Methods For Differential Equations Research Articles

RETRACTED: Using the method of differential equations by quadratic to solve the free vibrations of columns under the effect of axial load and column weight

This paper investigates various techniques used to solve the differential equations governing the free vibration of columns. The present work focuses on the study of the free vibration of Euler’s Bernoulli column of equal strength in compression, considering its own weight and the axial load in compression and tension while subjected to symmetrical boundary conditions. The investigation utilizes the differential quadrature method to examine the fifth natural frequency parameters of the column in different states of column boundary conditions and varying geometric section shapes, including pin-pin and clamp-clamp configurations. The results of this work contribute valuable insights for informed decisions on selecting the cross-section types and appropriate boundary conditions for ensuring the stability of such columns in civil constructions.

A Novel Shifted Jacobi Operational Matrix Method for Linear Multi-terms Delay Differential Equations of Fractional Variable-order with Periodic and Anti-periodic Conditions

This paper investigates the generalized linear multi-terms delay fractional differential equation of variable order with periodic and anti-periodic conditions. In this work, a novel shifted Jacobi operational matrix technique is applied to solve a class of these equations, so that the original problem becomes a system of algebraic equations that can be solved by numerical methods. The proposed technique is successfully applied to the aforementioned problem. Sufficient and complete numerical tests are presented to demonstrate the accuracy, generality, efficiency of presented technique and the flexibility of this scheme. The numerical results of this method are compared with other existing methods such as fractional backward differential formulas (FBDF). Comparing the outcomes of these schemes, as well as comparing the current technique (NSJOM) with the exact solution, demonstrates the efficiency and validity of this method. It should be noted that the implementation of current method is considered very easy and general for many numerical techniques. Furthermore, the error and its bound are estimated.

Semi-analytical iteration techniques for solving differential equations

The main objective of this work is to use and compare some techniques, namely the variational Iteration method (VIM), Decomposition Method (ADM), Differential Transform Method (DTM), and Temimi and Ansari Method (TAM) for differential equations of the first and second orders that appeared in physics and engineering. Notably, the results indicate that the VIM gives a solution in series form, which converges to an exact solution;. At the same time, DTM is easy to apply but requires transformation, ADM does not need any transformation except the calculation of Adomain polynomials. TAM solved nonlinear problems in much easier iteration steps, which converge to the true solution without applying any assumption. The results obtained converge rapidly to the exact solution.

Galerkin Spectral Method of Stochastic Partial Differential Equations Driven by Multivariate Poisson Measure

A considerable body of prior research has been dedicated to devising efficient and high‐order numerical methods for solving stochastic partial differential equations (SPDEs) driven by discrete or continuous random variables. The majority of these efforts have concentrated on introducing numerical approaches for SPDEs involving independent random variables, with limited exploration into solving SPDEs driven by dependent and discrete random variables. In this paper, we introduce a Galerkin spectral method tailored for addressing SPDEs driven by multivariate Poisson variables. Our novel numerical method is applied to solve the one‐ and two‐dimensional stochastic Poisson equations, as well as the one and two‐dimensional stochastic heat equation, stochastic KdV equation, and stochastic Burgers equation, respectively. Notably, all the presented stochastic equations are governed by two dependent Poisson random variables.

Solutions of random ordinary differential equations with laplace transform - Adomian decomposition method

Combine the Laplace Adomian Decomposition method (LTADM) for non-linear random ordinary differential equations is utilised in a novel way to obtain accurate solutions.The iterative implementation of the Laplace transform decomposition is employed to approximate the solutions of ordinary differential equations with random components. We use a Normal distribution to analyse the parameters and initial conditions of ordinary differential equations with random components. Using the Mathematica 13.3 programme, the graphs of the approximate solutions and absolute error are presented. An investigation is conducted into the effects of the normal distribution on the results of random component differential equations. According to the results of the numerical investigations, this method is extremely effective. Two applications are presented as examples of how the proposed technique can be utilised to obtain analytical or numerical solutions for certain kinds of random differential equations in order to demonstrate its efficacy and potential.

