Biograph is an interactive program that computes numerically, then graphs solutions to systems of ordinary differential equations (ODE's) and difference equations. The program is capable of solving systems of n simultaneous DE's or difference equations. Biograph is a menu driven program, but the menus are a little hard to understand. Also, the graphics are not very sophisticated. There is also great difficulty in creating a file. It has a lot of options such as saving data to a file, retrieving data previously saved, output data, changing parameters in already existing models, changing initial conditions, or determining the limits of the axes for a graph. You can also determine equilibrium points, get a phase plane, and graph every variable versus time axis. When compared to Models or Phaser, this software is easily the most difficult to execute and work with. The program itself would be good to solve differential and difference equations because it provides the user with great options in order to examine the graphs and solutions of the problems. However, it requires a great deal of time in order to be able to execute the program successfully.
Chaotic Dynamics Workbench allows the advanced college physics student or researcher to perform interactive numerical experiments on nonlinear systems modeled by ordinary differential equations. Using graphical displays and direct, real-time interaction, this software product allows exploration of such phenomena as the period-doubling route to chaos, strange attractors, sensitive dependence on initial conditions, Lyapunov exponents, the fractal dimension of strange attractors, and the effect of damping on the dimension of the attractor.
Chaos Demonstrations is a collection of eighteen demonstrations illustrating chaos in physical and biological systems. The goal of the program is to encourage an appreciation of the complexity and beauty of even simple systems, not only for the professional scientist, but also for the interested non-specialist. The demonstrations can be appreciated on many different levels, from the forefront of research in nonlinear dynamics to pure art. They can be used by an instructor in the classroom, or by individual students as a tutorial. You can sit back and watch the demonstrations cycle through an automatic sequence, or in the complete version of the program you can take control and examine in detail the effect of varying the parameters.
Chaotic Mapper is an exploratory tool that allows you to examine 22 one- and two-dimensional iterative maps and three-dimensional differential equations. You can enter your own system of equations and, depending on the map you choose, you can examine two-dimensional plots, parameter space plots, Poincare sections, convergence maps, basins of attraction, or the behavior of a group of points.
Depad graphs solutions of systems of differential equations, plot dynamically simulations in two and three dimensions, calculate and plot partial Fourier series and use such series in differential equations.
Application package Desir (Differential Equations Systems Integrator Research) is devoted to the numerical investigation of ordinary differential equations. It is represented by a collection of executed programs and libraries for Turbo Pascal language. The programs being executed are designed to the introduction of new information to the archive or to the choice of the existing system of differential equations, to generating the program in Pascal language for the given system of equations, to starting of the TPC translator of the version corresponding to the package libraries, and to starting of the created program of investigating the system of differential equations. The system under study should be given in the form: Y'(t)=F(t,Y(t),P) , where Y is the vector of N components (system's dimensionality, maximum value of which equals 8) and P is the vector of parameters. Besides the system itself, the matrix of linearized system (Jacobi matrix) should be introduced. The main features of the program are the following:
Differential Systems is an ODE solver which deals with a system of up to two first order differential equations. A variety of solving methods are available: fifteen methods including Euler, Runge-Kutta and Adams and adaptive Runge-Kutta methods.
The first goal of the Godess project is that of good solvers for IVPs implemented in a uniform way for several different methods. Since one purpose was to try out new methods it was concepted in the idea to be easy to add a new method and to test and verify it under certain conditions All solvers behave in a coherent way and have the same calling sequence. The solver has the broad functionality required by modern modeling environments and simulation tools. The Godess program is written in C++ in an object oriented fashion. EpODE design is very close to the requests of the Godess project.
Graphmatica is an equation plotter which supports five types of planar graphs (including polar, parametric, logarithmic, inequalities and limited implicit plots), plots planar vector fields and solutions of the corresponding differential equations, unlimited graphs on screen at once, saving setup information and lists of equations, flexible grid labeling, lock-on coordinate cursor and several ways to resize the grid. Its calculus options include symbolic differentiation, drawing of tangent lines, and numerical integration (you use a mouse to select points and areas). Offers on-line help and demonstration files.
