2,801 research outputs found
Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs
Algorithms are presented for the tanh- and sech-methods, which lead to
closed-form solutions of nonlinear ordinary and partial differential equations
(ODEs and PDEs). New algorithms are given to find exact polynomial solutions of
ODEs and PDEs in terms of Jacobi's elliptic functions.
For systems with parameters, the algorithms determine the conditions on the
parameters so that the differential equations admit polynomial solutions in
tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples
illustrate key steps of the algorithms.
The new algorithms are implemented in Mathematica. The package
DDESpecialSolutions.m can be used to automatically compute new special
solutions of nonlinear PDEs. Use of the package, implementation issues, scope,
limitations, and future extensions of the software are addressed.
A survey is given of related algorithms and symbolic software to compute
exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at
http://www.mines.edu/fs_home/whereman
Exact Solution of the Hyperbolic Generalization of Burgers Equation, Describing Travelling Fronts and their Interaction
We present new analytical solutions to the hyperbolic generalization of
Burgers equation, describing interaction of the wave fronts. To obtain them, we
employ a modified version of the Hirota method.Comment: 12 pages, 3 figure
Hybrid modelling of individual movement and collective behaviour
Mathematical models of dispersal in biological systems are often written in terms of partial differential equations (PDEs) which describe the time evolution of population-level variables (concentrations, densities). A more detailed modelling approach is given by individual-based (agent-based) models which describe the behaviour of each organism. In recent years, an intermediate modelling methodology – hybrid modelling – has been applied to a number of biological systems. These hybrid models couple an individual-based description of cells/animals with a PDEmodel of their environment. In this chapter, we overview hybrid models in the literature with the focus on the mathematical challenges of this modelling approach. The detailed analysis is presented using the example of chemotaxis, where cells move according to extracellular chemicals that can be altered by the cells themselves. In this case, individual-based models of cells are coupled with PDEs for extracellular chemical signals. Travelling waves in these hybrid models are investigated. In particular, we show that in contrary to the PDEs, hybrid chemotaxis models only develop a transient travelling wave
Travelling wave solutions in a negative nonlinear diffusion-reaction model
We use a geometric approach to prove the existence of smooth travelling wave
solutions of a nonlinear diffusion-reaction equation with logistic kinetics and
a convex nonlinear diffusivity function which changes sign twice in our domain
of interest. We determine the minimum wave speed, c*, and investigate its
relation to the spectral stability of the travelling wave solutions.Comment: 23 pages, 10 figure
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