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A three-point monotone difference scheme is proposed for solving a one-dimensional non-stationary convection-diffusion-reaction equation with variable coefficients. The scheme is based on a parabolic spline and allows to linearly reproduce…
Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we…
We consider stochastic impulse control problems where the process is driven by a general one-dimensional diffusion. We shall show a new mathematical characterization of the value function as a linear function in a certain transformed space.…
We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the…
We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak…
In this work, we study two-dimensional diffusion-wave equations with variable exponent, modeling mechanical diffusive wave propagation in viscoelastic media with spatially varying properties. We first transform the diffusion-wave model into…
We derive novel algorithms for optimization problems constrained by partial differential equations describing multiscale particle dynamics, including non-local integral terms representing interactions between particles. In particular, we…
In this paper we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual…
The solution of nonsymmetric but positive definite (NSPD) systems arising from advection-diffusion problems is an important research topic in science and engineering. Balancing domain decomposition by constraints with an adaptive coarse…
In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a…
This work is concerned with optimal control of partial differential equations where the control enters the state equation as a coefficient and should take on values only from a given discrete set of values corresponding to available…
This article concerns a systemic manifestation of small scale interfacial heterogeneities in large scale quantities of interest to a variety of diverse applications spanning the earth, biological and ecological sciences. Beginning with…
We study optimal design problems for stationary diffusion involving one or more state equations and mixtures of an arbitrary number of anisotropic materials. Since such problems typically do not admit classical solutions, we adopt a…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
We consider the response of a one-dimensional system with friction. S.W. Shaw (Journal of Sound and Vibration, 1986) introduced the set up of different coefficients for the static and dynamic phases (also called stick and slip phases). He…
We propose and analyze a one-dimensional multi-species cross-diffusion system with non-zero-flux boundary conditions on a moving domain, motivated by the mod- eling of a Physical Vapor Deposition process. Using the boundedness by entropy…
In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…
Traditionally, systems governed by linear Partial Differential Equations (PDEs) are spatially discretized to exploit their algebraic structure and reduce the computational effort for controlling them. Due to beneficial insights of the PDEs,…
Diffusion alignment aims to optimize diffusion models for the downstream objective. While existing methods based on reinforcement learning or direct backpropagation achieve considerable success in maximizing rewards, they often suffer from…