Universal zero solutions of linear partial differential operators

We use transfinite induction to show that if L is an epimorphism of a vector space V and maps a vector subspace W of V into a proper subspace of itself, then there is a smallest subspace E of V containing W such that L(E) E (or a minimal space of surjectivity or solvability) and we give examples where there are infinitely many distinct minimal spaces of solvability. We produce an example showing that if L 1 and L 2 are two epimorphisms of a vector space V which are endomorphisms of a proper subspace W of V such that LI(W) n L2(W is a proper subspace of W, then there may not exist a smallest subspace E of V containing W such that LI(E) E L2(E). While no nonconstant linear partial differential operator maps the field of meromorphic functions onto itself, we construct a locally convex topological vector space of formal power series containing the meromorphic functions such that every linear partial differential operator with constant coefficients maps this space linearly and continuously onto itself. Furthermore, we show that algebraically there is for every linear partial differential operator P(D) with constant coefficients a smallest extension E of the meromorphic functions in n complex variables, where

We study the case when a bivariate Linear Partial Differential Operator (LPDO) of orders three or four has several different factorizations. We prove that a third-order bivariate LPDO has a first-order left and right factors such that their symbols are co-prime if and only if the operator has a factorization into three factors, the left one of which is exactly the initial left factor and the right one is exactly the initial right factor. We show that the condition that the symbols of the initial left and right factors are co-prime is essential, and that the analogous statement "as it is" is not true for LPDOs of order four. Then we consider completely reducible LPDOs, which are defined as an intersection of principal ideals. Such operators may also be required to have several different factorizations. Considering all possible cases, we ruled out some of them from the consideration due to the first result of the paper. The explicit formulae for the sufficient conditions for the complete reducibility of an LPDO were found also.

A test in terms of invariants for the existence of a factorization of a bivariate, non-hyperbolic third-order Linear Partial Differential Operator (LPDO) which has a given factorization of its principal symbol is found. The invariants that are used are with respect to known gauge transformations, which is together with constructive factorization itself are essentially involved in modern exact integration algorithms. The invariants, and even a generating system of those were found in previous paper using Moving Frames methods. In order to find the expressions in terms of invariants that guarantee the existence of a factorization of a certain type, we show that the operation of taking the formal adjoint can be also defined in terms of invariants, that is for equivalence classes of LPDOs, and explicit formulae defining this operation in the space invariants are obtained. The operation of formal adjoint is highly interesting for factorization of LPDOs for if the initial operator has a factorization, its adjoint has also one, and they are related. (informally, the factorization types are symmetric).

Using a new definition of the generalized factorization of linear partial differential operators, we discuss possible generalizations of the Darboux integrability of nonlinear partial differential equations.

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

It has been shown that, under suitable hypotheses, boundary value problems of the form, $Ly+\lambda y=f,$ $BC y =0$ where $L$ is a linear ordinary or partial differential operator and $BC$ denotes a linear boundary operator, then there exists $\Lambda >0$ such that $f\ge 0$ implies $\lambda y \ge 0$ for $\lambda\in [-\Lambda ,\Lambda ]\setminus\{0\},$ where $y$ is the unique solution of $Ly+\lambda y=f,$ $BC y =0$. So, the boundary value problem satisfies a maximum principle for $\lambda\in [-\Lambda ,0)$ and the boundary value problem satisfies an anti-maximum principle for $\lambda\in (0, \Lambda ]$. In an abstract result, we shall provide suitable hypotheses such that boundary value problems of the form, $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y=f,$ $BC y =0$ where $D_{0}^{\alpha}$ is a Riemann-Liouville fractional differentiable operator of order $\alpha$, $1<\alpha \le 2$, and $BC$ denotes a linear boundary operator, then there exists $\mathcal{B} >0$ such that $f\ge 0$ implies $\beta D_{0}^{\alpha -1}y \ge 0$ for $\beta \in [-\mathcal{B} ,\mathcal{B} ]\setminus\{0\},$ where $y$ is the unique solution of $D_{0}^{\alpha}y+\beta D_{0}^{\alpha -1}y =f,$ $BC y =0$. Two examples are provided in which the hypotheses of the abstract theorem are satisfied to obtain the sign property of $\beta D_{0}^{\alpha -1}y.$ The boundary conditions are chosen so that with further analysis a sign property of $\beta y$ is also obtained. One application of monotone methods is developed to illustrate the utility of the abstract result.

