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There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…
The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad…
In this paper, we present a theoretical and computational workflow for the non-parametric Bayesian inference of drift and diffusion functions of autonomous diffusion processes. We base the inference on the partial differential equations…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to…
We present an algorithm for the numerical solution of the equations governing combustion in porous inert media. The discretization of the flow problem is performed by the mixed finite element method, the transport problems are discretized…
This paper provides a methodology of verified computing for solutions to 1-dimensional advection equations with variable coefficients. The advection equation is typical partial differential equations (PDEs) of hyperbolic type. There are few…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
Despite its numerical challenges, finite element method is used to compute viscous fluid flow. A consensus on the cause of numerical problems has been reached; however, general algorithms---allowing a robust and accurate simulation for any…
To obtain the highest confidence on the correction of numerical simulation programs for the resolution of Partial Differential Equations (PDEs), one has to formalize the mathematical notions and results that allow to establish the soundness…
In this note, we present a new numerical method for solving backward stochastic differential equations. Our method can be viewed as an analogue of the classical finite element method solving deterministic partial differential equations.
Computational fluid dynamics (CFD) is a specialised branch of fluid mechanics that utilises numerical methods and algorithms to solve and analyze fluid-flow problems. One promising avenue to enhance CFD is the use of quantum computing,…
Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…
Fluid-solid reactions exist in many chemical and metallurgical process industries. Several models describe these reactions such as volume reaction model, grain model, random pore model and nucleation model. These models give two nonlinear…
In this work, we obtain the numerical temperature field to a thermally developing fluid flow inside parallel plates problem with a quantum computing method. The physical problem deals with the heat transfer of a steady state,…
We discuss possibilities of application of Numerical Analysis methods to proving computability, in the sense of the TTE approach, of solution operators of boundary-value problems for systems of PDEs. We prove computability of the solution…
In this paper we present numerical methods - finite differences and finite elements - for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is…