Matlab has facilities for the numerical solution of ordinary differential equations (ODEs) of any order. In this document we first consider the solution of a first order ODE. Higher order ODEs can be solved using the same methods, with the higher order equations first having to be reformulated as a system of first order equations. Techniques for solving the first order and second order equation...

Under the traveling wave transformation, Camassa-Holm equation with dispersion is reduced to an integrable ODE whose general solution can be obtained using the trick of one-parameter group. Furthermore combining complete discrimination system for polynomial, the classifications of all single traveling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an aff...

Most sequence-learning studies have confounded different types of information, making it difficult to know precisely what is learned. Addressing many of the confounds, the current study shows that people can learn 1st-, 2nd-, and 3rd-order transition probabilities. Measures directly assessing awareness of the probabilities show that the knowledge is implicit early in training and becomes explic...

Nonlinear differential-algebraic equations (DAE) are typically solved using implicit stiff solvers based on backward difference formula or RADAU formula, requiring a Newton-Raphson approach for the nonlinear equations or using Rosenbrock methods specifically designed for DAEs. Consistent initial conditions are essential for determining numeric solutions for systems of DAEs. Very few systems of ...

Control of systems governed by Partial Differential Equations (PDEs) is an interesting subject area, as the classical tools of control theory are not directly applicable and PDEs can display enormously rich behaviour spatiotemporally. This paper considers the boundary control of a 2D heat flow problem. A solution to the control problem is obtained after a suitable model reduction. The considere...

An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ξ = F (x), η = P (x) y+Q(x)] where ξ ∂/∂x+η ∂/∂y is the symmetry generator is presented. To these linear symmetries one can associate an ODE class which embraces all first order ODEs mappable into separable through linear transformations {t = f(x), u = p(x) y + q(x)}. This single ODE class includes ...

Integrating various suppliers to satisfy market demand is of great importance for e ective supply chain management. In this paper, we consider the ODE-PDE model of supply chain and apply a classical explicit fourth-order Runge-Kutta scheme for the related ODE model of suppliers. Also, the convergence of the proposed method is proved. Finally a numerical example is studied to demonstrate the acc...

Nonsmoothness in dynamic process models can hinder conventional methods for simulation, sensitivity analysis, and optimization, and can be introduced, for example, by transitions in flow regime or thermodynamic phase, or through discrete changes in the operating mode of a process. While dedicated numerical methods exist for nonsmooth problems, these methods require generalized derivative inform...