Convergence and Stability of Compenseted Split-Step Theta Methods for Stochastic Pantograph Differential Equations with Jumps

Convergence and Stability of Compenseted Split-Step Theta Methods for Stochastic Pantograph Differential Equations with Jumps

Machine learning methods for autonomous ordinary differential equations

Machine learning methods for autonomous ordinary differential equations

STUDY ON THE INITIAL BOUNDARY VALUE PROBLEM FOR AFRACTIONAL DIFFERENTIAL EQUATION WITH A FRACTIONALDERIVATIVE OF VARIABLE ORDER

This article studies the convergence of a numerical method for solving an initial-boundary value problem of a fractional differential equation with a variable order of the fractional derivative. In the generalized fractional differential filtration equation with a transitional filtration law in heterogeneous porous media, it is assumed that the order of the fractional derivative depends on the spatial variable. The main attention is paid to the development and theoretical justification of a method that provides high accuracy and efficiency of calculations with a variable order of the fractional derivative. For the numerical solution, an approximation was developed that combines the finite difference method for the time derivative and the finite element method for the spatial variable. The fractional derivative of variable order in the sense of Caputo is approximated by a formula of second order in time. The convergence of the constructed method is proven with order O(τ2+hk+1) for the case α(x)ϵ(0,1). The results of computational experiments for various functions of the order of the fractional derivative are presented, confirming the reliability of the theoretical analysis. The conclusions drawn emphasize the importance and relevance of the further development of numerical methods for fractional differential equations of variable order in modern mathematics and applied sciences, including the modeling of complex processes.

Extension of Chaplygin's existence and uniqueness method for fractal-fractional nonlinear differential equations

<abstract><p>The existence and uniqueness of solutions to nonlinear ordinary differential equations with fractal-fractional derivatives, with Dirac-delta, exponential decay, power law, and generalized Mittag-Leffler kernels, have been the focus of this work. To do this, we used the Chaplygin approach, which entails creating two lower and upper sequences that converge to the solution of the equations under consideration. We have for each case provided the conditions under which these sequences are obtained and converge.</p></abstract>

A study of strong convergence of differential equations based on Euler’s algorithm

Abstract Differential equations have important applications in many fields, such as chemistry, biology, epidemiology, and finance. Most analytic solutions of differential equations are difficult to obtain. Therefore numerical solutions of differential equations become an important tool. The truncated Euler method is proposed in this paper, and we investigate how the truncated EM solution of the derived SDDE converges strongly under the local Lipschitz condition and the one-sided linear growth condition after relaxation. The basic strong convergence theorem is set up and the new notation X(t,x;s) is introduced as an analytic solution of the stochastic differential equation. Establish the assumption that the coefficients of the drift term and the coefficients of the diffusion term of the stochastic differential equation satisfy a contractionary monotonicity condition in order to prove, by induction, that the exact solution of this stochastic differential equation is bounded for a long time. Different examples are given to compare the simulated deviations between the numerical and analytical solutions of Euler’s algorithm and the modified Euler’s algorithm for the initial value problem of fractional order differential equations and to analyze the convergence of the two numerical solutions. The truncated Euler method is applied to highly nonlinear time-transformed stochastic differential equations by means of the dyadic principle, and it is proved that the order of convergence of the strong convergence of the truncated Euler-Maruyama method for time-transformed stochastic differential equations is min ( α , γ , 1 2 − ɛ ) \min \left( {\alpha ,\gamma ,{1 \over 2} - \varepsilon } \right) .

Functional substitution method for differential equations with physical applications

Functional substitution method for differential equations with physical applications

The Riccati-Bernoulli sub-optimal differential equation method for analyzing the fractional Dullin-Gottwald-Holm equation and modeling nonlinear waves in fluid mediums

<abstract><p>The present study investigates the fractional Dullin-Gottwald-Holm equation by using the Riccati-Bernoulli sub-optimal differential equation method with the Bäcklund transformation. By employing a well-established criterion, the present study reveals novel cusp soliton solutions that resemble peakons and offers valuable insights into their dynamic behaviors and mysterious phenomena. The solution family encompasses various analytical solutions, such as peakons, periodic, and kink-wave solutions. Furthermore, the impact of both the time- and space-fractional parameters on all derived solutions' profiles is examined. This investigation's significance lies in its contribution to understanding intricate dynamics inside physical systems, offering valuable insights into various domains like fluid mechanics and nonlinear phenomena across different physical models. The computational technique's straightforward, effective, and concise nature is demonstrated through introduction of some graphical representations in two- and three-dimensional plots generated by adjusting the related parameters. The findings underscore the versatility of this methodology and demonstrate its applicability as a tool to solve more complicated nonlinear problems as well as its ability to explain many mysterious phenomena.</p></abstract>

