Order Linear Partial Differential Equations Research Articles

The structure of pan-addition operator with pre-determined pan-multiplication

Based on the structure of general pan-multiplication operators and the theory of first order linear partial differential equations, we study the structure of pan-addition operators with pre-determined pan-multiplication operators. At first, we describe the structure of pan-addition operators under the ordinary multiplication. Then, with this structure in place, we derive the general representation for the structure of pan-addition operators.

Symmetry operators for Riemann’s method

Riemann’s method is one of the definitive ways of solving Cauchy’s problem for a second order linear hyperbolic partial differential equation in 2 variables. Chaundy’s equation, with 4 parameters, is the most general self-adjoint equation for which the Riemann function is known. Here we show that Chaundy’s equation possesses a two-dimensional vector space of second-order symmetry operators. Hence a new equivalence class of Riemann functions, admitting no first-order symmetries and obtainable only via a higher order symmetry, is found. A new 5 parameter Riemann function is then subsequently derived.

Taylor polynomial solutions of second order linear partial differential equations

The purpose of this study is to give a Taylor polynomial approximation for the solution of second order linear partial differential equations with two variables and variable coefficients. For this purpose, Taylor matrix method for the approximate solution of second order linear partial differential equations with specified associated conditions in terms of Taylor polynomials about any point. The method is based on, first, taking truncated Taylor series of the functions in equation and then substituting their matrix forms into the given equation. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Hence, the result matrix equation can be solved and the unknown Taylor coefficients can be found approximately.

Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —

This paper is concerned with the existence of the Gevrey asymptotic solutions for the divergent formal solution of singular first order linear partial differential equations of nilpotent type. By using the Gevrey version of Borel-Ritt's theorem, we can prove the existence of asymptotic solutions in a small sector unconditionally. However, when we require the Borel summability of the formal solution (that is, the existence of asymptotic solutions in an open disk), global analytic continuation properties for coefficients are demanded.

REGULARITY RESULTS FOR LINEAR ELLIPTIC PROBLEMS RELATED TO THE PRIMITIVE EQUATIONS

The authors study the regularity of solutions of the GFD-Stokes problem and of some second order linear elliptic partial differential equations related to the Primitive Equations of the ocean. The present work generalizes the regularity results in [18] by taking into consideration the non-homogeneous boundary conditions and the dependence of solutions on the thickness e of the domain occupied by the ocean and its varying bottom topography. These regularity results are important tools in the study of the PEs (see e.g. [6]), and they seem also to possess their own interest.

Gevrey Asymptotic Theory for Singular First Order Linear Partial Differential Equations of Nilpotent Type — Part II Klein-Gordon equations is studied in the Sobolev space Hs = Hs(

Gevrey Asymptotic Theory for Singular First Order Linear Partial Differential Equations of Nilpotent Type — Part II Klein-Gordon equations is studied in the Sobolev space Hs = Hs(

An explicit integration scheme based on recursion for the curved triangular finite elements

This paper is concerned with explicit formulae for computing integrals of rational functions of bivariate polynomial numerator with linear denominator over a unit triangle {( ξ, η)|0⩽ ξ, η⩽1, ξ+ η⩽1} in the local parametric space ( ξ, η). These integrals arise in a variety of applications governing the second order linear partial differential equations. Explicit evaluation of these integrals produce algebraic finite element relations, provided that the original element geometry is restricted to a triangle with two straight sides and one curved side. For a rational integral of nth degree bivariate polynomial numerator with a linear denominator there are {( n+1)( n+2)/2} rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these {( n+1)( n+2)/2} integrals can be computed in two ways. We have presented a recursive scheme to compute { n( n+1)/2} such integrals efficiently whenever ( n+1) integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulae. The analytical quadrature formulae from zeroth to sextic order due to different element geometry are, for clarity and reference, summarized in tabular forms. An integration formula is also presented for the evaluation of integrals that occur in the calculation of stiffness matrices when straight sided triangular finite elements are considered. Finally we have considered two application examples: First example is to compute the Prandtl stress function values and the torsional constant k for an elliptic cross-section with varying ratios of major and minor distances. The other example is concerned with numerical solution for the Poisson equation in 2D. It is observed in the calculation of components of element matrices by using the Gauss quadrature rule (e.g. 7-point and 13-point) a remarkable discrepancy occurs if the element geometry contains a concave curved side. A computer code based on the present integration scheme to obtain the element stiffness matrices for both the curved and straight sided elements is appended.