Hi-Q is a compiled language for developing numerical mathematical solution functions. Unlike other mathematics programs, it is based around an object oriented compiled language, rather than an interpreted script. No high-level solving tools are provided, instead the user has to generate them using the language. The language consists of a large selection of language commands for accomplishing mathematical tasks. As an example, instead of providing a complete ordinary differential equation solver, the tools for generating the solver are provided. This way, the solcer can be tailored to fit user's needs. The user combines these tools with basic input and output functions to develop the solver. This allows for producing functions tailor made for the type of solution desired. It also makes a less intuitive package. Hi-Q is more application oriented than other solvers, and as such is primarily designed for solving numeric, rather than symbolic problems. %The graphics support for Hi-Q is rudimentary. The newest version has prebuilt solvers included for ODEs.
Mafia (Math And Fun International Association) is a potpourri of programs; applications of various algorithms to give numerical solutions to various problems. Menu items, between others, include: Plot a given function, solve for a zero of a given function, solve system of N first order ODE, solve N -th order ODE (the method used for this problems is the Bulirsch-Stoer integration procedure with 4-th order Runge-Kutta step taken at points of non-convergence), numerical integration of a function using iterative methods ( Romberg's method, Adaptive Gauss-Legendre Quadrature), numerical integration of a function using various Gauss Quadratures ( Legendre Quadrature, Chebyshev Quadrature, Laguerre Quadrature, Hermite Quadrature), and analytical integration of one of the functions stored in the integration database.
Maple is an essential tool for anyone who needs to use or study mathematics. It is a powerful, sophisticated program which offers a variety of tools for solving differential equations. Its explicit solvers will significantly lessen the tedium of solving DE after DE by using cookbook methods. For numerical approximations, it offers techniques ranging from Euler's method to Lsode. Graphs and animations can be made in two or three dimensions. In Release 4 version, some procedures have been added to convert one or more DEs to a system of first-order DEs, and to solve by the method of reduction. The numerical dsolve procedure has added single and multi-step Gear methods as well as lsode to its suite of available solvers. Maple offers a simple structured programming language (similar to C and Pascal) which can aid greatly in solving complicated problems. Errors messages are often cryptic, and while Maple's one-line help is extensive, it is not always written clearly and can take some effort to understand.
Mathematica is a general symbolic and numeric solver, capable of performing a wide variety of mathematical computations. It uses its own programming language, which makes for a very flexible solver. The Mathematica programming language contains a rich set of commands, for solving calculus, algebra, differential equations, linear algebra, and trigonometric problems. Mathematica performs very well as a DE solver, presenting solutions to the n th order systems both numerically and graphically. Mathematica has a wide variety of graphing capabilities, including 3D graphs and plots. While this diversity makes Mathematica a very comprehensive solver system, it also makes Mathematica a particularly difficult program to use, requiring a fairly lengthy learning curve.
Matlab is the most famous numerically oriented multifunctional program system. Using Matlab, equations can be defined and evaluated in a very simple way; data and user-defined functions can be stored and re-used and the results of calculations can be displayed graphically. The primary field of Matlab applications is interactive numerical linear algebra: MATrix LABoratory. Matlab covers, in addition to matrix calculations, many other fields of numeric such as determination of the zeros of polynomials, the analysis of data using fast Fourier transforms, the numerical solution of initial value problems in ordinary differential equations etc. The graphical functionality makes it possible to plot two- and three- dimensional technical color graphics. Matlab has an interpreter and can be programmed to extend the functionality of the system in this way. Fortran and C programs can be called from Matlab. As a result, the numerical solution of computationally extensive problems can be speed up, and existing software can be re-used. A way to expand the Matlab functionality in the field of symbolic is to use the Symbolic Math Toolbox which uses the Maple kernel.
Mdep (Midshipman Differential Equations Program) is a program developed for use in differential equations and other applied mathematics courses. It provides:
Mlab, (for Modeling LABoratory), is a program for interactive mathematical and statistical modeling. It includes curve-fitting, differential equations, statistics and graphics as some of its major capabilities. MLAB provides more than thirty command types and more than four hundred built-in functions from the areas of elementary mathematics, transcendental functions, probability and statistics, linear algebra, optimization, cluster analysis, combinatorics, numeric input/output, and graphics. The usual low-level functions, e.g., sine, cosine, log, etc., are present, as well as functions performing more complex analyses, such as singular value decomposition, discrete Fourier transforms, solution of differential equation systems, and constrained non-linear optimization, among many others.
Models is a tool for defining and visualizing self-referential mathematical models, including dynamical systems and differential equations.
