In this paper, we discuss the accuracy and complexity of various numerical techniques to solve the stochastic Landau-Lifshitz-Gilbert-Slonczewski (s-LLGS) equation. The s-LLGS equation is widely used by researchers to study the temporal evolution of the macrospin subject to spin torque and thermal noise. The numerical simulation of the s-LLGS equation requires an appropriate choice of stochasti...

The finite-genus solutions for the Hirota's bilinear difference equation are constructed using the Fay's identities for the θ-functions of compact Riemann surfaces. In the present work I want to consider once more the question of constructing the finite-genus solutions for the famous Hirota's bilinear difference equation (HBDE) [1] which has been solved in [2] using the so-called algebraic-geom...

The asymptotic behaviour of a Sturm-Liouville differential equation with coefficient λq(s), s ∈ [s0,∞) is investigated, where λ ∈ R and q(s) is a nondecreasing step function tending to ∞ as s → ∞. Let S denote the set of those λ’s for which the corresponding differential equation has a solution not tending to 0. It is proved that S is an additive group. Four examples are given with S = {0}, S =...

For a simple model problem | the Laplace equation on the unit square with a Dirichlet boundary function vanishing for x = 0, x = 1, and y = 1, and equaling some suitable g(x) for y = 0 | we present a proof of convergence for the combination technique, a modern, eecient, and easily parallelizable sparse grid solver for elliptic partial diierential equations that recently gained importance in eld...

We introduce an extension of the l-reduced KP hierarchy, which we call the lBogoyavlensky hierarchy. Bogoyavlensky’s 2 + 1-dimensional extension of the KdV equation is the lowest equation of the hierarchy in case of l = 2. We present a group-theoretic characterization of this hierarchy on the basis of the 2-toroidal Lie algebra sl l . This reproduces essentially the same Hirota bilinear equatio...

Class Meeting # 24: Transport Equations and Burger’s Equation In these notes, we introduce a class of evolution PDEs known as transport equations. Such equations arise in a physical context whenever a quantity is “transported” in a certain direction. Some important physical examples include the mass density flow for an incompressible fluid, and the Boltzmann equation of kinetic theory. We discu...

We consider the generalized Burgers equation with and without a time delay when the boundary conditions are periodic with period 2π. For the generalized Burgers equation without a time delay, that is, ut = νuxx − uux + u+ h(x), 0 < x < 2π, t > 0, u(0, t) = u(2π, t), u(x,0) = u0(x), a Lyapunov function method is used to show boundedness and uniqueness of a steady state solution and global stabil...