We study local conservation laws of variable coefficient diffusion–convection equations of the form f(x)ut = (g(x)A(u)ux)x + h(x)B(u)ux. The main tool of our investigation is the notion of equivalence of conservation laws with respect to the equivalence groups. That is why, for the class under consideration we first construct the usual equivalence group G∼ and the extended one Ĝ∼ including tran...

In this paper, we propose a two-stage variable selection procedure for high dimensional quantile varying coefficient models. The proposed method is based on basis function approximation and LASSO-type penalties.We show that the first stage penalized estimator with LASSO penalty reduces the model from ultra-high dimensional to a model that has size close to the true model, but contains the true ...

In this paper, we use two different integral techniques, the first integral and the direct integral method, to study the variable coefficient nonlinear Schrödinger (NLS) equation arising in arterial mechanics. The application of the first integral method yielded periodic and solitary wave solutions. Using the direct integration lead to solitary wave solution and Jacobi elliptic function solutions.

We develop a solver for nonseparable, self adjoint elliptic equations with a variable coefficient. If the coefficient is the square of a harmonic function,a transformation of the dependent variable, results in a constant coefficient Poisson equation. A highly accurate, fast, Fourier-spectral algorithm can solve this equation. When the square root of the coefficient is not harmonic, we approxima...

We show that the integrable subclassess of the equations q,t = f(x, t) q,3 + H(x, t, q, q,1) are the same as the integrable subclassess of the equations q,t = q,3 + F (q, q,1).

In this paper, we develop a new variable selection procedure for quantile varying coefficient models with longitudinal data. The proposed method is based on basis function approximation and a class of group versions of the adaptive LASSOpenalty,which penalizes the Lγ norm of the within-group coefficients with γ ≥ 1. We show that with properly chosen adaptive group weights in the penalization, t...

For scalar time-dependent variable-coefficient PDE, it has been demonstrated that Krylov subspace spectral (KSS) methods achieve high-order accuracy and also possess highly desirable stability properties, especially considering that they are explicit. In this paper, we examine the generalization of these methods to systems of variable-coefficient PDE by selection of appropriate bases of trial a...

In this paper, we design a domain decomposition algorithm for the two-dimensional parabolic equation with variable coefficient by using a larger spacing at interface points and the implicit scheme at the interior points, hence get an algorithm with the relaxed stability bounds. Then we prove the stability and analyze the accuracy of the algorithm by using the idea of maximum principle. Some res...

We study the blow-up behavior for positive solutions of a reaction–diffusion equationwith nonnegative variable coefficient. When there is no stationary solution, we show that the solution blows up in finite time. Under certain conditions, we then show that any point with zero source cannot be a blow-up point. © 2012 Elsevier Ltd. All rights reserved.

In this article we consider variable coefficient time dependent wave equations in R×R. Using phase space methods we construct outgoing parametrices and prove Strichartz type estimates globally in time. This is done in the context of C metrics which satisfy a weak asymptotic flatness condition at infinity.