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We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion coefficients. Under certain type of Lipschitz and linear growth conditions, we develop a method to prove the existence and uniqueness of adapted solution to these B-SPDEs with jumps. Comparing with the existing discussions on conventional backward stochastic (ordinary) differential equations (BSDEs), we need to handle the differentiability of adapted triplet solution to the B-SPDEs with jumps, which is a subtle part in justifying our main results due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operator appeared in the diffusion coefficient. In addition, we also address the issue about the B-SPDEs under certain Markovian random environment and employ a B-SPDE with strongly nonlinear partial differential operator in the drift coefficient to illustrate the usage of our main results in finance.

The objective of this research was to develop a parallel computing version of Beam's object oriented C++ class library, PDESolve. The research carried out was the first year of a two-year research effort. In this contracting period the kernel of the PDESolve library was redesigned so as to enable efficient execution of parallel computing applications. This redesigned kernel is the core of a new release of the PDESolve library. The User's Guide for this latest release is enclosed as part of this report.

It has been previously shown that the temporal integration of hyperbolic partial differential equations may, because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic p.d.e's (linear and nonlinear) whose boundary treatment is done via the SAT-procedure. A methodology is present for recovery of the full order of accuracy, and has been applied to the case of a 4th order explicit finite difference scheme.

It has been previously shown that the temporal integration of hyperbolic partial differential equations may, because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic p.d.e's (linear and nonlinear) whose boundary treatment is done via the SAT-procedure. A methodology is present for recovery of the full order of accuracy, and has been applied to the case of a 4th order explicit finite difference scheme.

The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear ti surfaces. The spherical mean is the average of u over a constant ti surface. The conditions on the linear differential operator, L, and on the orthogonal coordinates (ti, eta, zeta) are established so that the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be determined directly as a problem with only space variable. Conditions are then established so that the spherical mean of the solution in one concial region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

The spherical means of the solutions of a linear partial differential equation Lu = f in a conical region are studied. The conical region is bounded by a surface generated by curvilinear ti surfaces. The spherical mean is the average of u over a constant ti surface. The conditions on the linear differential operator, L, and on the orthogonal coordinates (ti, eta, zeta) are established so that the spherical mean of the solution subjected to the appropriate boundary and initial conditions can be determined directly as a problem with only space variable. Conditions are then established so that the spherical mean of the solution in one concial region will be proportional to that of a known solution in another conical region. Applications to various problems of mathematical physics and their physical interpretations are presented.

Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and Techbychef orthogonal polynomials for spatiotemporal approximation of PDEs as a weighted sum of polynomials. We use collocation at some clustered grid points to generate a system of equations to approximate the weights for the polynomials. We finish the study by demonstrating approximate solutions of some PDEs in one space dimension.

Two- and three-dimensional surface normal grids are generated in Cartesian coordinates around a supersonic/hypersonic waverider configuration using parabolic partial differential equations. The elliptic partial differential equations for grid generation are parabolized in the xi direction in two dimesions, and in the xi and zeta directions in three dimensions. This is consistent with spatial marching flow solutions. The parabolized grid equations march in the xi direction for two dimensions and in both the xi and zeta directions for three dimensions, without iteration. The following problems are investigated: describing the boundary points, generating grids around the waverider's wing tip, using approximations to the elliptic grid generation equations too far downstream around a convex corner, and grid crossover in concave regions when orthogonality is specified. The degree of grid smoothing in the marching directions is related to the positioning of the approximations to the elliptic grid generation equations. Highly stretched surface orthogonal grids are accurately and efficiently generated without grid embedding for high Reynold's number flows. Keywords: Grids; Fluid mechanics; Navier Stokes equations; Three dimensional flow; Thesis.

This paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations (PDEs) in 1-D. The goal of the paper is to extend the successful framework of monotone, consistent, and stable finite difference methods for first order fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs such as Monge-Amp\`ere and Bellman type equations. New concepts of consistency, generalized monotonicity, and stability are introduced; among them, the generalized monotonicity and consistency, which are easier to verify in practice, are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear Hamilton-Jacobi equations. The main component of the proposed framework is the concept of "numerical operator", and the main idea used to design consistent, monotone and stable finite difference methods is the concept of "numerical moment". These two new concepts play the same roles as the "numerical Hamiltonian" and the "numerical viscosity" play in the finite difference framework for first order fully nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also are proved to be stable and the second class contains Godunov-like methods. Numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper.

