Order Linear Equations Research Articles

Approximating Higher Order Linear Fredholm Integro-Differential Equations by an Efficient Adomian Decomposition Method

Approximating Higher Order Linear Fredholm Integro-Differential Equations by an Efficient Adomian Decomposition Method

The Analysis and the Solution of Incubation Period in a Disease Model

This study deals with the analysis and the solution of incubation period in a disease model by adopting the mathematical model with incubation period of diseases and the mathematical model without the incubation period of diseases. In the model equations, we partitioned the population into Susceptible (S), Incubated (I), Infected (D) population. We have compared the model equations without incubation period with the model equation with incubation period by solving and incorporating the system of first order linear equations into fourth order Runge-kutta method which has better error accuracy for solving first order equations. Graphical results for incubation class show that the infectious diseases were fatal if immediate attention is not given to endemic villages and communities.Keywords: SID Model, Incubation period, Runge-kutta method, numerical simulation, transmission.

Impact of prominence type on the coarticulation of vowels following palatalized consonants

The quantitative impact of prominence (neutral vs emphatic stress) on the coarticulation of the Russian low [æ], mid [e] and high [i] front vowels is evaluated in CV sequences with palatalized consonants of various places of obstruction. The research is conditioned by the aims of speech synthesis and recognition of the Russian language, the current data in the field being insufficient. First order linear equations (locus equations) are used to evaluate the degree of CV coarticulation. The experiment is performed on CjVCjV nonsense words imbedded in the middle position of a carrier phrase «Vyros CVCV sjyljnym» («CjVCjV has grown up strong»). The first syllable of the target words was uttered both with neutral and emphatic stress. The Praat software was used to measure F2 formant values at vowel onsets and central parts. It was established that linear regression slopes are similar both in the Y-intercepts and slope coefficients for the palatalized consonants of various places of obstruction uttered with the same prominence. The degree of coarticulation was different depending upon the type of prominence (neutral or emphatic): the coefficients were lower in case of the emphatic stress compared to those of the neutral one. ANOVA analysis and Student’s t-test were performed to reveal the difference in F2 values of vowel onsets and middle parts as a function of the vowel openness and type of prominence. The results obtained were interpreted based on the degree of the openness of the vowels under study: an open vowel under emphatic stress tends to expand its vowel space by a backward displacement while close vowels tend to move forward in the mouth cavity when pronounced with emphatic stress.

Application of First Order differential Equations in R L Circuits

Abstract: The application of first order differential equation in Growth and Decay problems will study the method of variable separable and the model of Malthus (Malthusian population model), where we use the methods to find the solution to the population problems which are of use in mathematics, physics and biology especially in dealing with problems involving growth and decay problems that requires the use of Malthus model. Differential equations are fundamental importance in engineering mathematics because any physical laws and relations appear mathematically in the form of such equations. In this paper we discussed about first order linear homogeneous equations, first order linear non homogeneous equations and the application of first order differential equation in electrical circuits

On Zaremba problem for second - order linear elliptic equation with drift in case of limit exponent

We establish the unique solvability of the Zaremba problem with the homogeneous Dirichlet and Neumann boundary conditions for an inhomogeneous linear second order second order equation in the divergence form with measurable coefficients and lower order terms. The problem is considered in a bounded strictly Lipschitz domain. We suppose that the domain is contained in an $n$ - dimensional Euclidean space, where $n\ge2$. If $n>2$, then the lower coefficient belong to the Lebesgue space with the limiting summability exponent from the Sobolev embedding theorem. If $n=2$, then the lower coefficients are summable at each power exceeding two. Apart of the unique solvability, we establish an energy estimate for the solution.

8-Step Method for Third Order Fredholm Integro-Differential Equations with Vieta-Pell-Lucas as Approximating Function.

