A Single-Step Approach with Seven Hybrid Points for the Solution of Stiff Differential Equations

Preliminaries. Introduction. Initial Value Problem, Oscillation and Nonoscillation. Continuability and Boundedness. Some Basic Results for Second Order Linear Ordinary Differential Equations. Some Useful Criteria for First Order. Some Useful Results from Analysis and Fixed Point Theorems. Notes and General Discussions. References. Oscillations of Differential Equations with Deviating Arguments. Oscillation Theorems (I). Oscillation Theorems (II). Comparison Theorems for Second Order Functional Differential Equations. Oscillation of Functional Equations with a Damping Term. Oscillation of Second Order Linear Delay Differential Equations. Oscillation of Forced Functional Differential Equations. Oscillation of Functional Equations with Damping and Forcing Terms. Necessary and Sufficient Conditions for the Oscillation of Forced Equations. Oscillation for Perturbed Differential Equations. Asymptotic Behavior of Oscillatory Solutions of Functional Equations. Notes and General Discussions. References. Oscillation of Neutral Functional Differential Equations. Oscillation of Nonlinear Neutral Equations. Oscillation of Neutral Equations with Damping. Oscillation of Forced Neutral Equations. Oscillation of Neutral Equations with Mixed Type. Necessary and Sufficient Conditions for Oscillations of Neutral Equations with Deviating Arguments. Comparison and Linearized Oscillation Theorems for Neutral Equations. Existence of Nonoscillatory Solutions of Neutral Delay Differential Equations. Asymptotic Behavior of Nonoscillatory Solutions of Neutral Nonlinear Delay Differential Equations. Notes and General Discussions. References. Conjugacy and Nonoscillation for Second Order Differential Equations. Conjugacy of Linear Second Order Ordinary Differential Equations. Nonoscillation Theorems. Integral Conditions and Nonoscillations. Notes and General Discussions. References. Oscillation of Impulsive Differential Equations. Oscillation Criteria for Impulsive Delay Differential Equations. Oscillation of Second Order Linear Differential Equations with Impulses. Notes and General Discussions. References. Subject Index with Deviating Arguments. Oscillation of Neutral Functional Differential Equations. Conjugacy and Nonoscillation for Second Order Differential Equations. Oscillation of Impulsive Differential Equations.

This paper is concerned with the oscillation of first-order linear delay differential equations in which the coefficients are periodic functions with a common period and the delays are constants and multiples of this period. A necessary and sufficient condition for the oscillation of all solutions is established.

This study introduces a novel single-step hybrid block method with three intra-step points that attains fifth-order accuracy, offering an accurate and computationally economical tool for solving first-order differential equations. The method is specifically designed to handle first-order differential equations with efficiency and precision while employing a constant step size throughout the computation. To further enhance accuracy, interpolation techniques are incorporated to approximate function values at specific positions, addressing the fundamental properties of the method and verifying its mathematical soundness. These analyses confirm that the scheme satisfies the essential requirements of stability, consistency, and convergence, ensuring reliability in practical applications. In addition, the method demonstrates strong adaptability, making it suitable for a broad spectrum of problem settings that involve both stiff and non-stiff systems. Numerical experiments are carried out, and the results consistently demonstrate that the proposed method is robust and effective under various test cases. The outcomes further reveal that it frequently outperforms several existing numerical approaches in terms of both accuracy and computational efficiency.

An L-stable extended two-step method for the integration of ordinary differential equations

3 - Applications of First-Order Ordinary Differential Equations

This research shows the development and simulation of an implicit second-order block method for solving Betiss and Stiefel differential equations. The method's fundamental properties including order, consistency, and stability were rigorously analyzed, confirming its theoretical robustness under standard numerical analysis principles. Through comparative testing on oscillatory differential equations, the proposed method demonstrated enhanced accuracy and computational efficiency over existing approaches. The results reveal its superior performance in terms of error reduction and stability, making it a viable improvement for long-term simulations of stiff and oscillatory systems.

In this chapter we shall discuss some techniques which are different than those employed in the previous chapters to obtain oscillatory criteria for differential equations. In Section 6.1 we shall present oscillation and nonoscillation theorems for nonlinear second order differential equations by using the method of Olech, Opial and Wazewski. Section 6.2 is concerned with the oscillation of half-linear second order differential equations by employing the variational inequality given in Lemma 3.2.6. In Section 6.3 we shall begin with some preliminaries of Liapunov functions, and then apply Liapunov second method to obtain criteria for the oscillation of second order nonlinear equations.

On oscillation of difference equations with bounded phi-Laplacian

In this chapter, we discuss the major approaches to obtain analytical solutions of ordinary differential equations. We begin with the solutions of first-order differential equations. Several first-order differential equations can be transformed into two major solution approaches: the separation of variables approach and the exact differential approach. We start with a brief review of both approaches, and then we follow them with two sections on how to reduce other problems to either of these methods. First, we discuss the use of similarity transformations to reduce differential equations to become separable. We show that these transformations cover other well-known approaches, such as homogeneous-type differential equations and isobaric differential equations, as special cases. The next section continues with the search for integrating factors that would transform a given differential equation to become exact. Important special cases of this approach include first-order linear differential equations and the Bernoulli equations (after some additional variable transformation). Next, we discuss the solution of second-order differential equations. We opted to focus first on the nonlinear types, leaving the solution of linear second-order differential equations to be included in the later sections that handle high-order linear differential equations. The approaches we consider are those that would reduce the order of the differential equations, with the expectation that once they are first-order equations, techniques of the previous sections can be used to continue the solution process. Specifically, we use a change of variables to handle the cases in which either the independent variable or dependent variable are explicitly absent in the differential equation.