Octave, a public domain product, is very useful for solving practical numerical problems. Octave has the same basic designs and language constructs as Matlab.
Ode is a public domain program which solves numerically initial value problems for systems of ordinary differential equations such as arise in all branches of science. The solution is presented graphically or as a table of values.
Odepack is a Systematized Collection of ODE Solvers available from http://netlib.org. Odepack is a collection of Fortran solvers for the initial value problem for ordinary differential equation (ODE) systems. It currently includes six solvers, suitable for both stiff and nonstiff systems, and includes solvers for systems given in linearly implicit form as well as solvers for systems given in explicit form. The solvers are written in Fortran IV with a few exceptions, and with minimal machine dependencies. Each solver consists of a driver having the same name as the solver, and some number of subordinate routines. Lsode (Livermore Solver for Ordinary Differential Equations) is the basic solver of the collection. It solves stiff and nonstiff systems of the form y'(t) = f(y(t)) . In the stiff case, it treats the Jacobian matrix as either a full or a banded matrix, and as either user-supplied or internally approximated by difference quotients. It uses Adams methods (predictor-corrector) in the nonstiff case, and Backward Differentiation Formula (BDF) methods in the stiff case. The linear systems that arise are solved by direct methods (LU factor/solve). Lsode supersedes the older GEAR and GEARB packages, and reflects a complete redesign of the user interface and internal organization, with some algorithmic improvements.
Our software is similar to Odexpert. The main thrust of EpODE is also its automated identification of problem properties and method properties. Both applications (1) were designed especially for the numerical solution of ordinary differential equations of stiff type (many software packages can not integrate such equations), (2) have a friendly user interface for problem description, (3) perform automatic test as stiffness or linearity of the problem, (4) recommend the appropriate type of method to be used. Odexpert is dependent from Maple: the nonlinearity is verified by Maple, and the Jacobian is generated by Maple. EpODE has some internal procedures which are duing this tasks. Moreover, in EpODE, no supplementary code will be generated when a new method will be added (only a new text file with the user fields from the dialog boxes) and will be applied. Odexpert is designed to automatically select the appropriate solvers and to generate the corresponding source codes for calling the selected solvers; parallel/distributed codes can not be generated with Odexpert. On the other hand, EpODE do not use a rule-based programming language (the inference mechanism for Odexepert is implemented in OPS83). All the subroutines of the first version of EpODE are written in C.
Ode Workbench is designed to numerically solve an ordinary differential equation of arbitrary order n or a system of n first-order equations.
Phaser (An Animator/ Simulator for Dynamic Systems) solves systems of simultaneous autonomous differential or difference equations. It has a library of more well known dynamical systems and lets the user to add additional equations. Phaser provides three different algorithms (Euler, improved Euler or Henry, and fourth order Runge-Kutta) for solving differential equations. It can solve systems of up to nine first-order ODEs.
The Plot program is a graphing package. Equations can be graphed in four different modes: cartesian, polar, parametric, and implicit. Tables of values can be generated. Sums, products, compositions, etc. of functions as well as parameterized families of functions can be plotted. The program will also plot solutions of both two and three dimensional differential equations using a modified Euler method. An animation feature allows the user to create several images which are stored in RAM and then can be displayed in rapid succession.
SymbMath, (an abbreviation for Symbolic Mathematics), is not only a symbolic calculator but also an expert system that can solve symbolically mathematical problems. SymbMath performs symbolic formula, as well as exact numeric computation. It can manipulate complicated formulas and return answers in terms of symbols, formulas and exact numbers, not just floating-point numbers. Its capabilities include facilities to provide analytical and numerical answers for: Differentiation, Integration, Solution of equations (roots of a polynomial, systems of algebraic or differential equations), Manipulation of expressions, Calculation, Limits, Chemical calculation, Taylor series, lists, arrays, vectors, matrices, sum, product, etc.
VisualMethods is a shareware package which graphs functions (of a single variable), numerically integrates functions, shows the area, numerically integrates differential equations and performs interpolation.
WinPlot is a multi-purpose plotting program. Line fields of differential equations on the plane as well as solutions to initial value problems can be plotted. Parametric families of slope fields can be plotted and displayed. Solutions of differential equations on 3-dimensional space can also be plotted; such solutions can be rotated and displayed.