Purpose: In this paper, we derive multistep numerical methods for third order linear Fredholm integro-differential equation using Vieta-Pell-Lucas polynomial as approximating function. Methodology: These new method is implemented by embedding it with Trapezoidal, Booles and Simpson 1/3 numerical methods. The derivation led to a block of twenty-four discrete schemes which simultaneously solves the third order linear integro-differential equation. The qualitative properties of the schemes have been investigated. Findings: The analysis of the method reveals that the proposed method is of order seven and the method is found to be consistent and stable, hence convergent. Unique contributor to theory, policy and practice: Numerical experiments have been performed on some selected problems and the results show that the proposed method perform creditably well when compared with the exact solution of the selected examples

A time independent least squares algorithm for parameter identification of Turing patterns in reaction-diffusion systems.

Turing patterns arising from reaction-diffusion systems such as epidemic, ecology or chemical reaction models are an important dynamic property. Parameter identification of Turing patterns in spatial continuous and networked reaction-diffusion systems is an interesting and challenging inverse problem. The existing algorithms require huge account operations and resources. These drawbacks are amplified when apply them to reaction-diffusion systems on large-scale complex networks. To overcome these shortcomings, we present a new least squares algorithm which is rooted in the fact that Turing patterns are the stationary solutions of reaction-diffusion systems. The new algorithm is time independent, it translates the parameter identification problem into a low dimensional optimization problem even a low order linear algebra equations. The numerical simulations demonstrate that our algorithm has good effectiveness, robustness as well as performance.

The periodic solution of a second order linear equation

The periodic solution of a second order linear equation

Meromorphic solutions of linear q-difference equations

In this article, we construct explicit meromorphic solutions of first order linear q-difference equations in the complex domain and we describe the location of all their zeros and poles. The homogeneous case leans on the study of four fundamental equations, providing the previous informations in the framework of entire or meromorphic coefficients. The inhomogeneous situation, which stems from the homogeneous one and two fundamental equations, is also described in detail. We also address the case of higher-order linear q-difference equations, using a classical factorization argument. All these results are illustrated by several examples.

Heat Transfer in Solid by Using Linear Differential Equation

Abstract: Differential equations are fundamental importance in engineering mathematics because any physical law s and relations appear mathematically in the form of such equations. . The rate of heat conduc- tion in a specified direction is proportional to the temperature gradient,which isthe rate of change in temperature with distance in that direction. In this paper we discussed about first order linear homogeneous equations, first order linear non homogeneous and the application of first order differential equationto heat transfer analysis particularly in heat conduction in solids.

Restrictions in a distributed complex fractional order linear constitutive equations of viscoelasticity

We analyze a viscoelastic body in a linear stress state by the use of the distributed complex order fractional derivative in the constitutive equation. A model is formulated so that it takes into account all complex order derivatives of the form t0Dtα+iβ(α), α∈[α1,α2]⊂(0,1), β∈C1([α1,α2]). Using a new procedure, we derive restrictions on a model that follows from the Clausius–Duhem inequality under isothermal conditions. We apply our analysis to the creep test, as well as to generalized wave equation on R, with a specific constitutive equation of Kelvin–Voigt type.

Oscillation condition for first order linear dynamic equations on time scales

In this paper, we deal with the first-order dynamic equations withnonmonotone arguments\begin{equation*}y^{\Delta }(t)+\underset{i=1}{\overset{m}{\sum }}p_{i}(t)y\left( \tau_{i}(t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}}\end{equation*}where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},%\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{%T}},\mathbb{T}\right) $ and $\tau _{i}(t)\leq t,\ \lim_{t\rightarrow \infty}\tau _{i}(t)=\infty $ for $1\leq i\leq m$. Also, we present a new sufficient condition for theoscillation of delay dynamic equations on time scales. Finally, we give anexample illustrating the result.

NUMERICAL SOLUTIONS OF THIRD ORDER FREDHOLM INTEGRO DIFFERENTIAL EQUATION VIA LINEAR MULTISTEP-QUADRATURE FORMULAE

In this study, we present a 6-step linear multistep mehod in union with some newton-cote quadrature family for solving third order Linear Fredholm Integro-Differential Equation(LIDE). The schemes were derived using Vieta-Pell-Lucas Polynomial as the approximating function. The linear multi-step component is for the non integral part while the quadrature family is for the Integral part. The quaudrature methods Boole , Simpson 3/8 and Trapezoidal rule were separately combined with the linear multistep method. The qualititaive analysis of the scheme revealed that the method is consistence, stable and convergent. In order to further attest to the behavioural attribute of the methods, numerical experiments were carried out on some selected Initial Value Problems.The results from the tested problems and their absolute errors of deviation revealed that the new method is very suitbale for solution to the tested problems. The scheme when combined with Boole and Simpson 3/8 merhod, performed better with Tracendental function than when combined with Trapezoidal rule and vice –versa. The results further showed that the proposed method perform creditably well with lesser computional steps when compared with some existing methods when applied to the selected examples.

