ODEs in appropriate parameter regimes, with the embedded solitons occurring only on isolated curves in the part of parameter space where they exist. Future work will involve higher order Lagrangians, the resulting equations of motion, and their solitary wave solutions.

Global stability of traveling waves for delay reaction-diffusion systems without quasi-monotonicity

In this article, we consider the existence of positive solutions for a nonlinear system of fourth-order ordinary differential equations. By constructing a single cone \(P\) in the product space \(C[0, 1] \times C[0, 1]\) and applying fixed point theorem in cones, we establish the existence of positive solutions for a system in which the nonlinear terms are both superlinear or sublinear. In addition, by the construction of the product cone \(K_1 \times K_2 \subset C[0, 1] \times C[0, 1]\) along with the product formula of fixed point theory on a product cone, we investigate the existence of positive solutions involving nonlinear terms, one uniformly superlinear or sublinear, and the other locally uniformly sublinear or superlinear.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/45/abstr.html

Variable thermal conductivity and viscosity effects on non-Newtonian fluids flow through a vertical porous plate under Soret-Dufour influence

The purpose of this paper is to consider heat and mass transfer processes on two non-Newtonian fluids (Walters-B viscoelastic and Casson fluid) situated in a hot environment. This study considered concentrated fruit juice and tomato sauce as the type of Casson fluid and polymethyl methacrylate, chromatography, and ceramic processing as the type of Walters-B viscoelastic fluids. The model is governed by systems of partial differential equations. These equations were simplified with the help of appropriate similarity variables to obtain nonlinear couple fourth-order ordinary differential equations. The transformed equations were solved numerically using the spectral homotopy analysis method (SHAM). SHAM is rooted in the homotopy analysis method (HAM) and the Chebyshev spectral collocation method. SHAM is efficient and converges faster than HAM. This paper is unique because it explored variable viscosity and thermal conductivity in the presence of flow parameters such as Soret, Dufour, radiation, viscous dissipation, magnetic field, and so on. The applied magnetic field serves as an opposition to the flow and thereby slows down the motion of an electrically conducting fluid. The influence of pertinent flow parameters is extensively discussed and presented graphically. The permeability parameter is noticed to decrease the velocity profile and increase the temperature profile. The present results were found to be in good agreement with existing works in literature.

On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary value conditions: a fixed point theory approach

On the symmetric positive solutions of nonlinear fourth order ordinary differential equations with four-point boundary value conditions: a fixed point theory approach

Existence of Non-Spurious Solutions to Discrete Fourth-Order Lidstone Boundary Value Problems

In this paper, we study boundary value problems of difference equations which arise as discrete approximations to Lidstone boundary value problems of fourth-order ordinary differential equations. By developing the method of lower and upper solutions, we investigate the existence of solutions as well as the nonexistence of spurious solutions to the discrete problems.

The fourth-order ordinary differential equation with the fractional initial/boundary conditions

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Complete Classification of Scalar Fourth-Order Ordinary Differential Equations and Linearizing Algorithms

Complete Classification of Scalar Fourth-Order Ordinary Differential Equations and Linearizing Algorithms

Dependence of eigenvalues of fourth-order boundary value problems with transmission conditions

A general fourth-order regular ordinary differential equation with eigenvalue dependent boundary conditions and transmission conditions are considered. We prove that the eigenvalues depend continuously and smoothly on the coefficients of the differential equation and on the boundary and transmission matrices. We provide as well formulas for the derivatives with respect to each of these parameters.

An Implicit Collocation Method for Direct Solution of Fourth Order Ordinary Differential Equations

This paper presented a linear multistep method for solving fourth order initial value problems of ordinary differential equations. Collocation and interpolation methods are adopted in the derivation of the new numerical scheme which is further applied to finding direct solution of fourth order ordinary differentiation equations. This implementation strategy is more accurate and efficient than Adams–Bashforth Method solution. The newly derive scheme have better stabilities properties than that of the Adams-Bashforth Method. Numerical examples are included to illustrate the reliability and accuracy of the new methods.Keywords: Linear Multistep Methods, Region of Absolute Stability, Zero-Stability, Error Analysis, Collocation.

