General Partial Differential Equations Research Articles

Interest rate risk premium and equity valuation

The authors employ the recent stochastic-control-based approach to financial mathematics to solve a problem of determination of the risk premium for a stochastic interest rate model, and the corresponding problem of equity valuation. The risk premium is determined explicitly, by means of solving a corresponding partial differential equation (PDE), in two forms: one, time-dependent, corresponding to a finite time contract expiration, and the simpler version corresponding to perpetual contracts. As stocks are perpetual contracts, when solving the problem of equity valuation, the latter form of the risk premium is used. By means of solving the general pricing PDE, an efficient equity valuation method was developed that is a combination of some sophisticated explicit formulas, and a numerical procedure.

Meshless Method Based on Radial Basis Functions for Solving Parabolic Partial Differential Equations with Variable Coefficients

The method of approximate particular solutions is extended for solving initial-boundary-value problems for general parabolic partial differential equations (PDEs) with variable coefficients. The main idea is to reduce the parabolic PDEs into a series of elliptic PDEs and approximate the unknown solution by the closed-form particular solution using radial basis functions. Numerical experiments in two and three dimensions show that the proposed scheme is accurate and easy to implement.

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.

A Variational Method and Approximations of a Cauchy Problem for Elliptic Equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H1/2(Γ1 U Γ3) is assumed available on a larger part of the boundary Γ of the bounded domain Ω than the boundary portion Γ1 on which the Neumann data is is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps

In this work, we investigate stochastic partial differential equations with variable delays and jumps. We derive by estimating the coefficients functions in the stochastic energy equality some sufficient conditions for exponential stability and almost sure exponential stability of energy solutions, and generalize the results obtained by Taniguchi [T. Taniguchi, The exponential stability for stochastic delay partial differential equations, J. Math. Anal. Appl. 331 (2007) 191–205] and Wan and Duan [L. Wan, J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist. Probab. Lett. 78 (5) (2008) 490–498] to cover a class of more general stochastic partial differential equations with jumps. Finally, an illustrative example is established to demonstrate our established theory.

Mixed boundary value problems for higher-order complex partial differential equations

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

Flux Recovery and A Posteriori Error Estimators: Conforming Elements for Scalar Elliptic Equations

In this paper, we first study two flux recovery procedures for the conforming finite element approximation to general second-order elliptic partial differential equations. One is accurate in a weighted $L^2$ norm studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132-2156] for linear elements, and the other is accurate in a weighted $H(\mathrm{div})$ norm, up to the accuracy of the current finite element approximation. For the $L^2$ recovered flux, we introduce and analyze an a posteriori error estimator that is more accurate than the explicit residual-based estimator. Based on the $H(\mathrm{div})$ recovered flux, we introduce two a posteriori error estimators. One estimator may be regarded as an extension of the recovery-based estimator studied in [Z. Cai and S. Zhang, SIAM J. Numer. Anal., 47 (2009), pp. 2132-2156] to higher-order conforming elements. The global reliability and the local efficiency bounds for this estimator are established provided that the underlying problem is neither convection- nor reaction-dominant. The other is proved to be exact locally and globally on any given mesh with no regularity assumptions with respect to a norm depending on the underlying problem. Numerical results on test problems for these estimators are also presented.

Gevrey Order of Formal Power Series Solutions of Inhomogeneous Partial Differential Equations with Constant Coefficients

In an earlier paper, the first author showed that certain normalized formal solutions of homogeneous linear partial differential equations with constant coefficients are multisummable, with a multisummability type that can be determined from a Newton polygon associated with the PDE. In this article, some of the results obtained there are extended in several directions: First of all, arbitrary formal solutions of inhomogeous PDE are considered, and it is shown that, in some sense, they can be computed completely explicitly. Secondly, the Gevrey order of these formal solutions is determined. Finally, formal power series are discussed that, in general, do not satisfy a PDE with constant coefficients, but instead may be considered as solutions of singularly perturbed ODE, or integro-differential equations of a certain form. Introduction In [3, 6], the first author introduced and studied normalized formal solutions of a Cauchy problem for general homogeneous linear partial differential equations in two variables having constant coefficients. Multisummability of these formal power series was then investigated in [4]. In detail, it has been shown that, under the assumption that the initial condition used is holomorphic near the origin, one can determine a multisummability type corresponding naturally to the PDE under consideration. The normalized formal solution then is multisummable in a given multidirection, provided that the initial condition can be continued into finitely many (small) sectors, and in every such sector is at most of a certain exponential growth that, in general, depends upon the sector. The multisummability type, the location of the sectors, and the corresponding ∗Institut fur Angewandte Analysis, Universitat Ulm, D–89069 Ulm, Germany †Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima 739-8526 Japan

