scholarly journals Solution of Differential Equations with Applications to Engineering Problems

10.5772/67539 ◽  
2017 ◽  
Author(s):  
Cheng Yung Ming
Keyword(s):  
Differential Equations ◽  
Engineering Problems

New Operational Matrix for Solving Multiterm Variable Order Fractional Differential Equations

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
A. M. Nagy ◽  
N. H. Sweilam ◽  
Adel A. El-Sayed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Algebraic System ◽  
Order Differential Equation ◽  
Fundamental Problem ◽  
Operational Matrix ◽  
Variable Order ◽  
Engineering Problems ◽  
Fractional Differential ◽  
Fourth Kind

The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

scholarly journals Data Driven Solutions and Discoveries in Mechanics Using Physics Informed Neural Network

2020 ◽  
Author(s):  
Qi Zhang ◽  
Yilin Chen ◽  
Ziyi Yang
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Data Driven ◽  
Attractive Alternative ◽  
The Finite Element Method ◽  
The Neural Network ◽  
Engineering Problems ◽  
Free Parameters

Deep learning has achieved remarkable success in diverse computer science applications, however, its use in other traditional engineering fields has emerged only recently. In this project, we solved several mechanics problems governed by differential equations, using physics informed neural networks (PINN). The PINN embeds the differential equations into the loss of the neural network using automatic differentiation. We present our developments in the context of solving two main classes of problems: data-driven solutions and data-driven discoveries, and we compare the results with either analytical solutions or numerical solutions using the finite element method. The remarkable achievements of the PINN model shown in this report suggest the bright prospect of the physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters. More broadly, this study shows that PINN provides an attractive alternative to solve traditional engineering problems.

A Matrix-Vector Operation-Based Numerical Solution Method for Linear m-th Order Ordinary Differential Equations: Application to Engineering Problems

2013 ◽  
Vol 5 (03) ◽  
pp. 269-308 ◽  
Author(s):  
M. Aminbaghai ◽  
M. Dorn ◽  
J. Eberhardsteiner ◽  
B. Pichler
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Solution Method ◽  
Variable Coefficients ◽  
Exact Integration ◽  
Robust Solution ◽  
Vector Operation ◽  
Engineering Sciences ◽  
Engineering Problems ◽  
Matrix Vector

AbstractMany problems in engineering sciences can be described by linear, inhomogeneous,m-th order ordinary differential equations (ODEs) with variable coefficients. For this wide class of problems, we here present a new, simple, flexible, and robust solution method, based on piecewise exact integration of local approximation polynomials as well as on averaging local integrals. The method is designed for modern mathematical software providing efficient environments for numerical matrix-vector operation-based calculus. Based on cubic approximation polynomials, the presented method can be expected to perform (i) similar to the Runge-Kutta method, when applied to stiff initial value problems, and (ii) significantly better than the finite difference method, when applied to boundary value problems. Therefore, we use the presented method for the analysis of engineering problems including the oscillation of a modulated torsional spring pendulum, steady-state heat transfer through a cooling web, and the structural analysis of a slender tower based on second-order beam theory. Related convergence studies provide insight into the satisfying characteristics of the proposed solution scheme.

MESHFREE METHODS AND THEIR COMPARISONS

2005 ◽  
Vol 02 (04) ◽  
pp. 477-515 ◽  
Author(s):  
Y. T. GU
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Future Development ◽  
Meshless Method ◽  
Meshfree Methods ◽  
Shape Functions ◽  
Engineering Problems ◽  
Practical Engineering ◽  
Technical Issues

In recent years, one of the hottest topics in computational mechanics is the meshfree or meshless method. Increasing number of researchers are devoting themselves to the research of the meshfree methods, and a group of meshfree methods have been proposed and used to solve the ordinary differential equations (ODEs) or the partial differential equations (PDE). In the meantime, meshfree methods are being applied to a growing number of practical engineering problems. In this paper, a detailed discussion will be provided on the development of meshfree methods. First, categories of meshfree methods are introduced. Second, the methods for constructing meshfree shape functions are discussed, and the interpolation qualities of them are also studied using the surface fitting. Third, several typical meshfree methods are introduced and compared with each others in terms of their accuracy, convergence and effectivity. Finally, the major technical issues in meshfree methods are discussed, and the future development of meshfree methods is addressed.

scholarly journals On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2262
Author(s):  
Emilio Defez ◽  
Javier Ibáñez ◽  
José M. Alonso ◽  
Michael M. Tung ◽  
Teresa Real-Herráiz
Keyword(s):  
Differential Equations ◽  
Numerical Test ◽  
Test Problems ◽  
Science And Engineering ◽  
Third Order ◽  
Differential Problem ◽  
Order Matrix ◽  
Engineering Problems ◽  
Matrix Differential Equations ◽  
Matrix Differential

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y(3)(x)=f(x,Y(x)). Some numerical test problems are also included, whose solutions are computed by our method.

scholarly journals Numerical solution for a new fuzzy transform of hyperbolic goursat partial differential equation

2019 ◽  
Vol 16 (1) ◽  
pp. 292
Author(s):  
Nur Syazana Saharizan ◽  
Nurnadiah Zamri
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Numerical Method ◽  
High Accuracy ◽  
Fuzzy Mathematics ◽  
Fuzzy Transform ◽  
Physical Problems ◽  
Engineering Problems ◽  
Physical Phenomena

<p>The main objective of this paper is to present a new numerical method with utilization of fuzzy transform in order to solve various engineering problems that represented by hyperbolic Goursat partial differentical equation (PDE). The application of differential equations are widely used for modelling physical phenomena. There are many complicated and dynamic physical problems involved in developing a differential equation with high accuracy. Some problems requires a complex and time consuming algorithms. Therefore, the application of fuzzy mathematics seems to be appropriate for solving differential equations due to the transformation of differential equations to the algebraic equation which is solvable. Furthermore, development of a numerical method for solving hyperbolic Goursat PDE is presented in this paper. The method are supported by numerical experiment and computation using MATLAB. This will provide a clear picture to the researcher to understand the utilization of fuzzy transform to the hyperbolic Goursat PDE.</p>

Fuzzy Sumudu Transform Approach to Solving Fuzzy Differential Equations With Z-Numbers

2019 ◽  
pp. 18-48 ◽  
Author(s):  
Raheleh Jafari ◽  
Sina Razvarz ◽  
Alexander Gegov ◽  
Satyam Paul ◽  
Sajjad Keshtkar
Keyword(s):  
Nonlinear Systems ◽  
Theoretical Analysis ◽  
Differential Equations ◽  
Fuzzy Numbers ◽  
Euler Method ◽  
Uncertain Nonlinear Systems ◽  
Engineering Problems ◽  
Sumudu Transform ◽  
Simulation Results

Uncertain nonlinear systems can be modeled with fuzzy differential equations (FDEs) and the solutions of these equations are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs. In this book chapter, the solutions of FDEs are approximated by utilizing the fuzzy Sumudu transform (FST) method. Here, the uncertainties are in the sense of fuzzy numbers and Z-numbers. Important theorems are laid down to illustrate the properties of FST. This new technique is compared with Average Euler method and Max-Min Euler method. The theoretical analysis and simulation results show that the FST method is effective in estimating the solutions of FDEs.

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