This paper develops a new framework for designing and analyzing convergent finite difference methods for approximating both classical and viscosity solutions of second order fully nonlinear partial differential equations (PDEs) in 1-D. The goal of the paper is to extend the successful framework of monotone, consistent, and stable finite difference methods for first order fully nonlinear Hamilton-Jacobi equations to second order fully nonlinear PDEs such as Monge-Amp\`ere and Bellman type equations. New concepts of consistency, generalized monotonicity, and stability are introduced; among them, the generalized monotonicity and consistency, which are easier to verify in practice, are natural extensions of the corresponding notions of finite difference methods for first order fully nonlinear Hamilton-Jacobi equations. The main component of the proposed framework is the concept of "numerical operator", and the main idea used to design consistent, monotone and stable finite difference methods is the concept of "numerical moment". These two new concepts play the same roles as the "numerical Hamiltonian" and the "numerical viscosity" play in the finite difference framework for first order fully nonlinear Hamilton-Jacobi equations. In the paper, two classes of consistent and monotone finite difference methods are proposed for second order fully nonlinear PDEs. The first class contains Lax-Friedrichs-like methods which also are proved to be stable and the second class contains Godunov-like methods. Numerical results are also presented to gauge the performance of the proposed finite difference methods and to validate the theoretical results of the paper.

In this paper, we consider the asymptotic stability for equations via the Lyapunov-Razumikhin approach. For the case when Tau = 0, a Lyapunov function is constructed and conditions which reduce to the generalized Routh-Hurwitz criteria for uniform asymptotic stability are obtained. The constructed Lyapunov function is also converted to a Lyapunov functional. This functional is used to give necessary conditions on a, b and h for which equation (*) is asymptotically stable. (Author)

We construct the field A generated by n algebraically independent elements, and show that the linear space of derivations over this field is faithfully represented by the linear space of the n-th fold Cartesian product of this field acting through inner product on the gradient of this field. We prove also that functional independence of a set in this field is equivalent to linear independence of the gradient set in the space of Cartesian product. It is shown that every subfield S of A which is generated by (n-1) functionally independent elements defines an one-dimensional space of derivations, such that each member L of the latter subspace has S as its kernel. Each coset of the multiplicative subgroup S defines a non-homogeneous differential operator L+q whose kernel coincide with this coset. We prove also that every element of A defines a coset of the subgroup ker(L+q) in the additive group A, on which L+q is constant.

We construct the field A generated by n algebraically independent elements, and show that the linear space of derivations over this field is faithfully represented by the linear space of the n-th fold Cartesian product of this field acting through inner product on the gradient of this field. We prove also that functional independence of a set in this field is equivalent to linear independence of the gradient set in the space of Cartesian product. It is shown that every subfield S of A which is generated by (n-1) functionally independent elements defines an one-dimensional space of derivations, such that each member L of the latter subspace has S as its kernel. Each coset of the multiplicative subgroup S defines a non-homogeneous differential operator L+q whose kernel coincide with this coset. We prove also that every element of A defines a coset of the subgroup ker(L+q) in the additive group A, on which L+q is constant.

We develope a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $ L^p $-distance between the solution process of an SDE and an arbitrary It\^o process, which we view as a perturbation of the solution process of the SDE, by the $ L^q $-distances of the differences of the local characteristics for suitable $ p, q > 0 $. As application of our perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with non-globally monotone coefficients. As another application of our perturbation theory, we prove strong convergence rates for spectral Galerkin approximations of solutions of semilinear SPDEs with non-globally monotone nonlinearities including Cahn-Hilliard-Cook type equations and stochastic Burgers equations. Further applications of the perturbation theory include the regularity of solutions of SDEs with respect to the initial values and small-noise analysis for ordinary and partial differential equations.

We extend our discussion of effective actions for stochastic partial differential equations to systems that give rise to a Martin-Siggia-Rose (MSR) type of action. This type of action naturally arises when one uses the many-body formalism of Doi and Peliti to describe reaction-diffusion models which undergo transitions into the absorbing state and which are described by a Master equation. These models include predator prey models, and directed percolation models as well as chemical kinetic models. For classical dynamical systems with external noise it is always possible to construct an MSR action. Using a path integral representation for the generator of the correlation functions, we show how, by introducing a composite auxiliary field, one can generate an auxiliary field loop expansion for the effective action for both types of systems. As a specific example of the Doi-Peliti formalism we determine the effective action for the chemical reaction annihilation and diffusion process $A+A \rightarrow 0$. For the external noise problem we evaluate the effective action for the Cole-Hopf form of the Kardar-Parisi Zhang (KPZ) equation as well as for the Ginzburg Landau model of spin relaxation. We determine for arbitrary spatial dimension $d$, the renormalized effective potential in leading order in the auxiliary field loop expansion (LOAF) and also determine the renormalization group equation for the running of the reaction rate (coupling constant) for arbitrary $d$. We compare our results with known perturbative and non-perturbative results for the renormalization group equations.

Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be reduced to a normal type (total degree) algebraic system.