We consider the Fuchsian linear differential equation obtained (modulo a prime) for , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of and can be removed from and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth-order linear differential operator occurs as the left-most factor of the ‘depleted’ differential operator and it is shown to be equivalent to the symmetric fourth power of LE, the linear differential operator corresponding to the elliptic integral E. This result generalizes what we have found for the lower order terms and . We conjecture that a linear differential operator equivalent to a symmetric (n − 1) th power of LE occurs as a left-most factor in the minimal order linear differential operators for all 's.

Differential operators with constant coefficients Gevrey pseudo-differential operators of infinite order canonical transformations and classical analytic Fourier integral operators propagation of Gevrey singularities Gevrey hypoellipticity the Cauchy problem in the Gevrey classes local solvability in Gevrey classes.

On the global solvability in Gevrey classes on the n-dimensional torus

Let P(z, ∂) be a linear partial differential operator with holomorphic coefficients in a neighborhood Ω of z = 0 in ℂ^{d+1} . Consider the equation P(z, ∂)u(z) = f(z) , where u(z) admits singularities on the surface K = \{z_0 = 0\} and f(z) has an asymptotic expansion of Gevrey type with respect to z_0 as z_0 → 0 . We study the possibility of asymptotic expansion of u(z) . We define the characteristic polygon of P(z, ∂) with respect to K and characteristic indices. We discuss the behavior of u(z) in a neighborhood of K , by using these notions. The main result is a generalization of that in [6].

