Solutions Of First-order Differential Equations Research Articles

Solution of First-Order Differential Equation Using Fourth-Order Runge-Kutta Approach and Adams Bashforth Methods

In this research, we investigate the solution of first-order differential equations (DEs) using Runge- Kutta fourth-order method (RKM) and Adams-Bashforth methods (ABMs). In this work we consider fourth-order RKM and ABMs for solving first order DEs. The method proof to be simple, easy, accurate and efficient technique for solving first order DEs. Moreover, there are unlimited application of fourth-order RK4 and ABMs for solving first-order DE in science, engineering, economics, social science, biology and business. These play an important role in science and engineering. Some examples are giving and solved to support the efficiency of our methods which are demonstrated by figures.

Upper Bounds for the Distance between Adjacent Zeros of First-Order Linear Differential Equations with Several Delays

The distance between successive zeros of all solutions of first-order differential equations with several delays is studied in this work. Many new estimations for the upper bound of the distance between zeros are obtained. Our results improve many-well known results in the literature. We also obtain some fundamental results for the lower bound of the distance between adjacent zeros. Some illustrative examples are introduced to show the accuracy and efficiency of the obtained results.

A Fourth Order Symplectic and Conjugate-Symplectic Extension of the Midpoint and Trapezoidal Methods

The paper presents fourth order Runge–Kutta methods derived from symmetric Hermite–Obreshkov schemes by suitably approximating the involved higher derivatives. In particular, starting from the multi-derivative extension of the midpoint method we have obtained a new symmetric implicit Runge–Kutta method of order four, for the numerical solution of first-order differential equations. The new method is symplectic and is suitable for the solution of both initial and boundary value Hamiltonian problems. Moreover, starting from the conjugate class of multi-derivative trapezoidal schemes, we have derived a new method that is conjugate to the new symplectic method.

Factorization method for some inhomogeneous Lienard equations

The factorization of inhomogeneous Li\'enard equations is performed showing that through the factorization conditions involved in the method one can obtain forcing terms for which closed-form solutions exist. Because of the reduction of order feature of factorization, the solutions are simultaneously solutions of first-order differential equations with polynomial nonlinearities. Several illustrative examples of such solutions are presented, generically having rational parts and consequently singularities.

Singularities of Singular Solutions of First-Order Differential Equations of Clairaut Type

A first-order differential equation of Clairaut type has a family of classical solutions, and a singular solution when the contact singular set is not empty. The projection of a singular solution of Clairaut type is an envelope of a family of fronts (Legendre immersions). In these cases, the envelopes are always fronts. We investigate singular points of envelopes for first-order ordinary differential equations, first-order partial differential equations, and systems of first-order partial differential equations of Clairaut type, respectively.

Solving First-Order Differential Equations of Z-Numbers’ Initial Value Using Radial Basic Function

In this paper, a method was proposed based on RBF for numerical solution of first-order differential equations with initial values that are valued by Z-numbers. The proposed method consists of two parts. The first part has stated the amount of limitation of the fragmentation solution, while the second part has described the assurance of the first part. The limitation section also has two parts. The first part has included the initial condition of the problem, while the second part has included the RBF network. The confidence interval was also considered as a function based on the probability function, which has calculated the confidence level of the first part (limitation). The RBF network or the radial-base grid network has three distinct layers: the input layer that is the set of elementary nodes (sensory units); the second layer is the hidden layers with high dimensions, in which the output layer that has responded to the network response and the activation patterns used in the input layer. The advantage of using RBF is that the use of this technique does not require sufficient information. It only relies on the domain and the boundary. In an example, we have showed that our proposed approach could approximate the problem with acceptable confidence.

First integrals and general solutions of the Biswas-Milovic equation

The Biswas-Milovic equation for description of pulse propagation in optical fiber is considered. Solutions of the equation are looked for using the traveling wave reduction. The first integrals for a system of equations are found. These first integrals allow us to reduce the system of equations to nonlinear first-order differential equation. The general solution of first-order differential equation with three arbitrary constants is presented in the form of a quadrature. The parameter values of the equation are given for which the general solution is written in an analytical form via elementary functions and the Weierstrass and Jacobi elliptic functions.

Finite Element Approach for the Solution of First-Order Differential Equations

The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.

On the unique continuation of the germs of solutions of first-order differential equations along curves

For solutions of linear, weakly nonlinear, and quasilinear first-order differential equations, we obtain theorems on the unique unbounded continuation of germs along continuous curves contained in the integral submanifolds of the distribution induced by the major part of the equation.

On the Unique Continuation of the Germs of Solutions of First-Order Differential Equations along Curves

Для решений линейных, слабо нелинейных и квазилинейных дифференциальных уравнений первого порядка получены теоремы об однозначном неограниченном продолжении ростков вдоль непрерывных кривых, содержащихся в интегральных подмногообразиях распределения индуцированного старшей частью уравнения. Библиография: 14 названий.