A Comparison Between Differential Equation Solver Suites In MATLAB, R, Julia, Python, C, Mathematica, Maple, and Fortran

Some comparison and oscillation results for first-order differential equations and inequalities with a deviating argument

On the solvability of Hill's equation by quadrature

Existing methods to calculate transients in electrical circuits have a number of disadvantages. Simple finite-difference methods have low orders of accuracy and are prone to swinging parasitic oscillations. High-precision methods based on complicated mathematics are poorly adapted to nonlinear and parametric circuits, as well as to circuits with frequent switching. The author has previously developed a new family of high-precision implicit numerical methods for calculating transients in linear electrical circuits. The new methods are based on the decomposition of the Pade approximation of the matrix exponent into the simplest fractions. They have no restrictions on the integration step, and are extremely simple, versatile, and require few computational work. The methods are suitable to calculate conventional circuits, as well as circuits with stiff and oscillatory equation systems. So far, when developing the new methods mentioned above, only circuits with simple branches consisting of serial connection of a single passive element with voltage source have been considered. There were no magnetic couplings in the circuits. The purpose of the work is to adapt the author’s high–precision methods to calculate transients based on Pade approximation of the matrix exponent to apply them in circuits with composite branches and magnetic couplings. Application of the equations for composite branches allows to reduce the order of the solved differential-algebraic system of circuit equations and simplify its forming. Calculation of the circuits with arbitrary number of magnetic couplings is necessary for qualitative expanding possibilities of the methods Materials and methods. The initial material for the study is a system of differential algebraic equations of an electric circuit with composite branches and magnetic couplings. The main instrument to solve the problem is the new developed by the author method for calculation transients in the linear electrical circuits based on Cauchy formula to solve the system of ordinary differential equations, Pade approximation of the matrix exponent, the e-embedding method and the Schur complementary method. Results. The first-order differential equations describing their dynamics were obtained for four types of composite branches (RL-series, RL-parallel, RC-series, RC-parallel). They were integrated into the structure of calculation formulas of the methods developed by the author earlier, without their cardinal complication. It is shown that the use of composite branches reduces the dimension of the solved system of equations by 2N + M, where N is the number of series branches and M is the number of parallel branches, compared with the model where each passive element is represented by a separate branch. It is demonstrated that accounting of magnetic couplings leads to replacement of the diagonal quasi-resistance matrix with a symmetrical block-diagonal one. Moreover, this change does not violate the overall structure of the algorithm and preserves its computational efficiency, especially with a small number of couplings. An example of a numerical calculation of a circuit containing all types of composite branches is given. The received results using Pade approximations of different orders were compared with the solution obtained analytically. It is shown that in order to achieve high accuracy (relative error of the order of 10–⁵-10–⁶), only a few integration points are required for the period of high-frequency oscillations. Conclusions. The application scope of the new methods has been expanded to include circuits with composite branches and magnetic couplings. The calculation of circuits with composite branches reduces the number of equations and unknowns, the amount of calculations and the calculation time, and simplifies the formation of circuit equations. Accounting of magnetic couplings complements the possibilities of new method qualitatively. Composite branches and magnetic couplings only slightly change the matrices in the calculation formulas.

A stiff equation is a differential equation for which certain numerical methods are not stable, unless the step length is taken to be extraordinarily small. The stiff differential equation includes few terms that could result in speedy variation in the solution. When integrating a differential equation numerically, the requisite step length should be incredibly small. In the solution curve, much variation can be observed where the solution curve straightens out to approach a line with slope almost zero. The phenomenon of stiffness is observed when the step-size is unacceptably small in a region where the solution curve is very smooth. A lot of work on solving the stiff ordinary differential equations (ODEs) have been done by researchers with numbers of numerical methods that currently exist. Extensive research has been done to unveil the comparison between their rate of convergence, number of computations, accuracy, and capability to solve certain type of test problems. In the present work, an advanced Feed Forward Neural Network (FFNN) and Bayesian regularization algorithm-based method is implemented to solve first order stiff ordinary differential equations and system of ordinary differential equations. Using proposed method, the problems are solved for various time steps and comparisons are made with available analytical solutions and other existing methods. A problem is simulated using proposed FFNN model and accuracy has been acquired with less calculation efforts and time. The outcome of the work is showing good result to use artificial neural network methods to solve various types of stiff differential equations in near future.

This work considers the use of the simulation to solve differential equations in MatLab/Simulink and its results. The MatLab software package is designed to provide analytical and numerical solutions for various mathematical problems and simulate complex technical objects and systems. The Simulink app is one of the tools within the MatLab package. It has major structured, object-oriented, and visual programming capabilities. The system can solve linear algebra problems, integral and differential equations, and perform Laplace and Fourier transformations. We reviewed both the simplest equations (a first-order differential equation) and more complex linear and non-linear differential equations of the first and second orders. The results of structured modeling (S-model) in Simulink were compared to program code results (M-file) in MatLab. For the first-order linear differential equation, we have a complete correlation of modeling results. When solving the second-order non-linear differential equation, we observed small result deviations due to the method selected and the integration step. When discussing the results, we used the example of the practical application of the MatLab/Simulink environment to calculate the parameters of an electric circuit, namely to obtain the relationship between the current and the modeling time in the series-oscillatory circuit with a harmonic alternating voltage source. It is reduced to solving a nonhomogeneous linear differential equation with constant second-order coefficients describing the forced current change in the oscillatory circuit following the second Kirchhoff’s law. In the conclusion, we note the versatility of the MatLab/Simulink modeling environment for the solution of differential equations including those with quotient derivatives.