Mathematics of Wave Propagation

9R3. Mathematics of Wave Propagation. - JL Davis (consultant). Princeton UP, Princeton. 2000. 395 pp. ISBN 0-691-02643-2. $49.50.Reviewed by LN Sankar (Sch of Aerospace Eng, Georgia Inst of Tech, Atlanta GA 30332-0150).There are a number of textbooks and references that have been published on wave propagation in fluids. One may also find numerous books that individually address the propagation of waves in solids, surface waves in water, and tidal waves. There are very few books that treat all of these related phenomena in a clear, unified manner. This book does this challenging job admirably well. The first chapter deals with the physics of propagating waves. Many classical problems such as the transverse oscillations of a string, acoustic waves in a duct, and compression waves in a bar are discussed. Topics such as the Doppler effect, dispersion, and group velocity are also covered. The mathematical aspects of these problems are not emphasized in this chapter. The second chapter deals with the theory behind hyperbolic partial differential equations. The method of characteristics is discussed and illustrated with increasingly complex model problems: 1D advection equation, 2D wave equation, and a system of first order linear and nonlinear partial differential equations, etc. Chapter 3 deals with how the partial differential equations developed in Chapter 2 may be solved. Classical mathematical techniques such as the separation of variables and Laplace transforms, as well as numerical methods that solve ODEs or algebraic equations along characteristic lines are covered. Numerous applications ranging from the vibration of a rectangular membrane to the propagation of current in an LC circuit are discussed. Having established the common mathematical foundation and the solution techniques for these problems, this book moves on to individual topics: wave propagation in viscous and inviscid fluids, elastic, viscoelastic and thermoelastic solids, and water waves. Each of these chapters may be read individually without reference to others. All the governing equations are derived from first principles, making it easier for the reader to understand and appreciate the physics behind these problems. This book ends with variational calculus based approaches (eg, Hamilton’s variational principle, Hamilton-Jacobi theory) to deriving the governing equations. Asymptotic approaches that rely on expansion of the solution for large wave numbers or frequency are also briefly discussed. There are numerous worked-out examples throughout the book. There are also exercises at the end of most of the chapters. These features make this work a suitable textbook for a senior-level or a graduate-level course on the theory of waves. In summary, Mathematics of Wave Propagation is an excellent book that covers seemingly diverse wave phenomena in a unified, coherent manner. Students and practicing engineers and physicists will find this book a useful addition to their collections.

Stabilized continuation methods for boundary value problems

Inspired by the classical control theory for linear dynamical systems, we present an analytical continuation algorithm with inherent error stabilization. Numerical error is stabilized quite the same way as controlling the tracking error dynamics in system theory. Simulations demonstrate the efficacy of the proposed method wherein the stabilized continuation method always converges to the unique solution (if it exists) or any feasible solution (in the case of non-unique solutions). Further, we find new applications for this method in solving second order linear and non-linear elliptic partial differential equations with linear and non-linear boundary conditions. Illustrative numerical examples deal with stiff non-linear systems arising out of various physical applications.

A-Quasiconvexity: Relaxation and Homogenization

Integral representation of relaxed energies and of Γ-limits of functionals (u,v)\mapsto \int_\Omega f( x,u(x),v(x))\,dx are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell's equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W1,p , are recovered.

Synthetic division based integration of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0⩽ ξ, η⩽1, ξ+ η⩽1} in the local parametric space ( ξ, η)

The domain of real problems in mechanics often contains curved boundaries. Curved boundaries are often more accurately modelled by curved finite elements than by straight edged elements, as straight sides are perfectly satisfactory if the domain has a polygonal boundary. Because fewer curved elements are required, the effort needed to obtain a solution is usually reduced. If some parts of the boundary are curved, however, elements with at least one curved side are desirable. Our aim in this paper is to consider the triangular element with two straight sides and one curved side. This paper is concerned with explicit formulae for evaluating integrals of rational functions of bivariate polynomial numerators with linear denominators over a unit triangle {0⩽ξ, η⩽1, ξ+η⩽1} in the local parametric two dimensional space ( ξ, η). These integrals arise in finite element formulations of second order linear partial differential equations by use of triangular element with two straight sides and one curved side of quadratic variation which often require relatively large numerical effort to integrate. The curved elements considered here are the four node, six node and ten node triangular elements with one curved side of quadratic variation and the other two sides have straight edges under the isoparametric and subparametric transformations, respectively. We have shown that by use of a method similar to synthetic division, the rational integrals of nth order bivariate polynomial numerator with a linear denominator having ( n+1)( n+2)/2 integrals can be reduced to rational integrals of nth order polynomial numerator in one variate with the same linear denominator having ( n+1) integrals and a simple integral of bivariate polynomial expression containing n( n+1)/2 terms (which is free from denominator), and this amounts to a substantial reduction in the numerical effort for such integrals. The explicit analytical integration formulae (obtained) up to sextic polynomial numerator in bivariates ξ, η due to different element geometry are, for clarity and reference, summarised in tables. Finally three application examples are also considered. For the first example we have used integration formulae derived in all theorems of this paper and explained the detailed computational scheme. For the other two examples computational scheme follows in a similar way. It is observed in the solution of all problems that the displacements and torsional constants are satisfactory when components of all element matrices are calculated by analytical integration formulae derived in this paper. It is also observed that in the calculation of components of element matrices by using numerical integration formulae (e.g., 7-point and 13-point rule) much discrepancy occurs if the element geometry is coarse with a concave curve side. But it is found that the analytical integration technique is always consistent. Therefore the symbolic integration formulae presented in this paper are reliable and may lead to an easy and systematic incorporation of element matrices required in the finite element solution procedure.

