Once ReΨ and ImΨ are found from these two coupled di erential equations, the time-dependent probability density |Ψ| for a particle represented by the wave function Ψ can be calculated from |Ψ| = Ψ∗Ψ = (ReΨ) + (ImΨ), where the * denotes the complex conjugate. Our task, then, is to solve Equations (2) and (3) numerically, and this computational approach requires that the partial derivatives be ap...

In this paper, we use a deep mathematical result (namely, a minor modi cation of Kolmogorov's solution to Hilbert's 13th problem) to explain why fundamental physical equations are of second order. This same result explain why all these fundamental equations naturally lead to nonsmooth solutions like singularity. Formulation of the problem. Most physical phenomena are described in terms of parti...

We establish a close relation between higher order Riccati equations and Faá di Bruno polynomial respectively Ramanujan’s differential equations connected to modular forms.

Continuous system dynamics can be described by possibly large systems of di erential equations These can be either ordinary di erential equations ODEs or contain algebraic constraints as well to form di erential and algebraic equations DAEs Complex systems such as aircraft often operate in di erent modes of continuous operation and when mode changes occur the continuous dynamics change abruptly...

| Four schemes for linearizing the transconductance of the basic di erential pair in subthreshold CMOS are examined. They are: source degeneration via (1) diode-connected transistors, (2) a single di usor and (3) symmetric di usors, and (4) multiple asymmetric di erential pairs. We derive equations for the di erential output current for each linearizing technique. We apply a maximally at optimi...

A transformation is introduced to e!ectively remove level e!ects, i.e. the state dependency of the di!usion function, in a restricted class of multivariate stochastic di!erential equations such that the general continuous}discrete-time nonlinear "ltering problem may be solved using new or existing implementations of the extended kalman "lter (EKF). An implementation of a quasi-maximum likelihoo...

In this paper we consider the pair di usion process in more than two spatial dimensions. In this case we are able to prove just a local existence result, since it is not possible to deduce global a priori estimates for the equations as it can be done in the two dimensional case. The model includes a nonlinear system of reaction{ drift{di usion equations, a nonlinear ordinary di erential equatio...

The equivalent, second equivalent and (simply) modi®ed equations for the implicit midpoint rule are shown to be asymptotically equivalent in the sense that an asymptotic analysis of these equations with the time step size as small parameter yields exactly the same results; for linear problems with constant coecients, they are also equivalent to the original ®nite di€erence scheme. Straightforw...

1. Introdu tion. There are plenty of appli ation papers in whi h kernels or radial basis fun tions are su essfully used for solving partial di erential equations by meshless methods. The usage of kernels is typi ally based on spatial interpolation at s attered lo ations, writing the trial fun tions entirely in terms of nodes [2℄. For stationary partial di erential equations, the dis retization ...

We classify all partial di¤erential equations with polynomial coe¢ cients in x and y of the form A(x)uxx + 2B(x; y)uxy + C(y)uyy +D(x)ux + E(y)uy = nu; which has weak orthogonal polynomials as solutions and show that partial derivatives of all order are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial di¤erential equations in d-variables.