Self-similar Solution of Population Balance Equation for Aggregation with Constant Kernel

Self-similar solution of population balance equation includes the number density function which remains invariant or contains a part that is invariant. This paper describes self-similar solution of aggregation population balance model with constant kernel. Using this constant kernel, moment of the population balance system achieves the form μ_i (t)∝t^(i-1) and the aggregation population balance equation with constant kernel reduces to a first order linear ordinary integro-differential equation whose solution is exponential function with negative power.

Lie algebra and conservation laws for the time-fractional heat equation

The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that the numbers of symmetries and Lie brackets are reduced significantly as compared to the integer order for all dimensions. In fact for integer order linear heat equation the number of solution symmetries is equal to the product of the order and space dimension, whereas for the fractional case, it is half of the product on the order and space dimension. The Lie algebras for the integer and fractional order equations are mentioned using the subsequent computations of Lie brackets and by inspection. Interestingly, it is observed that for the one-dimensional fractional heat equation, the Lie algebra obtained by inspection of symmetries is similar to the result obtained by computation of Lie brackets, which is . The Lie algebra using the symmetries of the two-dimensional heat equation is observed to be , whereas using the Lie brackets the algebra is deduced to be . Hence, it can be concluded that the Lie algebra obtained from the nonzero Lie brackets can be conflated to the algebra which is obtained by inspection. Further, the subsequent Lie algebras are mentioned for the three and four-dimensional integer and fractional equations and the conservation laws are explicitly stated.

SODES: Solving ordinary differential equations step by step

In this paper, we introduce SODES (Stepwise Ordinary Differential Equations Solver) a new solver for Ordinary Differential Equations (ODE). SODES can optionally provide the solution displaying all the steps needed to obtain it. This way, SODES is an important tool not only for researchers who need solving ODE but also constitutes an important tool for the teaching and learning process of ODE. SODES has been developed using programming with a Computer Algebra System (CAS). Specifically, we use the CAS Derive but it can be easily adapted to any other CAS supporting programming.SODES provides, step by step, the solution of the following types of ODE: separable, homogeneous, exact, integrating factors, linear, Bernoulli, Riccati, first order ODE of nth degree, Cauchy’s problems of first order ODE, higher order linear homogeneous equations with constant coefficients, Lagrange’s method for particular solutions of higher order linear equations with constant coefficients, higher order linear equations with constant coefficients and Cauchy’s problems of higher order linear equations with constant coefficients. SODES also deals with two generic programs which determine the type or types of a given ODE and provides the solution.In this paper we will also introduce a draft of a Graphical User Interface (GUI) for SODES in a local web application using programming in Python (using its CAS module SymPy) which is a more portable and free CAS. This draft can be used in English, French and Spanish, and can be easily extended to other languages.The code of SODES and the GUI are freely available so that it can be used by users who also will be able to adapt it to their needs.

NON-POLYNOMIAL CUBIC SPLINE APPROACH FOR NUMERICAL APPROXIMATION OF SECOND ORDER LINEAR KLEIN-GORDON EQUATION

Most of the fundamental theories and mathematical models of engineering and physical sciences are expressed in terms of partial differential equations (PDEs). Several studies were carried out for the numerical approximation of the second order linear Klein-Gordon equation. This study constructed a new numerical technique for the numerical approximation of second order linear Klein-Gordon equation. The new constructed scheme was based on employing non-polynomial cubic spline method (NPCSM). The second order time derivatives involved in the linear Klein-Gordon equation were decomposed into the first order derivatives. The decomposition generated a linear system of PDEs, where the first order time derivatives were approximated by the central finite differences of . Three test problems were considered for the numerical illustration of the developed scheme. For different values of spatial displacement , step size , and time step , the developed numerical technique produced encouraging results which were very much close to the analytical solution. For , , and , the best observed accuracy was close to the machine precision.

