Nonhomogeneous Ordinary Differential Equations Research Articles

Solution homogeneous and non-homogeneous fractional ordinary differential equations by ZMA-transform

In this view, a modern vision to discuss homogeneous and non-homogeneous fractional ordinary differential equations (HFODEs)- (NHFODEs) with initial condition (IC) by employment ZMA-transform (ZMAT). Out of exhibit basic properties for theorems of ZMA-transform (ZMAT) and fractional calculus(FC). Moreover presented ZMA-transform (ZMAT) of Riemann-Liovill derivative the consequent application of ZMA-transform (ZMAT) of fractional derivatives and to understand the task more, some applications has been solved.

Solution of System of First Order Nonlinear Non-Homogeneous Fuzzy Ordinary Differential Equations by Embedding Method

In this study, a system of first order nonlinear non-homogeneous fuzzy ordinary differential equations will be examine in fuzzy environment and solved using embedding method. The results of nonlinear non-homogeneous fuzzy ordinary differential equations are established which followed the form of SxS matrices and all the components of the matrices are real functions of time denoted by t. The accuracy of the results obtained is tested on some constructed example and recommended that further study should consider odd and even system of nonlinear non-homogeneous fuzzy ordinary differential equations by embedding method.

Nonlinear analysis of sound transmission loss through cylindrical shell considering companion modes

This paper investigates the nonlinear vibro-acoustic performance of simply supported cylindrical shell under oblique incident plane sound wave including companion modes participation. The cylindrical shell equation of motion is obtained employing Donnell’s nonlinear shallow shell theory. The Galerkin method is applied to the equations of motion in order to obtain the system of nonlinear nonhomogeneous second order ordinary differential equations. The considered mode shapes are two asymmetric driven and companion modes and one axisymmetric mode. Multiple Scales Method is used to determine the response of resultant nonlinear differential equations. Next, the effects of incident sound characteristics on the transmission loss factor are studied. According to the results, effect of companion mode on the transmission loss (TL) through the shell can be seen only in high frequency of the incident sound wave. In addition, at high incident angle of acoustic wave, the companion mode effect on the TL of the system is very small.

Photo-thermoelastic wave propagation with changing thermal conductivity on excited pulsed laser nanoscale microelongated semiconductor material

Indroduction: The novel model of a non-local elastic semiconductor material that is microelongated is created. The process of photothermal transfer is responsible for the stimulation of the material. Photo-thermoelastic theories are used when the thermal conductivity is changed for the non-local medium. Under the influence of laser pulses, the effective framework describes the nanoscale microelongation instance as well as the interference between the photo-thermoelastic propagation waves in the non-local medium. It is possible to think of thermal conductivity as a linear function of temperature when electronic and thermoelastic deformation mechanisms are described. Two-dimensional deformation (2D) is used to extract the main fields, which obtained in non-dimension.Methods: Harmonic wave analysis, which is described by the normal mode, has been used to convert the basic equations into non-homogeneous higher-order ordinary differential equations. Applying a small subset of the possible non-local semiconductor surface conditions leads to exhaustive solutions.Result and Discussion: The outcomes of numerical simulations for silicon (Si) are graphically shown. There are comparisons made and explanations given for the investigated physical factors like their thermal conductivity, laser pulses and microelongation parameters.

Uniform stability and chaotic dynamics in nonhomogeneous linear dissipative scalar ordinary differential equations

The paper analyzes the structure and the inner long-term dynamics of the invariant compact sets for the skewproduct flow induced by a family of time-dependent ordinary differential equations of nonhomogeneous linear dissipative type. The main assumptions are made on the dissipative term and on the homogeneous linear term of the equations. The rich casuistic includes the uniform stability of the invariant compact sets, as well as the presence of Li-Yorke chaos and Auslander-Yorke chaos inside the attractor.

Modified Variational Iteration Method Approach for Fuzzy Initial Value Problem

In this paper, an application of  modified variational iteration method (MVIM) for solving homogeneous and nonhomogeneous ordinary differential equations with fuzzy initial value conditions (FDEs) are examined. This modified form is about a procedure some changes in the general formula of the variational iteration method (VIM) . MVIM showed a qualitative and excellent improvement over the standard VIM. To illustrate how this modified form is works we picked some numerical examples and solved them. The efficacy and validity of MVIM have been confirmed through results obtained.  

Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching

Abstract The article uses ordinary differential to solve inhomogeneous equations by the constructivist learning concept. We use the equivalent equations to study the n-th order non-homogeneous linear ordinary differential equations with constant coefficients and get the method of solving this equation. Then we use the Filippov transformation and comparison theorem to prove the boundedness of all system trajectories. Numerical results show that the calculation formula is effective for solving stiff ordinary differential equations.

New integral “Kuffi-Abbas-Jawad” KAJ transform and its application on ordinary differential equations

In this work, a new integral transform called the “Kuffi-Abbas-Jawad” KAJ integral transform has been proposed and applied to solve ordinary differential equations. The proposed KAJ integral transform is a modification of the well-known (Sadiq-Emad-Eman) SEE integral transform. It is a novel method that could be used to solve ordinary differential equations. The KAJ integral transform has been tested and proven for its excellent ability to solve homogeneous and non-homogeneous higher-order ordinary differential equations.

Solving Non-Homogeneous Linear Ordinary Differential Equations Using Memristor-Capacitor Circuit

Inhomogeneous linear ordinary differential equations (ODEs) and systems of ODEs can be solved in a variety of ways. However, hardware circuits that can perform the efficient analog computation to solve them are rarely in the literature. To address such problems, this paper proposes a general method of using a memristor-capacitor (M-C) circuit to solve inhomogeneous linear ODEs and systems of ODEs of any order in initial value problems. The M-C circuit can match the coefficients of the equations sought by adjusting the memristor resistance value according to the coefficient formula proposed in the paper, which has higher programmability. Then, some ODEs and systems of ODEs are given in the paper as examples to evaluate the proposed method. According to the comparison results based on MATLAB software simulation and the simulation based on OrCAD software, the designed M-C circuit has an effective improvement in speed and the accuracy exceeds 99.95% in software simulation. Based on practical verification, this paper gives the actual M-C circuit experiment based on PCB. Moreover, the proposed method can be used to quickly solve the object motion state in the spring mass damping system in actual engineering, and the accuracy can reach 99.98%.

Mathematical Modelling in Engineering with Integral Transforms via Modified Adomian Decomposition Method

In this work three integral transforms through modified Adomian decomposition method (ADM) are proposed to obtain the approximate analytical solution of different types of mathematical models arising in physical problems. These transformations are applied for both homogeneous and non-homogeneous linear differential equations. The efficiency and accuracy of the proposed methods are implemented through higher order non-homogeneous ordinary differential equations. Numerical tests are reported for applicability of the current scheme based on different transformations and compared with exact solutions.

Scaled boundary finite element method for hydrodynamic bearings in rotordynamic simulations

The behavior of rotor systems with hydrodynamic bearings can be affected crucially by the nonlinear bearing forces. These forces are incorporated in rotordynamic simulations by solving the Reynolds equation in every step of a time integration scheme. Since closed-form analytical solutions of the Reynolds equation are known only for simplified boundary conditions, numerical methods and look-up table approaches are applied. To obtain accurate and convergent results, numerical solutions require a fine discretization, while the look-up tables require a fine grid of interpolation points. In both cases, the numerical effort of the rotordynamic simulation is dominated by the computation of the bearing forces.In this study, a semi-analytical solution of the Reynolds equation based on the Scaled Boundary Finite Element Method (SBFEM) is developed to reduce the numerical effort of the overall simulation. In contrast to a purely numerical solution, the two-dimensional pressure field is discretized in only one coordinate direction, whereas in the other direction, an analytical formulation is used. This results in a system of non-homogeneous ordinary differential equations, which is solved under consideration of the boundary conditions. Since the complexity of the SBFEM model and the resulting numerical effort depend on the modeling of the oil supply groove, three different models are analyzed in this paper. The equations are derived, the solutions are verified, and their numerical efficiency is investigated in comparison to an FEM solution, which is one of the standard numerical approaches.

COMPARATIVE STUDY OF SOLUTION METHODS OF NON-HOMOGENEOUS LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT

The methods to obtain a solution of non-homogeneous linear ordinary differential equations with constant coefficients vary from type of output function. The present paper addresses the undetermined coefficients, variations of parameters methods and discussed the rewards and drawbacks of these methods with examples. The method of undetermined coefficients is derived from the variation of parameters when the out function is an exponential function.