A SHORT SOLUTION OF THE LOCAL FRACTIONAL (2+1)-DIMENSIONAL DISPERSIVE LONG WATER WAVE SYSTEM

In this paper, a local fractional Riccati differential equation method is applied. A new travelling wave solution to the nonlinear local fractional [Formula: see text]-dimensional dispersive long water wave system is investigated. After travelling wave transformation, the governing model studied is converted into nonlinear ordinary differential equation. Some properties with the strain conditions are also reported.

Numerical Mean-square and Asymptotic Stability Analysis for the Weak Simpson Method

A weak Simpson method has order of weak convergence one in general and has order of weak convergence three under certain additional assumptions. The proposed method has the potential to overcome some of the numerical instabilities that are often experienced when using explicit Euler methods. This work aims to determine the mean-square stability region of the weak Simpson method for linear stochastic differential equations with multiplicative noises. In this work, a mean-square stability region of the weak Simpson scheme is identified, and step-sizes for the numerical method where errors propagation are under control in a well-defined sense are given. The main results are illustrated with numerical examples.

The traveling wave solutions to a variant of the Boussinesq equation

In this study, we study a variant of the Boussinesq equation called as \(B( n+1,\,1,\,n )\) equation, and construct some traveling wave solutions by using an effective approach called the extended trial equation method. Thus, the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions, which show the existence of various mathematical and physical structures and events in the fundamental equation considered, have been constructured. In order to make a more detailed examination of the physical behavior of these solutions, two- and three-dimensional graphs of some solution functions were drawn with the help of the Mathematica package program. In the section of Discussion, we suggest a more general version of the trial equation method for nonlinear differential equations.

NEW IMPLICIT MULTISTEP METHOD FOR ODE’S

A new class of multistep methods for stiff ordinary differential equations is presented. The method is based on the transformation of the arguments of the original system into purely algebraic combinations of the solutions of previous steps. The scheme differs from the classical multistep methods in that the state variables, instead of functions of them, are approximated by means of linear combination of previous steps. A family of coefficients is deduced from a combined analysis of convergence order and stability. Numerical results are presented for three test problems.

A super-localized generalized finite element method

AbstractThis paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method’s basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method’s applicability for challenging high-contrast channeled coefficients.

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

An Adaptive ANOVA Stochastic Galerkin Method for Partial Differential Equations with High-dimensional Random Inputs

ФОРМУВАННЯ УМІНЬ МАТЕМАТИЧНОГО МОДЕЛЮВАННЯ ПРИКЛАДНИХ ЗАДАЧ МЕТОДАМИ ДИФЕРЕНЦІАЛЬНИХ РІВНЯНЬ

The article focuses attention on the need to learn how to solve applied problems, substantiates the need to master the skills of mathematical modelling as a universal method of solving applied and practical problems, presents the experience of students of higher education in forming the skills of mathematical modelling of applied problems using the methods of differential equations, reveals interdisciplinary connections, examples of solving geometric and physical problems from the topic “Differential Equations” are given. Problem solving was carried out in three stages, namely: building a mathematical model, studying the model and interpreting the result.

A New Approximation Method for Multi-Pantograph Type Delay Differential Equations Using Boubaker Polynomials

In this paper, a new approaching technique is offered to unravel multi-pantograph-type delay differential equations. The suggested new method is a collocation method based on integration and Boubaker polynomials. As the main idea of the method, the process starts by approaching the first derivative function in the equation in the form of truncated Boubaker series. Then this approximating form is composed in the matrix form. The unknown function is then obtained by integrating the approximate derivative function and expressing it as a matrix. Using the approximation, the matrix forms for the proportionally delayed terms in the equation are derived. In addition, operational matrix forms are constructed for convenience in the method. By using these matrix forms and matrix operations, the problem is reduced to a system of algebraic linear equations. The method is illustrated through numerical implementations and compared with existing techniques in the literature. The results demonstrate the effectiveness and reliability of the proposed approach, highlighting its superiority over other methods.