Order Linear Ordinary Differential Equations Research Articles

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Developing the Third Order Linear and Nonlinear Fuzzy Ordinary Differential Equations in Hilbert Space by the Extension Principle.

In this study, a third nonlinear fuzzy ordinary differential equation is defined and established in Hilbert space using extension principle. The results obtained gives clear distinction between this work and the existing ones in the literature an example is formulated which shows that a closed subset of a Hilbert space is a Hilbert Space. It is recommended that future study should consider the development of system of third order linear and nonlinear fuzzy ordinary differential equations in Hilbert space by the extension principle.

Algorithmic realization of exact three‐point difference scheme for singular Sturm–Liouville problem

AbstractA new algorithmic realization of exact three‐point difference schemes for a class of singular Sturm–Liouville problems is presented. It is shown that to compute the coefficients of the exact scheme in an arbitrary grid node, it is necessary to solve two auxiliary initial value problems for the second order linear ordinary differential equations: one problem on the interval (forward) and one problem on the interval (backward). Finally, the coefficient stability of the exact three‐point difference scheme is proved.

Phase function methods for second order linear ordinary differential equations with turning points

Phase function methods for second order linear ordinary differential equations with turning points

Generalized Hukuhara Derivatives of Solving of Second Order Linear Homogeneous ODEs by Generalized Triangular Fuzzy Number Using Fuzzy Laplace Transform

Second order linear homogeneous ordinary differential equations are solved using fuzzy Laplace transform method with generalized Hukuhara differentiability concept. The results obtained is in this study is in the form of generalized triangular fuzzy number. Existence and uniqueness of solution are also obtained. However, based on the cases presented, the results have shown that relationship existed between the FLT of second order and its kth derivative for k ≥1.

Cost Analysis of Solar Water Pumping System for Small Town Water Supply

This study investigates the economics of a solar water pumping system for usage in small towns. The system consists of two solar panels, an inverter, a water pumping mechanism, and two parallel tanks. Using the transition diagram, a system of first order linear ordinary differential equations is generated and used to generate a statement of system availability and profit. To validate the expressions of system availability and profit, numerical experiments in the MATLAB software in the form of tables and graphs are utilized. According to the findings, the system's profitability increases as the value of repair rates across all subsystems increases, while it decreases when the value of failure rates across all subsystems increases.

Numerical solution for high-order ordinary differential equations using H-ELM algorithm

PurposeThis paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).Design/methodology/approachThe proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.FindingsTo demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.Originality/valueThe H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

An analysis of errors for pre-service teachers in first order ordinary differential equations

Literature has established that some learners encountered difficulties solving first order ordinary differential equations (ODEs). The use of error analysis in teaching ODEs is believed to make essential contribution towards calculus knowledge development. This paper therefore focuses on analyzing pre-service teachers’ (PSTs) errors and misconceptions apropos of first order ODEs. The paper analyzed the nature of errors made in a test which was written by PSTs on the above topic. The test comprised various types of first order differential equations such as ODEs with separable variables, exact ODEs, ODEs that needed integrating factors, linear ODEs, and homogeneous ODEs. The purpose was to investigate the challenges faced by PSTs in various types of ODEs and the nature of misconceptions that they had in each particular type. This is a qualitative study that involved 63 PSTs who wrote a test on ODEs after being taught the topic for two weeks. The authors marked the work in order to ascertain the misconceptions and errors exhibited by the participants in the test. The PSTs’ performance in the test was analyzed using the SOLO taxonomy and the Newman’s theory mistake analysis. The study established that the topic was rather difficult for PSTs due to various reasons that included, among others, knowledge gaps in integration rules, algebraic computations and, in rare cases, differentiation, as well as misapplication of the rules of natural logarithms. This research therefore recommends that mathematics teacher educators ought to rather focus on the concept of integration and basic algebra before introducing the topic on ODEs to teachers on training.

Concept of gH Differentiability in Solving Second Order Linear Homogeneous ODEs Based on the Relation between FLT and Its kth Derivative for the Case in Fuzzy Environment

In this study, a fuzzy Laplace transform is used to solve second order linear homogeneous ordinary differential equations. The solution obtained is based on the concept of gH differentiability and the relation between the fuzzy Laplace transform and its derivative for is obtained. Examples are constructed for the existence and uniqueness of solutions of second order FODE.

Surfaces of revolution with prescribed mean and skew curvatures in Lorentz-Minkowski space

In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew ($S$) and mean ($H$) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that $H$ and $S$ can be interpreted in terms of the expected value and standard deviation of the normal curvature seen as a random variable, we address the problem of prescribed curvatures for surfaces of revolution. For surfaces with a non-lightlike axis and prescribed $H$, the strategy consists in rewriting the equation for $H$, which is initially a nonlinear second order Ordinary Differential Equation (ODE), as a linear first order ODE with coefficients in a certain ring of hypercomplex numbers along the generating curves: complex numbers for curves on a spacelike plane and Lorentz numbers for curves on a timelike plane. We also solve the problem for surfaces of revolution with a lightlike axis by using a certain ODE with real coefficients. On the other hand, for the skew curvature problem, we rewrite the equation for $S$, which is initially a nonlinear second order ODE, as a linear first order ODE with real coefficients. In all the problems, we are able to find the parameterization for the generating curves in terms of certain integrals of $H$ and $S$.

