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An example of a nonlinear delay differential equation; applications in number theory, distribution of primes, and control theory [5] [6] [7] Chrystal's equation: 1 + + + = Generalization of Clairaut's equation with a singular solution [8] Clairaut's equation: 1
MATLAB (an abbreviation of "MATrix LABoratory" [18]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
Full API for Java and Matlab (the latter via add-on product) PyMFEM (Python) Python, Scilab or Matlab Python bindings to some functionality Python Other: Predefined equations: Yes, many predefined physics and multiphysics interfaces in COMSOL Multiphysics and its add-ons. A large number of Bilinear and Linear forms
Name Dim Equation Applications Landau–Lifshitz model: 1+n = + Magnetic field in solids Lin–Tsien equation: 1+2 + = Liouville equation: any + = Liouville–Bratu–Gelfand equation
SageMath, an open-source application that uses a Python-like syntax with a wide range of capabilities spanning several branches of mathematics. SciPy, a Python package that includes an ODE integration module. Chebfun, an open-source package, written in MATLAB, for computing with functions to 15-digit accuracy.
In practical terms, the distinction between DAEs and ODEs is often that the solution of a DAE system depends on the derivatives of the input signal and not just the signal itself as in the case of ODEs; [3] this issue is commonly encountered in nonlinear systems with hysteresis, [4] such as the Schmitt trigger. [5]
As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs). + = where is constant and is a function of and . We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form
For example, the second-order equation y′′ = −y can be rewritten as two first-order equations: y′ = z and z′ = −y. In this section, we describe numerical methods for IVPs, and remark that boundary value problems (BVPs) require a different set of tools. In a BVP, one defines values, or components of the solution y at more than one ...