Ohmic heating of dairy fluids—effects of local electric field on temperature distribution

AbstractThis paper presents the heat transfer model of a continuous flow ohmic heating process. The model fluid used was a mixture of reconstituted skimmed milk and whey protein concentrate solution. Two‐dimensional numerical simulations of an annular ohmic heater were performed using a general purpose partial differential equation solver, FlexPDE. The momentum, energy, and electrical equations were solved for a laminar flow regime. Two models were used to determine the volumetric heating rate, one taking into account the local electric field by solving the Laplace equation while another model assumes an average voltage gradient applied between the two electrodes. Results show that while the wall temperature distribution is different for the two cases, the bulk fluid temperature and the average outlet temperature are the same. The predicted temperatures generally agree well with the measured temperatures. Copyright © 2009 Curtin University of Technology and John Wiley & Sons, Ltd.

Quadratic spline collocation for one-dimensional linear parabolic partial differential equations

New methods for solving general linear parabolic partial differential equations (PDEs) in one space dimension are developed. The methods combine quadratic-spline collocation for the space discretization and classical finite differences, such as Crank-Nicolson, for the time discretization. The main computational requirements of the most efficient method are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth order. The stability and convergence properties of some of the new methods are analyzed for a model problem. Numerical results demonstrate the stability and accuracy of the methods. Adaptive mesh techniques are introduced in the space dimension, and the resulting method is applied to the American put option pricing problem, giving very competitive results.

Analysis of a transiently propagating crack in functionally graded materials under mode I and II

Crack tip stress and displacement fields for a transiently propagating crack along gradient in functionally graded materials (FGMs) with a linear variation of shear modulus are developed. The higher order terms of the transient stress and displacement fields at crack tip were obtained by transforming the general partial differential equations of the dynamic equilibrium into Laplace’s equations whose solutions have harmonic functions. Thus, the fields can be expressed very simply. Using these stress components, isochromatics and the first invariant at crack tip are generated. The results show that the isochromatics (constant maximum shear stress) for mode I crack tilt backward around the crack tip with an increase of crack tip acceleration c ˙ ( d c / d t ) , and tilt forward around the crack tip with an increase of rate of change of dynamic mode I stress intensity factor K ˙ I ( d K I / d t ) . The isochromatics for mixed mode crack move to upper direction with an increases of K ˙ I and K ˙ II , and lower direction with an increase of c ˙ . Contours of the first stress invariant for mode I crack enlarge around the crack tip with an increase of c ˙ , and decrease around the crack tip with an increase of K ˙ I . As K ˙ I ( II ) decreases at crack initiation, the predicted kinking angles increase. As c ˙ increases, the predicted kinking angles also increase.

Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps

The existence, uniqueness and some sufficient conditions for stability in distribution of mild solutions to stochastic partial differential delay equations with jumps are presented. The principle technique of our investigation is to construct a proper approximating strong solution system and carry out a limiting type of argument to pass on stability of strong solutions to mild ones. As a consequence, stability results of Basak et al . ( Basak et al . 1999 J. Math. Anal. Appl. 202 , 604–622) and Yuan et al . ( Yuan et al . 2003 Syst. Control Lett. 50 , 195–207) are generalized to cover a class of much more general stochastic partial differential delay equations with jumps in infinite dimensions. In contrast to the almost sure exponential stability in Ichikawa ( Ichikawa 1982 J. Math. Anal. Appl. 90 , 12–44) and Luo & Liu ( Luo & Liu 2008 Stoch. Proc. Appl. 118 , 864–895) and the moment exponential stability in Luo & Liu, we present a new result on the stability in distribution of mild solutions. Finally, an example is given to demonstrate the applicability of our work.

Mixed finite element methods for geometric modeling using general fourth order geometric flows

A variational formulation of a general form fourth order geometric partial differential equation is derived, and based on which three versions of the mixed finite element method are developed. Several surface modeling problems, including surface denoising, surface blending, hole filling and surface mesh refinement with the G 1 continuity, are taken into account. The used geometric partial differential equation is universal, containing several well-known geometric partial differential equations as its special cases. The proposed method is general which can be used to construct surfaces for geometric design as well as simulate the behaviors of various geometric partial differential equations. Experimental results show that the method is efficient and gives very desirable results.