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \sigma (u_t(x))\dot F(t,x)\quad{t>0}\quad\text{and}\quad x\in R^d. \end{equation*} The operator $\mathcal{L}$ is the generator of a strictly stable process and $\dot F$ is the random forcing term which is assumed to be Gaussian. Under some additional conditions, most notably on $\sigma$ and the initial condition, we show non-existence of global random field solutions. Our results are new and complement earlier works.

A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains the Cauchy-Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.

A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains the Cauchy-Riemann equations and the cofactor pair systems, included as special cases. The multiplication provides a method for generating, in a pure algebraic way, large classes of non-trivial solutions that can be constructed by forming convergent power series of trivial solutions.

An introductory exposition is given covering the solution of some partial differential equations by means of the method of finite elements. Special attention is given to the means of getting numerical approximations to the answer. (Author)

Most of the methods discussed in this course: separation of variables, Fourier Series, Green�s functions (later) can only be applied to linear PDEs. However, the method of characteristics can be applied to a form of nonlinear PDE.

We now consider D�Alembert�s solution for a �nite string.

A second order finite difference method has been developed for elliptic interface problems that involve discontinuous coefficients, singular source terms, non-smooth or even discontinuous coefficients across an arbitrary interface. The method is based on Cartesian grids and coupled with a multigrid solver. The new method will preserve the discrete maximum principle. The convergence of the new method has been proved. Two different finite element spaces and corresponding Galerkin methods for elliptic interface problems using Cartesian grids have been developed. Some theoretical results about these finite element methods are also obtained. We believe that we are the first to develop these new finite element spaces over Cartesian grids. Some public subroutines of fast solvers for Helmholtz and Poisson equations on irregular domains either exterior or interior with various boundary conditions have been developed and made public In collaboration with J. K. Hunter of UC Davis and H. K. Zhao of UC Irvine, we formulate a model for the spreading on a surface of a drop that deposits an autophobic monolayer of surfactant We present numerical solutions of the model equations using an immersed interface method and a level set method.

The Dirichlet problem for the non-linear elliptic partial differential equation a(x,y,u(x,y))u, sub xx + c(x,y,u(x,y))u, sub yy - gamma(x, y,u(x,y))u = O is studied. It is assumed that the coefficients are strictly positive and Lipschitz in the argument u(x,y). It is then proved that the solution may be uniformly approximated by the solution to the associated difference equation provide ed that a certain inequality, relating bounds on the coefficients, is satisfied.

The Dirichlet problem for the non-linear elliptic partial differential equation a(x,y,u(x,y))u, sub xx + c(x,y,u(x,y))u, sub yy - gamma(x, y,u(x,y))u = O is studied. It is assumed that the coefficients are strictly positive and Lipschitz in the argument u(x,y). It is then proved that the solution may be uniformly approximated by the solution to the associated difference equation provide ed that a certain inequality, relating bounds on the coefficients, is satisfied.

"On a System of Linear Partial Differential Equations of the Hyperbolic Type" is an article from The Annals of Mathematics, Volume 20 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-1967120

The method of equivariant moving frames on multi-space is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finite-dimensional symmetry groups. Invariant numerical schemes for a heat equation with a logarithmic source and the spherical Burgers equation are obtained. Numerical tests show how invariant schemes can be more accurate than standard discretizations on uniform rectangular meshes.

This article introduces the splitting method to systems responding to rough paths as external stimuli. The focus is on nonlinear partial differential equations with rough noise but we also cover rough differential equations. Applications to stochastic partial differential equations arising in control theory and nonlinear filtering are given.

This article introduces the splitting method to systems responding to rough paths as external stimuli. The focus is on nonlinear partial differential equations with rough noise but we also cover rough differential equations. Applications to stochastic partial differential equations arising in control theory and nonlinear filtering are given.

Book digitized by Google from the library of the New York Public Library and uploaded to the Internet Archive by user tpb.

This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon a multi-modes representation of the solution as a power series of the perturbation parameter, and the Monte Carlo technique for sampling the probability space. One key feature of the proposed method is that the governing equations for all the expanded mode functions share the same deterministic diffusion coefficients, thus an efficient direct solver by repeated use of the $LU$ decomposition matrices can be employed for solving the finite element discretized linear systems. It is shown that the computational complexity of the whole algorithm is comparable to that of solving a few deterministic elliptic partial differential equations using the $LU$ director solver. Error estimates are derived for the method, and numerical experiments are provided to test the efficiency of the algorithm and validate the theoretical results.

