Approximate Solutions Of Differential Equations Research Articles

Obtaining Approximate Solutions of Differential Equations Using the Method of Moving Nodes

Obtaining Approximate Solutions of Differential Equations Using the Method of Moving Nodes

A Chebyshev polynomial approach to approximate solution of differential equations using differential evolution

Most of the problems associated with physical, biological and engineering systems are complex and are usually modeled with the help of linear as well as non-linear differential equations. In general, the exact solutions of non-linear differential equations may not be obtained by analytical methods and thus, in this paper, a numerical method is proposed to find approximate solutions of differential equations using Chebyshev polynomials and metaheuristic optimization algorithms. In this paper, Ordinary differential equations (ODEs) are modeled as optimization problems to determine their approximate solutions by considering Chebyshev polynomials as the base approximation function, taking initial and/or boundary conditions as constraints. The unknown coefficients of the polynomials are calculated using Differential Evolution (DE) in order to get the approximate solution with minimal error. The proposed method is applied to find the approximate solution to both Initial and Boundary Value Problems. For all test problems considered in this paper, the algorithms have been programmed using MATLAB software to run on computer. The effectiveness of the method is demonstrated by graphical comparison of the computed approximate solutions with that of the exact ones. Further, the performance of the method proposed in this paper is shown by finding out the Root Mean Square Error (RMSE) between the exact and approximate solution, which is found to be better than that of the related existing papers.

Stochastically analysis by using fixed point approach of pendulum with rolling wheel via translational and rotational motion

This paper presents an analytical solution for a complex mechanical system consisting of a pendulum with a rolling wheel, which combines translational and rotational motion. The system’s dynamics are described by a set of coupled differential equations that are challenging to solve analytically using traditional methods. To overcome this challenge, the variation iteration method (VIM) is employed to derive an analytical solution. VIM is a powerful technique that allows for the approximate solution of differential equations by constructing a series solution iteratively. The Lagrange multiplier, a crucial component in the solution process, is determined for the first time using the Elzaki transformation. Interestingly, the Lagrange multiplier obtained through the Elzaki transformation matches the result obtained from Laplace transformation, which is a fundamental finding of this paper. Also we compared its results with the by employing the VIM and the derived Lagrange multiplier, a comprehensive analytical solution for the complex pendulum with a rolling wheel system is obtained. The solution provides insights into the system’s behavior, such as the oscillation amplitudes, angular velocities and other relevant dynamic parameters. The proposed approach demonstrates the efficacy of the VIM in tackling complex mechanical systems and showcases the equivalence between the Lagrange multiplier derived through Elzaki transformation and the well-established Laplace transformation. The results obtained from this study contribute to the understanding and analysis of coupled translational and rotational systems, providing a valuable tool for researchers and engineers working in the field of mechanical dynamics. Sketches are made of the phase portraits close to the equilibrium points.

APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS BY EULER AND RUNGE-KUTTA METHODS

APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS BY EULER AND RUNGE-KUTTA METHODS

Externally applied magnetic force effect on the Couette flow of a viscous incompressible fluid with heat source/sink

Purpose This study aims to investigate the effect of externally applied magnetic force and heat transfer with a heat source/sink on the Couette flow with viscous dissipation in a horizontal rotating channel. The magnetic force is added to the governing equations. The effects of fluid flow parameters are observed under the applied magnetic force. In this system, the magnetic force is applied perpendicular to the plane of the fluid flow. In recent years, the magnetic field has renewed interest in aerospace technology. The physical and theoretical approach in the multidisciplinary field of magneto fluid dynamics (MFD) is applied in the field of aerospace vehicle design. Design/methodology/approach Authors use the perturbation method to solve and find the approximate solutions of differential equations. First, convert the partial differential equation to ordinary differential equation and calculate the approximate solutions in two cases. The first solution got by assuming heat generating in the fluid and the second one got when heat absorbing. After applying the external magnetic force, the effects of various fluid parameters velocity, temperature, skin friction coefficient and Nusselt number are found and discussed using tables and graphs. Findings It is found that the velocity of the fluid has decreased tendency when the rotation of the fluid and magnetic force on the fluid increases. The temperature of the fluid, Prandtl value and Eckert number increased when the heat source generated heat. When heat absorbs the heat, sink parameter increases and the temperature of the fluid decreases. Also, while heat absorbs, the temperature increases when the Prandtl value and Eckert number increase. Originality/value The skin friction coefficient on the surface increases, when the rotation parameter and the magnetic force parameter of the fluid increase. In the case of heat generating, the Nusselt number increased, while the Eckert number and Prandtl numbers increased. Also, the Nusselt number has larger values when the heat source parameter has near the constant temperature, and it has smaller values when the temperature varies. In the case of heat-absorbing, the Nusselt number decreased when the Eckert and Prandtl numbers increased. Also, the Nusselt number varies up and down while the heat absorbing parameter increases.