The Adomian decomposition method for numerical solution of first-order differential equations

In this paper, the Adomian decomposition method (ADM) is a powerful method which considers the approximate solution of a non-linear equation as an infinite series which usually converges to the exact solution. In this paper, this method is proposed to solve some first-order differential equations. It is shown that the series solutions converge to the exact solution for each problem. It is observed that the method is particularly suited for initial value problems with oscillatory and exponential solutions.

Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations

The construction of exponentially fitted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of first-order differential equations is investigated. The revised EFTDRK methods proposed, with equation-dependent coefficients, take into consideration the errors produced in the internal stages to the update. The local truncation errors and stability of the new methods are analyzed. The numerical results are reported to show the accuracy of the new methods.

Existence of solutions of first-order differential equations via a fixed point theorem for discontinuous operators

AbstractWe use a recent Schauder-type result for discontinuous operators in order to look for solutions for first-order differential equations subject to initial functional conditions. We show how this abstract fixed-point result allows us to consider a nonlinearity which can be strongly discontinuous. Some examples of applications and comparison with recent literature are included.

Unbounded solutions of the equation [formula omitted

Asymptotic behavior of solutions of first-order differential equation with deviating arguments in the formẏ(t)=∑i=1nβi(t)y(t-δi)-y(t-τi)is discussed for t→∞. A criterion for representing solutions in exponential form is proved. Inequalities for solution estimation are given. Sufficient conditions for the existence of unbounded solutions are derived. A relevant illustrative example is given as well. Known results are discussed and compared.

Almost periodic solutions for first-order differential equations

We study here the almost periodic solutions of first-order differential equations. We give sufficient conditions for the existence and uniqueness. The method relies on penalization and a priori estimates. One of the main difficulties consists of verifying that the limit of the sequence of perturbed solutions remains almost periodic. We introduce the notions of minimal/maximal solutions.

Defect structures in sine-Gordon like models

We investigate several models described by real scalar fields, searching for topological defects. Some models are described by a single field, and support one or two topological sectors, and others are two-field models, which support several topological sectors. Almost all the defect structures that we find are stable and finite energy solutions of first-order differential equations that solve the corresponding equations of motion. In particular, for the double sine-Gordon model we show how to find small and large BPS solutions as deformations of the BPS solution of the ϕ 4 model. And also, for most of the two field models we find the corresponding integrating factors, which lead to the complete set of BPS solutions, nicely unveiling how they bifurcate among the several topological sectors.

Exponential solutions of equation

Asymptotic behaviour of solutions of first-order differential equation with two deviating arguments of the form y ̇ (t)=β(t) y(t−δ)−y(t−τ) is discussed for t →∞. A criterion for representing solutions in exponential form is proved. As consequences, inequalities for such solutions are given. Connections with known results are discussed and a sufficient condition for existence of unbounded solutions, generalizing previous ones, is derived. An illustrative example is considered, too.

Exponential solutions of equation [formula omitted

Asymptotic behaviour of solutions of first-order differential equation with two deviating arguments of the form y ̇ (t)=β(t) y(t−δ)−y(t−τ) is discussed for t→∞. A criterion for representing solutions in exponential form is proved. As consequences, inequalities for such solutions are given. Connections with known results are discussed and a sufficient condition for existence of unbounded solutions, generalizing previous ones, is derived. An illustrative example is considered, too.

FORMULAS OF BENDIXSON AND ALEKSEEV FOR DIFFERENCE EQUATIONS

A well-known formula of Bendixson states that solutions of first-order differential equations, as functions of their initial conditions, satisfy a certain partial differential equation. A consequence is Alekseev’s nonlinear variation of parameters formula. In this paper, corresponding results are proved for difference equations. To achieve this, use is made of the recently introduced concept of alpha derivatives, rather than of differences or of the usual derivatives. This technique allows the results to be generalized to alpha dynamic equations, which include among others ordinary differential and difference equations.

Problem Solving in Physical Chemistry with the TI-89 Calculator

The Texas Instruments TI-89 calculator is an advanced scientific calculator that has both graphing and programming capabilities, and advanced-mathematics software. In this article the TI-89 has been used for solving a variety of problems in physical chemistry. The applications in this paper include calculations with units, solving higher-order equations that are set up in chemical equilibrium problems, differential- and integral-calculus-based calculations in thermodynamics and quantum mechanics, numerical and analytic solutions of first-order differential equations in chemical kinetics, and regression analysis of data collected in a kinetics experiment. The appropriate calculator keystrokes are included for all the examples in this paper. Many complex and interesting problems can be studied with relative ease, thus allowing teachers to introduce modern scientific techniques in the classroom.