Divergence Property of Formal Solutions for Singular First Order Linear Partial Differential Equations

This paper is concerned with the study of the convergence and the divergence of formal power series solutions of the following first order singular linear partial differential equation with holomorphic coefficients at the origin: P(x,D)u(x) = \sum^d_{\iota=1}a_\iota(x)D_\iota u(x)+b(x)u(x) = f(x), with f(x) holomorphic at the origin. Here the equation is said to be singular if a_\iota(0)=0 ( \iota = 1, ...,d ). In this case, it is known that under the so-called Poincare condition, if \{a_\iota(x)\}^d_{\iota=1 } generates a simple ideal, every formal solution is convergent. However if we remove these conditions, we shall see that the formal solution, if it exists, may be divergent. More precisely, we will characterize the rate of divergence of formal solutions via Gevrey order of formal solutions determined by a Newton Polyhedron, a generalization of Newton Polygon which is familiar in the study of ordinary differential equations with an irregular singular point.

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.

Periodic systems for the higher-dimensional Laplace transformation

We consider overdetermined systems of linear partial differentialequations of the form $y_{,k\l}+a_{k\l}^ky_{,k}+a_{k\l}^\ly_{,\l}+c_{k\l}y=0\ ,\quad 1\le k\ne\l\le n\ , $ where the coefficients are smooth functions satisfying certainintegrability conditions. Generalizing the classical theory ofsecond order linear hyperbolic partial differential equation in theplane, we consider higher-dimensional Laplace invariants of a systemof the above class. These invariants are characterized as functionswhich must satisfy a set of differential equations. We establish anormal form for any system of the above class in terms of theseinvariants. Moreover, we solve the periodicity problem for thehigher-dimensional Laplace transformation applied to such systems,generalizing a classical theorem of Darboux which shows that for$n=2$ a 1-periodic equation is equivalent to the Klein-Gordonequation.

The Mizohata complex

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ ω \omega at 0 ∈ R n + 1 0\in \mathbb {R}^{n+1} of a C \mathbb {C} -valued, formally integrable ( ω ∧ d ω = 0 \omega \wedge d\omega =0 ), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for ( q − 1 ) (q-1) -forms u u , of the equation d u ∧ ω = η ∧ ω , \begin{equation*}du\wedge \omega =\eta \wedge \omega ,\end{equation*} where η \eta is a given q q -form satisfying d η ∧ ω = 0 d\eta \wedge \omega =0 is estimated in terms of the De Rham cohomology relative to ω \omega

A boundary element method for a second order elliptic partial differential equation with variable coefficients

A boundary element method is derived for solving a class of boundary value problems governed by an elliptic second order linear partial differential equation with variable coefficients. Numerical results are given for a specific test problem.

A class of stable difference schemes for linear elliptic PDEs and their asynchronous parallel computation

This paper improves and generalizes the two difference schemes presented in paper [1] and gives a new difference scheme for second order linear elliptic partial differential equations, its difference matrix is a M-matrix and because of the stability of the M-matrix, it is convergent by the asynchronous iterative method on multiprocessors. Then this paper gives a class of difference schemes for linear elliptic PDEs so that their difference matrixes are all M-matrixes and their asynchronous parallel computation are convergent.