Fourier Integral Transformation Method for Solving Two Dimensional Elasticity Problems in Plane Strain Using Love Stress Functions

The Fourier integral method was used in this work to determine the stress fields in a two dimensional (2D) elastic soil mass of semi-infinite extent subject to line and strip loads of uniform intensity acting on the boundary. The two dimensional plane strain problem was formulated using stress-based method. The Fourier integral was used to transform the biharmonic stress compatibility equation to a fourth order linear ordinary differential equation (ODE) in terms of the stress function. The ODE was solved subject to the boundedness condition to obtain the bounded stress function. Cartesian stress components were obtained using the Love stress functions. Application of the stress boundary conditions for the case of line load of uniform intensity and the cases of uniformly distributed load on a strip of finite width gave the respective unknown constants of the Love stress functions; and hence the complete determination of the Cartesian stress components for the two cases considered. Inversion of the Fourier integral expressions obtained for the normal and shear stresses in the Fourier parameter gave respective expressions for the normal and shear stress fields for line and finite strip loads of finite width in the physical domain variables. The results obtained agreed with the results from previous studies which used displacement based methods.

Solving polynomials with ordinary differential equations

In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More concretely, it satisfies several simple separated variables ODE, a first order generalized Abel ODE of degree n−1 and an (n−1)-th order linear ODE. Although some of our results are not new, our approach is simple and self-contained. For n=2,3 and 4 we recover, from these ODE, the classical formulas for solving these polynomials.

Model Persamaan Diferensial Orde Satu Untuk Masalah Kinematika Garis Lurus

Dalam sejarah pemodelan matematika, persamaan diferensial muncul sebagai salah satu alat untuk menyelesaikan berbagai masalah dalam kehidupan manusia termasuk pengembangan teori dalam matematika. Dalam penelitian ini akan dikonstruksi model-model persamaan diferensial untuk masalah kinematika garis lurus. Metode yang digunakan adalah metode kajian pustaka. Model-model persamaan diferensial yang diperoleh adalah (1) Persamaan diferensial dengan syarat awal adalah model persamaan diferensial untuk menyatakan masalah perubahan posisi benda setiap saat waktu t. Integrasi model ini akan menghasilkan persamaan posisi suatu benda, (2) persamaan dengan syarat awal yang menyatakan perubahan kecepatan setiap saat t. Integrasi dari model ini akan menghasilkan persamaan kecepatan suatu benda yang bergerak pada lintasan lurus, (3) Persamaan untuk menyatakan hukum Newton Kedua dan dengan syarat awal maka dengan integrasi akan menghasilkan persamaan kecepatan yang mempertimbgkan massa benda dan resultan gaya yang diberikan, (4) persamaan dengan syarat awal sehingga dengan integrasi akan menghasilkan persamaan kecepatan untuk benda jatuh bebas dan (5) persamaan diferensial biasa linear orde satu dan persamaan diferensial Bernoulli untuk menyatakan perubahan kecepatan suatu benda jatuh bebas yang dipengaruhi oleh gaya gesek udara.

Approximating non linear higher order ODEs by a three point block algorithm.

Differential equations are commonly used to model various types of real life applications. The complexity of these models may often hinder the ability to acquire an analytical solution. To overcome this drawback, numerical methods were introduced to approximate the solutions. Initially when developing a numerical algorithm, researchers focused on the key aspect which is accuracy of the method. As numerical methods becomes more and more robust, accuracy alone is not sufficient hence begins the pursuit of efficiency which warrants the need for reducing computational cost. The current research proposes a numerical algorithm for solving initial value higher order ordinary differential equations (ODEs). The proposed algorithm is derived as a three point block multistep method, developed in an Adams type formulae (3PBCS) and will be used to solve various types of ODEs and systems of ODEs. Type of ODEs that are selected varies from linear to nonlinear, artificial and real life problems. Results will illustrate the accuracy and efficiency of the proposed three point block method. Order, stability and convergence of the method are also presented in the study.

Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation

In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E²), maximum absolute error (E^∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.

A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.

Extraction of Some New Solutions of Linear Differential Equations by Dual Space

In this study, the solutions of second order linear homogeneous ordinary differential equations being variable and constant coefficients have been obtained by using power series method which is used frequently in dual space. The solutions of second order homogeneous ordinary differential equations being constant and variable coefficients in dual space by using power series have given and some applications have implemented.

Basic Control Theory for Linear Fractional Differential Equations With Constant Coefficients

In this paper we present an analogous result of the famous Kalman controllability criterion for first order linear ordinary differential equations with constant coefficients that applies to the case of linear differential equations of fractional order with constant coefficients. We present some simple examples, including a linear fractional harmonic oscillator, to illustrate our results. Moreover, another simple linear system of incommensurate fractional orders is suggested as an open problem.

A Block Diagram of Electromagnetoelastic Actuator for Control Systems in Nanoscience and Nanotechnology

The block diagram and the transfer functions of the electromagnetoelastic actuator are received for control systems in nanoscience and nanotechnology. The block diagram of the electromagnetoelastic actuator is reflected the transformation of electrical energy into mechanical energy, in contrast to Cady’s and Mason’s electrical equivalent circuits of piezotransducer. The electromagnetoelasticity equation and the second order linear ordinary differential equation with boundary conditions are solved for calculations the block diagram of the electromagnetoelastic actuator. The block diagram of the piezoactuator is obtained with using the reverse and direct piezoelectric effects. The back electromotive force is determined from the direct piezoelectric effect equation. The transfer functions of the piezoactuators are obtained for control systems in nanoscience and nanotechnology.

On Modelling and Simulation of Electric Circuit Problems

Differential equations are of fundamental importance in Mathematics, Physical Sciences and Engineering Mathematics. Many mathematical relations and physical laws appeared in the form of such equations. This paper reviewed an application of these equations in solving mathematical model on electric circuit problems using the First order linear differential equation. The analytical approached in solving the equations confirmed that solving electric circuits using first order linear ordinary differential equations gives accurate and reliable result. Therefore, the application is of importance and great need. However, complex problems need higher order differential equations, which are nonlinear and have entirely different approach in finding their solutions.