New Exact Operational Shifted Pell Matrices and Their Application in Astrophysics

In this work, the exact operational matrices for shifted Pell polynomials are achievable; so one can integrate and product the vector of basic functions s. The general form of the matrix of integration P is established, the dual matrix of integration Q is derived with general formulation, and the general form of the matrix derived from the product of two shifted Pell polynomials has been given. This idea is implemented on shifted Pell basis vector. Using such exact matrices, then the resident function of the equation is reached which can be written as R.P(x), where R is an algebraic equation vector and P(x) is the shifted Pell basis vector. The presented matrices can be utilized to find the approximate solution of differential equations, integral equation and the calculus of variations problems. An investigation for the convergence and error analysis of the proposed shifted Pell expansion is performed. Numerical treatment for problems in physics are included in this work to demonstrate the accuracy, easy to implement as well as accurate and satisfactory results with a small number of shifted Pell basis. Using operational matrices and the spectral technique are used together for solving Lane-Emden equation.

Gradient Descent Optimization Control of an Activated Sludge Process based on Radial Basis Function Neural Network

Most systems in science and engineering can be described in the form of ordinary differential equations, but only a limited number of these equations can be solved analytically. For that reason, numerical methods have been used to get the approximate solutions of differential equations. Among these methods, the most famous is the Euler method. In this paper, a new proposed control strategy utilizing the Euler and the gradient method based on Radial Basis Function Neural Network (RBFNN) model have been used to control the activated sludge process of wastewater treatment. The aim was to maintain the Dissolved Oxygen (DO) level in the aerated tank and have the substrate concentration Chemical Oxygen Demand (COD5) within the standard limits. The simulation results of DO show the robustness of the proposed control method compared to the classical method. The proposed method can be applied in wastewater treatment systems.

Modeling of fundamental electronic circuits by the Euler method using the Python programming language

In this work three examples of textbook circuits (resistor-capacitor, resistor-inductor and resistor-inductor-capacitor) have been modeled by employing the Euler method for the approximate solution of differential equations using algorithms implemented in the Python programming language. The aim of this work was to demonstrate how simple, yet highly effective, algorithms can be neatly implemented in one of the currently most popular programming languages for scientific applications. At the same time, the visualization of the results allows for a detailed introduction to the properties of the circuits most frequently commented on in all the relevant textbooks. Our code, as provided to the reader, includes detailed comments, and can easily be modified to model other similar systems. Instructions on how to prepare tasks for students have been also provided.

Complete in phase method for problems in chemistry

A new complete in phase method for the approximate solution of differential equations in Quantum Chemistry is proposed in this paper.

A Particle Swarm Optimization-Based Method for Numerically Solving Ordinary Differential Equations

The Euler method is a typical one for numerically solving initial value problems of ordinary differential equations. Particle swarm optimization (PSO) is an efficient algorithm for obtaining the optimal solution of a nonlinear optimization problem. In this study, a PSO-based Euler-type method is proposed to solve the initial value problem of ordinary differential equations. In the typical Euler method, the equidistant grid points are always used to obtain the approximate solution. The existing shortcoming is that when the iteration number is increasing, the approximate solution could be greatly away from the exact one. Here, it is considered that the distribution of the grid nodes could affect the approximate solution of differential equations on the discrete points. The adopted grid points are assumed to be free and nonequidistant. An optimization problem is constructed and solved by particle swarm optimization (PSO) to determine the distribution of grid points. Through numerical computations, some comparisons are offered to reveal that the proposed method has great advantages and can overcome the existing shortcoming of the typical Euler formulae.

