A user-friendly Fortran BVP solver

Textbooks on differential equations often give the impression that most differential equations can be solved in closed form, but experience does not bear this out. It remains true that solutions of the vast majority of first order initial value problems cannot be found by analytical means. Therefore, it is important to be able to approach the problem in other ways. Today there are numerous methods that produce numerical approximations to solutions of differential equations. Here, we introduce the oldest and simplest such method, originated by Euler about 1768. It is called the tangent line method or the Euler method. It uses a fixed step size h and generates the approximate solution.

Numerical methods are very significant as it provides approximate solutions to initial value problems (IVPs) in ordinary differential equations (ODEs) where analytical methods fail. This research work considers the implementation of four widely-used numerical methods: Euler's Method, Runge-Kutta Fourth-Order Method, Heun's Method, and Milne's Predictor-Corrector Method, using MATLAB, a powerful tool for technical computing. This work aims to serve as a practical guide for students and researchers, illustrating the seamless integration of theoretical concepts with computational techniques. It provides a structured framework that can be used for any initial value problems (IVPs) in ordinary differential equations (ODEs). Using a single example problem, we demonstrate the step-by-step MATLAB programming of each method, emphasizing computational efficiency, accuracy and error analysis. This research underscores MATLAB's capacity to simplify complex numerical computations and offers recommendations for future enhancements. By bridging theoretical foundations and practical applications, this work contributes to the broader understanding and accessibility of numerical methods in scientific computing.

2014 ASEE Annual Conference & Exposition Proceedings

Ordinary Differential Equations (ODE) are used to model a wide range of physical processes. An ODE is an equation containing a function of one independent variable and its ordinary derivatives. This paper presents the development and application of a practical teaching module introducing java programming techniques to electronics, computer, and bioengineering students before they encounter digital signal processing and its applications in junior and senior level courses. This paper will focus primarily on how to solve ODEs using Java and Matlab programming tools. There are two basic types of boundary condition categories for ODEsinitial value problems and two-point boundary value problems. Initial value problems are simpler to solve because you only have to integrate the ODE one time. The solution of a two-point boundary value problem usually involves iterating between the values at the beginning and end of the range of integration. Runge-Kutta schemes are among the most commonly used techniques to solve initial-value problem ODEs. Matlab also presents several tools for modeling linear systems. These tools can be used to solve differential equations arising in such models, and to visualize the input-output relations. This paper attempts to describe how to use Java programming tool to solve initial value problems of ordinary differential equations (ODEs) using the Runge-Kutta scheme. It will also discuss how to represent initial value problems and demonstrate how to apply Matlab's ODE solvers to such problems. It will also explain how to select a solver and how to specify solver options for efficient, customized execution. This paper provides an introduction to the Ordinary Differential Equations(ODEs). After a quick overview of selected numerical methods for solving differential equations using Matlab, we will briefly give an account of Euler and modified Euler methods for solving first order differential equations. This will be followed by numerical method for systems specially Runge-Kutta schemes and applications of second order differential equations in mechanical vibrations and electric circuits by leveraging the power of Java and Matlab. This paper will explain how this learning and teaching module is instrumental for progressive learning of students; the paper will also demonstrate how the numerical and integral algorithms are derived and computed through leverage of the java data structures. As a result, there will be a discussion concerning the comparison of Java and Matlab programming as well as students' feedback. The result of this new approach is expected to strengthen the capacity and quality of our undergraduate degree programs and enhance overall student learning and satisfaction.

2012

Ordinary differential equations (ODEs) are used throughout engineering, mathematics, and science to describe how physical quantities change. Hence, effective simulation (or prediction) of such systems is imperative. This paper explores the ability of MATLAB/Simulink ® to achieve this feat with relative ease-either by writing MATLAB code commands or via Simulink for linear Initial Value Problems (IVPs). Also, solutions to selected examples considered in this paper were approached from the standpoint of a numerical and exact solution for the purpose of comparison. Since no single numerical method of solving a model suffices for all systems, choice of a solver is of utmost important. Thus, experimenting between fixed-step and variable-step solver was also explored. For the selected examples, variable-step solvers out-performed the fixed-step counterpart.

Numerical methods are commonly used for solving mathematical problems that are formulated in science and engineering where it is di?cult or even impossible to obtain exact solutions. Only a limited number of di?erential equations can be solved analytically. Numerical methods, on the other hand, can give an approximate solution to (almost) any equation. An ordinary differential equation (ODE) is an equation that contains an independent variable , a dependent variable, and derivatives of the dependent variable. Literal implementation of this procedure results in Euler's method, which is, however , not recommended for any practical use. There are other methods more sophisticated than Euler's. Among them, there are four types of practical numerical methods for solving problems of ODEs. They are (i) Runge-Kutta Methods, (ii) Heun's Method, (iii) RK4 Method, and (iv)RKF5 Method. MATLAB is an integrated technical computing environment that combines numeric computation, advanced graphics and visualization and a high-level programming language. MATLAB was founded by Jack Little Cleve Moler and Steve Bangert in 1984. Color graphics makes it possible to display curves, surfaces, and solids in two and three dimensions in a way that it is both more e?ective and more engaging. The easy syntax of MATLAB makes it possible for users to get comfortable results quickly. MATLAB graphics are excellent and easy to use. It is easy to program in MATLAB, making it possible to do numerical calculations using simple steps. We can also use the symbolic capability of MATLAB to get analytical solutions. MATLAB is becoming the most popular software package in Mathematics. • In first chapter, we study the preliminaries needed to use MATLAB to solve problems in ODEs. We see basic definitions based on Di?erential Equation and Partial Di?erential Equation. We introduce MATLAB plotting to simulate the solution of Di?erential Equation problems. • In second chapter, we review the software package MATLAB for high-performance numerical computation and visualization which provides an interactive environment with hundreds of built-in-functions for technical computation, graphics and animation. • In third chapter, we discuss the various Numerical Methods available for solving DEs and they are (1) Euler's method, (2) Heun's method,(3) RK4 method and (4) RKF5 method. • In fourth chapter, we study the MATLAB programs for Solving Differential Equations. Also we study Euler methods and Runge-Kutta methods in MATLAB. • In fifth chapter, we compare various methods and their approximation for 1st order di?erential equation and a system of ODEs. We also test the problem with the above methods. In sixth chapter, we attempt to simulate the solution of few partial di?erential equations using MATLAB.

Acm Computing Surveys, 1985

Mathematical models when simulating the behavior of physical, chemical, and biological systems often include one or more ordinary differential equations (ODES). To study the system behavior predicted by a model, these equations are usually solved numerically.

The Mathematical Gazette, 2005

SIAM J. SCI. COMP, 1993

A popular approach for the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge-Kutta formulas, known as mono-implicit Runge-Kutta (MIRK) formulas, which can be implemented substantially more e ciently than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather only a discrete approximation at certain points within the problem interval is obtained. However recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge-Kutta formula. These ideas have recently been extended to develop interpolants for the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in a code for the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on these interpolants, as an alternative to the standard use of global error estimates, as the basis for termination and mesh redistribution criteria.

2000

In this paper we describe the development of parallel software for the numerical solution of boundary value ordinary differential equations (BVODEs). The software, implemented on two shared memory, parallel architectures, is based on a modification of the MIRKDC package, which employs discrete and continuous mono-implicit Runge-Kutta schemes within a defect control algorithm. The primary computational costs are associated with the

2015

We discuss how polynomial collocation can be used to solve ordinary differential equations with singularities and differential-algebraic equations of higher index. We introduce the open domain MATLAB code bvpsuite and apply it to solve the Korteweg-de Vries equation.