A dual reciprocity boundary element method based on global shapes functions belonging to a complete set to transform domain integrals into boundary ones is presented. Governing equations of Poisson and Helmholtz type are tested. It is demonstrated that the complete set shape functions are more efficient and accurate in approximating domain integrals than those based on incomplete set. INTRODUCTION The application of Boundary Element Method to linear partial differential operators, or to nonlinear and time-dependent ones, often leads to domain integrals that are cumbersome to handle. The dual reciprocity method pioneered by Nardini and Brebbia* is a powerful numerical technique which converts domain integrals into boundary ones using a set of global shape functions to approximate the integrand. It has been successfully applied in many problems involving dynamic, diffusion and nonlinear type operators. See Partridge, et al. for a comprehensive bibliographical review. The success of the dual reciprocity hinges on the ability to express the global shape functions into the Laplacian of a set of known functions such that the divergence theorem can be applied to resolve the domain integral into a boundary one. Such a constraint has so far limited the selection of shape functions to a set of radial functions of the form 1 + ar -f br^ -f • • •• Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 344 Boundary Elements These radial functions do not form a complete set hence may lack convergence properties. Although practical evidences have indicated that those shape functions work quite satisfactorily, it is still desirable to use shape functions from a set that is complete in the mathematical sense. This paper present a dual reciprocity BEM based on global shape functions derived from a polynomial series. Perhaps due to its mathematical completeness, we are able to demonstrate that the numerical results are more efficient and accurate than the traditional radial set. DUAL RECIPROCITY For many non-standard, transient, or nonlinear partial differential equations it is possible to extract a standard, time-independent, linear partial differential operator C and lump the remainder to the right-hand-side: jC% = 6(z, ?/, Z, f , %, . . . , %,., %„ , %,, %,, %= . • •) (1) For the purpose of BEM implementation, these non-standard portions can be formally regarded as body forces. By the application of generalized Green's theorem = / Jr (2) where v is a conjugate variable, £* is the adjoint operator of £, B» and B* are operators resulting from integration by parts, which may be viewed as generalized normal derivatives, F is the solution boundary, and fi the solution domain, a boundary integral equation can be formed as: cu = / (u*q uq*) j bu* dSl (3) where quantities denoted by asterisk are free-space Green's functions satisfying £*u* = —6, with 6 a Dirac delta function, c is a constant dependent on the Cauchy principal value integration of the singularity at the base point, q and q* are generalized normal derivatives of u and u*. To circumvent the domain integration, the function b can be approximated by an expansion based on a set of global shape functions /, such that &*fX£ (4) j=l in which a, are coefficients determined by collocation, and n^ is the number of collocation nodes. The shape functions are chosen such that particular solutions of the set of inhomogeneous equations Ciij = /,• (5) Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 345 are explicitly known. The domain integral in (3) can be transformed as [ / j^bu da = i ŷ /ju (6) j=l L ^1 J The last expression was a consequence of the application of (2). Substitution of the above into (3) yields / (u*q uq} dY = Y a, \cu, I (%*& titf) dT (7) •/r fr(. L 7r J cu We hence in principal have achieved the elimination of the domain integral. However, the detail of the implementation is somewhat dependent on the type of body force terms. We shall illustrate below the implementation by examples. EXAMPLES Operator of V^u = b(x, y) type For the governing equation of Poisson type, q in (7) is du/dn, and u* is — lnr/2%for 2-D or l/4%r for 3-D geometries. For simplicity, only two-dimensional problems will be studied herein. If the right-hand-side 6(x, y) is a mathematical expression whose particular solution is known as Up, the solution of (8) can be expressed as % = %& + %? (9) where u^ is the homogeneous solution and V2%& = 0 (10) It is apparent that Uh = u — Up satisfies the boundary integral equation c(u «„) = / [u(q q,} -(uu,)q'] (11) If we approximate the right-hand-side b(x,y) as (4), and find the particular solution term by term as V = /,(12) Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 346 Boundary Elements it is clear that j=l By the above interpretation, we have provided an alternative way for deriving (7), which is rewritten below with explicit reference to coordinates as c(x)u(x)f(u*(x',x)q(x')-u(x')q-(x',x)] dT(x') j r

Linear Partial Differential Operators

We present a survey of our work about applications of symbol calculus to quantum mechanics. Symbol calculus embodies the rules of algebra for the linear partial differential operators with variable coefficients, when such operators are expressed as functions of the non-commuting quantities xj and ~/~xj (j=l ..... n). In quantum mechanics the symbol calculus generates S~-C~sica£ expansions, i.e. perturbative series in powers of Planck's constant ~, that are both numerically useful and mathematically interesting. As the complexity of such expansions increases very rapidly with the order, algebraic computer systems have proven most useful in their study [l] . We present here a few of our results, obtained with the extensive use of the formula manipulation language REDUCE (except when specified) : in particular we have found new algorithms to solve the stationary SchrBdinger equation formally in powers of ~. Basic facts about quantum mechanics and semi-classical methods are found in [3] any textbook such as . More precise references, about symbol calculus for instance, are listed in our earlier articles [4j.~ ~

Let \(P = P(\partial ) = \sum\nolimits_{|a| \le m} {C_\alpha \partial ^\alpha }\) be a linear partial differential operator in R n with constant coefficients, as introduced in (7.5). A fundamental solution of P is a distribution \(E\ \in\ \mathcal {D}^\prime({\rm \mathbf R}^n )\ {\rm such\ that}\ P\,E - \delta,\) the Dirac measure at the origin.KeywordsHarmonic FunctionOpen SubsetFundamental SolutionWave OperatorLinear Partial Differential OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let P be a linear partial differential operator with coefficients in the Gevrey class Gs. We prove first that if P is s-hypoelliptic then its transposed operator tP is s-locally solvable, thus extending to the Gevrey classes the well-known analogous result in the C∞class. We prove also that if P is s-hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s-hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.