A fast variational method for the construction of resolution adaptive [formula omitted]-smooth molecular surfaces

We present a variational approach to smooth molecular (proteins, nucleic acids) surface constructions, starting from atomic coordinates, as available from the protein and nucleic acid data banks. Molecular dynamics (MD) simulations traditionally used in understanding protein and nucleic acid folding processes, are based on molecular force fields, and require smooth models of these molecular surfaces. To accelerate MD simulations, a popular methodology is to employ coarse grained molecular models, which represent clusters of atoms with similar physical properties by psuedo-atoms, resulting in coarser resolution molecular surfaces. We consider generation of these mixed-resolution or adaptive molecular surfaces. Our approach starts from deriving a general form second order geometric partial differential equation in the level-set formulation, by minimizing a first order energy functional which additionally includes a regularization term to minimize the occurrence of chemically infeasible molecular surface pockets or tunnel-like artifacts. To achieve even higher computational efficiency, a fast cubic B-spline C 2 interpolation algorithm is also utilized. A narrow band, tri-cubic B-spline level-set method is then used to provide C 2 smooth and resolution adaptive molecular surfaces.

A note on solutions of wave, Laplace’s and heat equations with convolution terms by using a double Laplace transform

In this study we consider general linear second-order partial differential equations and we solve three fundamental equations by replacing the non-homogeneous terms with double convolution functions and data by a single convolution.

A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal $(k+1)$-th order of accuracy when using piecewise $k$-th degree polynomials, under the condition that $k+1$ is greater than or equal to the order of the equation.

Existence and Smoothness of the Density for Spatially Homogeneous SPDEs

In this paper, we extend Walsh’s stochastic integral with respect to a Gaussian noise, white in time and with some homogeneous spatial correlation, in order to be able to integrate some random measure-valued processes. This extension turns out to be equivalent to Dalang’s one. Then we study existence and regularity of the density of the probability law for the real-valued mild solution to a general second order stochastic partial differential equation driven by such a noise. For this, we apply the techniques of the Malliavin calculus. Our results apply to the case of the stochastic heat equation in any space dimension and the stochastic wave equation in space dimension d=1,2,3. Moreover, for these particular examples, known results in the literature have been improved.

Domains in infinite jet spaces: -spectral sequences

Domains in infinite jets present the simplest class of diffieties with boundary. In this note some basic elements of geometry of these domains are introduced and an analogue of the C-spectral sequence in this context is studied. This, in particular, allows cohomological interpretation and analysis of initial data, boundary conditions, etc, for general partial differential equations and of transversality conditions in calculus of variations. This kind applications and extensions to arbitrary diffieties will be considered in subsequent publications.

Historical development of soil-water physics and solute transport in porous media

The science of soil-water physics and contaminant transport in porous media began a little more than a century ago. The first equation to quantify the flow of water is attributed to Darcy. The next major development for unsaturated media was made by Buckingham in 1907. Buckingham quantified the energy state of soil water based on the thermodynamic potential energy. Buckingham then introduced the concept of unsaturated hydraulic conductivity, a function of water content. The water flux as the product of the unsaturated hydraulic conductivity and the total potential gradient has become the accepted Buckingham-Darcy law. Two decades later, Richards applied the continuity equation to Buckingham's equation and obtained a general partial differential equation describing water flow in unsaturated soils. For combined water and solute transport, it had been recognized since the latter half of the 19th century that salts and water do not move uniformly. It wasn't until the middle of the 20th century that scientists began to understand the complex processes of diffusion, dispersion, and convection and to develop mathematical formulations for solute transport. Knowledge on water flow and solute transport processes has expanded greatly since the early part of the 20th century to the present.

A multiscale stochastic finite element method on elliptic problems involving uncertainties

In this study a multiscale stochastic finite element method (MsSFEM) is developed to resolve scale-coupling stochastic elliptic problems, based on formulations of a stochastic variational approach and scale-bridging multiscale shape functions. By employing polynomial chaos expansions and Lagrange polynomials, stochastic Galerkin method and stochastic collocation method are introduced and combined with deterministic multiscale methods, i.e. Variational Multiscale Method and Multiscale Finite Element Method. The resulting spectral and pseudo-spectral stochastic finite element methods are incorporated into the fine- and coarse-mesh computation of the MsSFEM. A benchmark multiscale model and the numerical experiment based on the formulated MsSFEM are provided for illustration. It is expected that the proposed MsSFEM can act as a paradigm for solving of general stochastic partial differential equations involving multiscale stochastic data.