Universal computational algorithms and their justification for the approximate solution of differential equations

The paper is devoted to the problem that determines the typical characteristics of computing equipment associated with the amount of work needed to obtain a result at a given point in the computation domain. The use of grid methods is associated with the need for continuous processing and storage of data arrays determined by the number of grid elements, which is directly proportional to the performance of the systems used. We consider alternative approaches for the construction and justification of computational methods that are not focused on the grid structure of the approximations. The substantiation of the convergence of kinetic approximations to the solution of the Cauchy problem is obtained.

Universal Computational Algorithms and Their Justification for the Approximate Solution of Differential Equations

The paper is devoted to the problem that determines the typical characteristics of computing equipment associated with the amount of work needed to obtain a result at a given point in the computation domain. The use of grid methods is associated with the need for continuous processing and storage of data arrays determined by the number of grid elements, which is directly proportional to the performance of the systems used. We consider alternative approaches for the construction and justification of computational methods that are not focused on the grid structure of the approximations. The substantiation of the convergence of kinetic approximations to the solution of the Cauchy problem is obtained.

Squeezing flow of nanofluids of Cu–water and kerosene between two parallel plates by Gegenbauer Wavelet Collocation method

Operational matrices of Gegenbauer wavelets have significant role for approximate solution of differential equations. In the present study, approximate solutions of the squeezing nanofluids of Cu–kerosene and Cu–water between parallel plates with magnetic field are obtained by GW Collocation Method. The governing nonlinear PDEs may be turned into the nonlinear ODEs by similarity transformation. These nonlinear equations are turned into the set of linear ODEs by quasilinearization technique. The effective thermal conductivity and the effective dynamic viscosity of nanofluids have been taken as models of Maxwell–Garnetts and Brinkman. The effects of physical parameters have been displayed by graphs and tables.

Approximate Solution of Differential Equations with the Help of Rational Spline Functions

For twice continuously differentiable functions on an interval and for their derivatives up to the second order, estimates are obtained for their joint uniform approximations by rational interpolation splines and their corresponding derivatives. These estimates are used to construct approximate twice differentiable solutions of boundary value problems and an initial value problem for some second-order linear differential equations.

О приближенном решении рациональными сплайн-функциями дифференциальных уравнений с особенностями

О приближенном решении рациональными сплайн-функциями дифференциальных уравнений с особенностями

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

Применение метода конечных элементов при расчете трехслойных балок

authors of this article note that one of the widely used in practice methods of strength calculation of structures is the finite element method (FEM). FEM is a procedure of approximate solution of differential equations. It was originally designed for solving problems related to the calculation of strength of structures, that is, to calculate forces, stresses and strains in solids.

Constructing approximate solutions of differential equations using fundamental functions

Constructing approximate solutions of differential equations using fundamental functions

Analytical solution of system of differential equations by variational iteration method

In this paper, Varitational Iteration Method using He’s Polynomials is used to construct the exact as well as approximate solutions of differential equations. From the obtained numerical results, it has been observed that this proposed technique is very efficient and reliable for the solution of the linear and non-linear system of differential equations. Numerical results and graphical representation reflect the accuracy and effectiveness of the proposed modification.BIBECHANA 13 (2016) 77-86

Adaptive algorithm for stochastic Galerkin method

We introduce a new tool for obtaining efficient a posteriori estimates of errors of approximate solutions of differential equations the data of which depend linearly on random parameters. The solution method is the stochastic Galerkin method. Polynomial chaos expansion of the solution is considered and the approximation spaces are tensor products of univariate polynomials in random variables and of finite element basis functions. We derive a uniform upper bound to the strengthened Cauchy-Bunyakowski-Schwarz constant for a certain hierarchical decomposition of these spaces. Based on this, an adaptive algorithm is proposed. A simple numerical example illustrates the efficiency of the algorithm. Only the uniform distribution of random variables is considered in this paper, but the results obtained can be modified to any other type of random variables.