We derive bounds for the minimum and maximum eigenvalues and the spectral condition number of matrices for isogeometric discretizations of elliptic partial differential equations in an open, bounded, simply connected Lipschitz domain $\Omega\subset \mathbb{R}^d$, $d\in\{2,3\}$. We consider refinements based on mesh size $h$ and polynomial degree $p$ with maximum regularity of spline basis functions. For the $h$-refinement, the condition number of the stiffness matrix is bounded above by a constant times $ h^{-2}$ and the condition number of the mass matrix is uniformly bounded. For the $p$-refinement, the condition number grows exponentially and is bounded above by $p^{2d+2}4^{pd}$ and $p^{2d}4^{pd}$ for the stiffness and mass matrices, respectively. Rigorous theoretical proofs of these estimates and supporting numerical results are provided.

This paper is an assessment of the current state of controllability and observability theories for linear partial differential equations, summarizing existing results and indicating open problems in the area. (Author)

We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on the compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropy flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multidimensional steady compressible fluids. We then analyze the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.

In this paper, we solve the initial value problems of variable-coefficient generalized wave equations associated with trees and a large family of linear constant-coefficient partial differential equation by algebraic methods. Moreover, we find all the polynomial solutions for a 3-dimensional variable-coefficient flag partial differential equation of any order, the linear wave equation with dissipation and the generalized anisymmetrical Laplace equation. Furthermore, the polynomial-trigonometric solutions of a generalized Klein-Gordan equation associated with 3-dimensional generalized Tricomi operator $\ptl_x^2+x\ptl_y^2+y\ptl_z^2$ are also given. As applications to representations of Lie algebras, we find certain irreducible polynomial representations of the Lie algebras $sl(n,\mbb{F}), so(n,\mbb{F})$ and the simple Lie algebra of type $G_2$.

In this paper, we investigate stochastic partial differential equations driven by multi-parameter anisotropic fractional Levy noises, including the stochastic Poisson equation, the linear heat equation, and the quasi-linear heat equation. Well-posedness of these equations under the fractional noises will be addressed. The multi-parameter anisotropic fractional Levy noise is defined as the formal derivative of the anisotropic fractional Levy random field. In doing so, there are two folds involved. First, we consider the anisotropic fractional Levy random field as the generalized functional of the path of the pure jump Levy process. Second, we build} the Skorohod integration with respect to the multi-parameter anisotropic fractional Levy noise by white noise approach.

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.

Families of energy operators and generalized energy operators have recently been introduced in the definition of the solutions of linear Partial Differential Equations (PDEs) with a particular application to the wave equation [Montillet, 2014, doi: 10.1007/s10440-014-9978-9]. To do so, the author has introduced the notion of energy spaces included in the Schwartz space $\mathbf{S}^-(\mathbb{R})$. In this model, the key is to look at which ones of these subspaces are reduced to {0} with the help of energy operators (and generalized energy operators). It leads to define additional solutions for a nominated PDE. Beyond that, this work intends to develop the concept of multiplicity of solutions for a linear PDE through the study of these energy spaces (i.e. emptiness). The main concept is that the PDE is viewed as a generator of solutions rather than the classical way of solving the given equation with a known form of the solutions together with boundary conditions. The theory is applied to the wave equation with the special case of the evanescent waves. The work ends with a discussion on another concept, the duplication of solutions and some applications in a closed cavity.

This document describes research activity supported under this grant from its inception (June, 1982) to the present. (KR)

This document describes research activity supported under this grant from its inception (June, 1982) to the present. (KR)

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.

A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and E. Vanden-Eijnden, Comm. Pure Appl. Math., 58(11):1544--1585, 2005]. The class of problems that we consider are SPDEs with quadratic nonlinearities that were studied in [D. Blomker, M. Hairer, and G.A. Pavliotis, Nonlinearity, 20(7):1721--1744, 2007.] For such SPDEs an amplitude equation which describes the effective dynamics at long time scales can be rigorously derived for both advective and diffusive time scales. Our method, based on micro and macro solvers, allows to capture numerically the amplitude equation accurately at a cost independent of the small scales in the problem. Numerical experiments illustrate the behavior of the proposed method.

In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A function that has a prescribed value on the domain in which a differential equation is valid and smoothly but rapidly varying values on the boundary where boundary conditions are imposed is used to modify the original differential equations. The mathematical derivations are straight forward, and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions, (2) the diffusion equation with surface diffusion, (3) the mechanical equilibrium equation, and (4) the equation for phase transformation with additional boundaries. The solutions for a few of these cases are validated against corresponding analytical and semi-analytical solutions. The potential of the approach is demonstrated with five applications: surface-reaction diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces, and a self-propelling droplet.

The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations implies the linearizability of systems of partial differential equations corresponding to those complex ordinary differential equations. The invertible complex transformations can be used to obtain invertible real transformations that map a system of nonlinear partial differential equations into a system of linear partial differential equation. Explicit invariant criteria are given that provide procedures for writing down the solutions of the linearized equations. A few non-trivial examples are mentioned.