Lidstone-type problems on the whole real line and homoclinic solutions applied to infinite beams

This work provides sufficient conditions for the existence of solutions to fourth-order nonlinear ordinary differential equations with Lidstone-type boundary conditions on the real line. Using Green’s functions, we formulate a modified integral equation and correspondent integral operators, in which fixed points are the solutions of the initial problem. Moreover, it is proved that every solution of the Lidstone problem on the whole real line is an homoclinic solution.

A Step by Step Guide on Derivation and Analysis of a New Numerical Method for Solving Fourth-order Ordinary Differential Equations

This manuscript presents a step by step guide on derivation and analysis of a new numerical method to solve initial value problem of fourth order ordinary differential equations. The method adopted hybrid techniques using power series as the basic function. Collocation of the fourth derivatives was done at both grid and off-grid points. The interpolation of the approximate function is also taken at the first four points. The complete derivation of the new technique is introduced and shown here, as well as the full analysis of the method. The discrete schemes and its first, second, and third derivatives were combined together and solved simultaneously to obtain the required 32 family of block integrators. The block integrators are then applied to solve problem. The method was tested on a linear system of equations of fourth order ordinary differential equation in order to check the practicability and reliability of the proposed method. The results are displaced in tables; it converges faster and uses smaller time for its computations. The basic properties of the method were examined, the method has order of accuracy p=10, the method is zero stable, consistence, convergence and absolutely stable. In future study, we will investigate the feasibility, convergence, and accuracy of the method by on some standard complex boundary value problems of fourth order ordinary differential equations. The extension of this new numerical method will be illustrated and comparison will also be made with some existing methods.

Approximate of solution of a fourth order ordinary differential equations via tenth step block method

Approximate of solution of a fourth order ordinary differential equations via tenth step block method

Approximate of solution of a fourth order ordinary differential equations via tenth step block method

This paper carries a different approach of collection and interpolation to develop a tenth block method for the numerical solution of linear or nonlinear ordinary differential equations of fourth order with initial conditions. The method has been implemented at the selected mesh points to generate a direct tenth block method through Taylor series. Some critical properties of this method such as zero stability, order of the method, and convergence have been analysed. Two numerical tests have taken to make a comparison of the approximate results with exact as well as results of other authors.

Численное интегрирование матричным методом и оценка порядка аппроксимации разностных краевых задач для неоднородных линейных обыкновенных дифференциальных уравнений четвертого порядка с переменными коэффициентами

Использование многочлена Тейлора второй степени при аппроксимации производных конечными разностями приводит ко второму порядку аппроксимации традиционного метода сеток при численном интегрировании краевых задач для неоднородных линейных обыкновенных дифференциальных уравнений второго порядка с переменными коэффициентами. В работе при исследовании краевых задач для неоднородных линейных обыкновенных дифференциальных уравнений четвертого порядка с переменными коэффициентами рассмотрен предложенный ранее метод численного интегрирования, использующий средства матричного исчисления, в котором аппроксимация производных конечными разностями не использовалась. Согласно указанному методу, при составлении системы разностных уравнений может быть выбрана произвольная степень многочлена Тейлора в разложении искомого решения задачи в ряд Тейлора. В работе возможные граничные условия дифференциальной краевой задачи записаны как в виде производных степеней от нуля до трех, так и в виде линейных комбинаций этих степеней. Краевая задача названа симметричной, если количества граничных условий в левой и правой границах совпадают и равны двум; в противном случае задача названа несимметричной. Для дифференциальной краевой задачи составлена ее аппроксимирующая разностная краевая задача в виде двух подсистем: в первую подсистему вошли уравнения, при построении которых не были использованы граничные условия краевой задачи; во вторую подсистему вошли четыре уравнения, при построении которых были использованы граничные условия задачи. Теоретически выявлены закономерности между порядком аппроксимации разностной краевой задачи и степенью используемого многочлена Тейлора. Установлено следующее: а) порядок аппроксимации первой и второй подсистем пропорционален степени используемого многочлена Тейлора; б) порядок аппроксимации первой подсистемы меньше степени многочлена Тейлора на две единицы при ее четном значении и меньше на три единицы при ее нечетном значении; в) порядок аппроксимации второй подсистемы меньше степени многочлена Тейлора на три единицы независимо как от четности или нечетности этой степени, так и от степени старшей производной в граничных условиях краевой задачи. Вычислен порядок аппроксимации разностной краевой задачи со всеми возможными комбинациями граничных условий. Теоретические выводы подтверждены численными экспериментами.