On the Iterative Solution of Hermite Collocation Equations

Collocation methods based on bicubic Hermite piecewise polynomials have been proven to be effective techniques for solving general second order linear elliptic partial differential equations (PDEs) with mixed boundary conditions [ACM Trans. Math. Software, 11 (1985), pp. 379–412]. The corresponding system of discrete collocation equations is generally nonsymmetric and nondiagonally dominant. Methods for their iterative solution are not known and are currently solved using Gauss elimination with scaling and partial pivoting. Point iterative methods like those in ITPACK [Tech. Report CNA-216, Center for Numerical Analysis, Univ. of Texas at Austin, April 1988] do not converge even for the collocation equations obtained from the discretization of model PDE problems. The development of efficient iterative solvers for these collocation equations is necessary for the case of three-dimensional PDE problems and their parallel solution, since direct solvers tend to be space bound and their parallelization is difficult. In this paper block iterative methods are developed and analyzed for the collocation equations corresponding to elliptic PDEs defined on a rectangle and subject to uncoupled mixed boundary conditions. For these types of PDE problems certain boundary degrees of freedom of the collocation approximation can be predetermined symbolically [Houstis, Mitchell, and Rice]. The remaining equations are called “interior” collocation equations. The system of all discrete equations is referred to as “general” collocation equations. Papatheodorou [Math. Comp., 41 (1983), pp. 511-525] was first to determine the exact parameters of accelerated overrelaxation (AOR)-type iterative methods for the case of “interior” collocation equations associated with a model problem. This paper generalizes the results of Papatheodorou for the “interior” collocation equations and presents new results for a particular class of “general” collocation equations. Specifically, in the case of a model elliptic PDE problem with uncoupled mixed boundary conditions, analytic expressions are derived for the eigenvalues of the block Jacobi iteration matrix based on a new partitioning of the interior collocation matrix, and the optimal overrelaxation factors are determined for the block successive overrelaxation (SOR) iterative method. A number of numerical results are presented to verify the theoretical analysis of the block SOR method and to compare its convergence behavior with those of the block Jacobi, Gauss–Seidel and the optimal AOR of Papatheodorou. Furthermore, the authors compare the time and memory complexity of the block SOR, UNPACK Band GE, and generalized minimal residual (GMRES) mathematical software for solving the Hermite collocation equations obtained from the discretization of several PDE problems. The numerical results indicate that the block SOR is an efficient method for solving these equations.

Fluiddynamik in der Cigarette und deren Einfluß auf den Zugwiderstand

Abstract The objective of the theoretical study is to present a model that simulates the instationary flow characteristics within cigarettes which are adjusted to a routine smoking machine. By assuming laminar fluid flow the well knownKozeny-Carman equation is used to evaluate the flow impedance of cigarettes. The model parameters are determined by five variations of routine pressure-drop measurements yielding the entire set of flow impedances, which are necessary for describing a steady state Fluid-Network. Using this relation further elements of the fluid dynamics, e.g. the Fluid-Inductor, a storage element of kinetic energy (fluid inertor), and the Fluid-Capacitor, a storage element of potential energy (fluid volume), are evaluated. In addition, the electrical and fluid dynamical analogies are demonstrated. The model uses a set of distributed parameters including pneumatic impedances, capacitors and inductors, to represent filtered cigarettes and a second order linear partial differential equation to describe the fluid flow within the cigarettes. A simple presentation of the Fluid-Network with lumped parameters is analysed and its significant time constants are determined. A characteristical value for the time constant of the cigarette fluid-inertor is given by TL ≈ 0.02 ms and depending on the fluid volume of cigarette their dominant time parameter is given by TC ≈ 2 ms. Finally the transient fluid flow within cigarettes in a conventional smoking machine is considered including a simulation of their fluid dynamics. A simple estimation gives the time response of the system (consisting of cigarettes and routine smoking machine) with TcR ≈ 70 ms as a product of dead and sweep volume times the pressure drop of unencapsulated cigarettes. Variation of parameters caused by cigarette pressure drops and dead volumes of the smoking machine are simulated and discussed.

Transformation of nonlinear systems in observer canonical form with reduced dependency on derivatives of the input

The transformation of nonlinear multi-input-multi-output systems x ̇ = f(x, u) , y = h(x, u) into an observer canonical form with reduced dependency on derivatives of the input is studied. Necessary and sufficient conditions for its existence and a straightforward algorithm for obtaining the canonical model are derived. The proposed method involves the solution of a nonlinear algebraic equation system and systems of first order linear partial differential equations. The nonlinear canonical form obtained permits global observer error linearization and it is a stage in the design of nonlinear observers. The method is illustrated by an example.