Dirichlet problems for second order linear elliptic equations with $ L^{1} $-data

We consider the Dirichlet problems for second order linear elliptic equations in non-divergence and divergence forms on a bounded domain $ \Omega $ in $ \mathbb{R}^n $, $ n \ge 2 $: \begin{document}$ -\sum\limits_{i, j = 1}^n a^{ij}D_{ij} u + b \cdot D u + cu = f \;\;\text{ in }~~ \Omega \quad \text{and} \quad u = 0 \;\;\text{ on }~~ \partial \Omega $\end{document} and \begin{document}$ - {\rm div} \left( A D u \right) + \text{div}\, (ub) + cu = \text{div}\, F \;\;\text{ in }~~ \Omega \quad \text{and} \quad u = 0 \;\;\text{ on }~~ \partial \Omega , $\end{document} where $ A = [a^{ij}] $ is bounded, uniformly elliptic, and of vanishing mean oscillation (VMO). The main purpose of this paper is to study unique solvability for both problems with $ L^1 $-data. We prove that if $ \Omega $ is of class $ C^{1} $, $ \text{div}\, A + b\in L^{n, 1}( \Omega; \mathbb{R}^n) $, $ c\in L^{\frac{n}{2}, 1}( \Omega) \cap L^s( \Omega) $ for some $ 1<s<\frac{3}{2} $, and $ c\ge0 $ in $ \Omega $, then for each $ f\in L^1 (\Omega ) $, there exists a unique weak solution in $ W^{1, \frac{n}{n-1}, \infty}_0 ( \Omega) $ of the first problem. Moreover, under the additional condition that $ \Omega $ is of class $ C^{1, 1} $ and $ c\in L^{n, 1}( \Omega) $, we show that for each $ F \in L^1 ( \Omega ; \mathbb{R}^n ) $, the second problem has a unique very weak solution in $ L^{\frac{n}{n-1}, \infty}( \Omega) $.

Numerical Treatment for n th Order Linear Functional-Differential Equations Using Nyström's Method

تقدم الورقة طريقة مقترحة مع خوارزميات مكتوبة في Matlab لغة لحل نظام المعادلات التفاضلية الوظيفية الخطية عدديًا باستخدام طريقة نيستروم من الدرجة الخامسة ذات المرحلة السادسة. هذه الطريقة لها تم تطويره لإيجاد التطورات العددية لثلاثة أنواع (متخلفون ، محايد ومختلط) من المعادلات التفاضلية الوظيفية الخطية من الرتبة n. تم إعطاء مقارنة بين النتائج العددية والدقيقة لثلاثة أمثلة عددية لحل ثلاثة أنواع من التفاضل الوظيفي الخطي المعادلات. أخيرًا ، النتائج مرتبة في شكل مجدول ومناسب الرسوم البيانية أعطيت للأمثلة.

Operator estimates for non-periodically perforated domains with Dirichlet and nonlinear Robin conditions: vanishing limit

We consider a general second order linear elliptic equation in a finely perforated domain. The shapes of cavities and their distribution in the domain are arbitrary and non-periodic; they are supposed to satisfy minimal natural geometric conditions. On the boundaries of the cavities we impose either the Dirichlet or a nonlinear Robin condition; the choice of the type of the boundary condition for each cavity is arbitrary. Then we suppose that for some cavities the nonlinear Robin condition is sign-definite in certain sense. Provided such cavities and ones with the Dirichlet condition are distributed rather densely in the domain and the characteristic sizes of the cavities and the minimal distances between the cavities satisfy certain simple condition, we show that a solution to our problem tends to zero as the perforation becomes finer. Our main result are order sharp estimates for the \(L_2\)- and \(W_2^1\)-norms of the solution uniform in the \(L_2\)-norm of the right hand side in the equation.