This paper deals with an inverse problem of simultaneously determining the dispersion coefficients and the space-dependent source magnitude in 2D advection dispersion equation with finite observations at the final time. The forward problem is solved by using the alternating direction implicit (ADI) finite difference scheme, and then the optimal perturbation algorithm with the regularization par...

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection– diffusion equation in the unbounded domain Rn is converted to one on the bounded domain [−1,1]n , and the Laplace transform is applied to eliminate time dependency. The consequent boundary value ...

A large class of physically important nonlinear and nonhomogeneous evolution problems, characterized by advection-like and diffusion-like processes, can be usefully studied by a time-differential form of Kolmogorov’s solution of the backward-time Fokker-Planck equation. The differential solution embodies an integral representation theorem by which any physical or mathematical entity satisfying ...

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri ...

As a category of challenging flow problems, flows with moving boundaries and interfaces, includes fluid–particle, fluid–object and fluid–structure interactions; free-surface and two-fluid flows; and flows with moving mechanical components. To address the challenges involved in computation of this category of problems, we developed a number of interface-tracking and interface-capturing technique...

We describe an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators we consider do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we...

A general framework for constructing constraint-preserving numerical methods is presented and applied to a multidimensional divergence-constrained advection equation. This equation is part of a set of hyperbolic equations that evolve a vector field while locally preserving either its divergence or curl. We discuss the properties of these equations and their relation to ordinary advection. Due t...

We show that human mental states are unresolvable by suggesting a mathematical function that describes human mental states in relation to parallel universe theory. The function is a solution to a multi-dimensional advection equation; representing a situation a person is faced with, and its timederivative showing the mental state in that situation. This function has interesting characteristics t...

The propagation of a signal in a continuous medium and the associated evolution of error is of prime importance in many applications of applied physics. There have been many efforts in analyzing error dynamics, using method attributed to von Neumann [1,2], that is readily applied for linear equations and in quasilinearized form for non-linear equations. The main assumption for linear problems i...

We describe briefly how a third-orderWeighted Essentially Nonoscillatory (WENO) scheme is derived by coupling aWENO spatial discretization scheme with a temporal integration scheme. The scheme is termed WENO3. We perform a spectral analysis of its dispersive and dissipative propertieswhenused to approximate the 1D linear advection equation anduse a technique of optimisation to find the optimal ...