Homogeneous and Non Homogeneous Ordinary Differential Equations with the Second Order

Since Newton time, Differential Equations have been used to comprehend most of physical, geometrical, and vital science in addition to their participation in the study of arithmetic analysis. Furthermore, they have been used in economical and social aspects. They have been developed and got significant position in various sciences. This paper presents for the non-homogeneous ordinary differential equations with the second order. This idea starts in chapter one which talks about the notion of those equations, their orders, in addition to the study of the linear differential equation with the first order and its solution. Whereas the second chapter studies the ordinary differential equations (homogeneous and non-homogeneous) and their solution. Each chapter is supported by solved examples that cover most of the aspects of the topic and we hope that this paper will make a good reference for those who would like to study this topic furtherly.

A semi-analytical approach based on the variational iteration method for static analysis of composite beams

Using the Variational Iteration Method (VIM) the 3D static deflection problem of composite beams subject to concentrated tip and uniformly distributed loads is analysed, resulting in a system of coupled non-homogeneous ordinary differential equations. Using a general Lagrange multiplier, identified by variational theory, a special type of functional is constructed. By making an initial approximation in the form of a Maclaurin series and by using successive iterations, the solution in the form of convergent series is obtained. The results based on VIM are compared against those of the exact solution and Chebyshev Collocation Method (CCM) for different layups and boundary conditions and good agreement is observed between them. These results show the applicability and effectiveness of VIM for the static analysis of composite beams.

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping. Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory. Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function. The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes. By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations. The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency. The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system. It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points. The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data. The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes.

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.

Three-dimensional analytical solution of arbitrarily supported cylindrical panels with weak interfaces using the extended Kantorovich method

This is the first time, a three-dimensional (3D) analytical solution of arbitrarily supported cross-ply cylindrical panel with weak interfaces is developed using multi-term extended Kantorovich method (EKM). A generalized variational based mathematical model is developed by considering the Reissner-type mixed variational principle in conjunction with weak interface model. A linear spring layer model is implemented for depicting the behaviour of weak interfaces. Recently developed EKM is applied to obtain two sets of non-homogeneous ordinary differential equations (ODEs) along radial and circumferential directions, respectively. A suitable solution algorithm is developed for the present case. The accuracy of the presently developed solution is verified by comparing with 3D finite element model using ABAQUS. The multi-term EKM solution is efficient as just two terms yield the accurate results. Further, displacement and stress variations in a laminate at the edges are obtained and some new benchmark results are also presented for arbitrarily supported cylindrical panels with weak interfaces. Effect of the weak interface parameter and other characteristic parameters are investigated to reflect their design aspect. This development may help the researchers to develop analytical models for viscoelastic cases.

Exploring Students’ Difficulties in Solving Nonhomogeneous 2nd Order Ordinary Differential Equations with Initial Value Problems

This research aims to explore students’ difficulties in resolving Nonhomogeneous 2nd Order Ordinary Differential Equations with initial value problems. The method that can be used to solve this equation is the undetermined coefficient and the Laplace transformation. This research is used descriptive method. The subjects of this study were 73 students in the second year of the Mathematics Education. Data is collected through tests and interviews. Data were analyzed descriptive qualitative. The results of data analysis show that in undetermined coefficient method, students difficult in determining the particular solution of non-homogeneous second-order ordinary differential equations. This is due to student errors in the first step especially in determining the characteristics equation. Whereas, for the Laplace transformation method, students most difficulties are in the step of solving the subsidiary equation. This is due to the weakness of students in completing arithmetic operations in the form of fractions and partial fractions.

On Discrete Fractional Solutions of Non-Fuchsian Differential Equations

In this article, we obtain new fractional solutions of the general class of non-Fuchsian differential equations by using discrete fractional nabla operator ∇ η ( 0 < η < 1 ) . This operator is applied to homogeneous and nonhomogeneous linear ordinary differential equations. Thus, we obtain new solutions in fractional forms by a newly developed method.

On a New Modification of Homotopy Analysis Method for Solving Nonlinear Nonhomogeneous Differential Equations

In this paper, new powerful modification of homotopy analysis technique (NMHAM) was submitted to create an approximate solution of nonhomogeneous nonlinear ordinary and partial differential equations. The NMHAM is a combination of the new technique of homotopy analysis method(NHAM) [4] and the new technique of homotopy analysis method(nHAM) [7].Three illustrative examples are employed to illustrate the accuracy and computational proficiency of this approach. The outcomes uncover that the NMHAM is more accurate than the NHAM and nHAM.