Adjoint Differential Operators Research Articles

Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

<p style='text-indent:20px;'>We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.</p>

Numerical solution of Korteweg–de Vries equation using discrete least squares meshless method

In this study, a meshless approach called Discrete Least Squares Meshless is used to solve the third-order nonlinear Korteweg–de Vries equation numerically. In shallow water, the Korteweg–de Vries equation illustrates the unidirectional propagation of waves. For function approximation, the suggested technique uses moving least squares shape functions. Furthermore, the least squares functional constructed on certain auxiliary points known as sampling points is minimized in order to generate the system of algebraic equations. This approach may be considered meshless since it does not need a mesh for field variable approximation or system of equations creation; it also gives a symmetric and positive definite matrix even for non-self adjoint differential operators. To demonstrate the applicability of the current meshless approach, the propagation of a single soliton and multi-solitons is explored. The approach may be applied directly to the Korteweg–de Vries equation because to the high degrees of continuity of the moving least squares shape function when an exponential weight function is utilized. Several benchmark numerical instances are explored and the resulting results are compared to earlier research to test the capabilities of the suggested approach for the solution of Korteweg–de Vries equation.

K-Version of Finite Element Method for BVPs and IVPs

The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work.

Evolution of Time-Harmonic Electromagnetic and Acoustic Waves Along Waveguides

We study time-harmonic electromagnetic and acoustic waveguides, modeled by an infinite cylinder with a non-smooth cross section. We introduce an infinitesimal generator for the wave evolution along the cylinder and prove estimates of the functional calculi of these first order non-self adjoint differential operators with non-smooth coefficients. Applying our new functional calculus, we obtain a one-to-one correspondence between polynomially bounded time-harmonic waves and functions in appropriate spectral subspaces.

Three-dimensional deformations of a curved circular beam subjected to thermo-mechanical loading using green's function method

This article deals with the exact thermo-mechanical static responses of a curved circular beam by Green's function method. To find the solution, the related governing differential equations are found using the Hamilton's principles. They are combined into two six-order ordinary inhomogeneous differential equations. To form the Green's problem, the adjoint differential operator along with corresponding boundary conditions is found from the inner product calculations. It is noteworthy that the lateral temperature distribution leads to both in-plane and out-of-plane deformations. This important point has not been mentioned in the previous studies. Moreover, the radially distributed temperature results in no out-of-plane displacements. Finally, the answers of some numerical tests confirmed the accuracy of the proposed solution method.

Optimization of higher order differential inclusions with initial value problem

This paper concerns the sufficient conditions of optimality for initial value problem with higher order differential inclusions (HODIs) and free endpoint constraints. Formulation of the transversality conditions plays a substantial role in the next investigations without which hardly any necessary or sufficient conditions would be obtained. In terms of Euler–Lagrange and Hamiltonian forms the sufficient conditions of optimality both for convex and “non-convex” HODIs are based on the apparatus of locally adjoint mappings. Moreover, by applying the main result to a Bolza problem described by a polynomial differential operator with constant coefficients in terms of the adjoint differential operator the sufficient condition of optimality is obtained.

Three-dimensional analytical solution for transport problem during convection drying using Green's function method (GFM)

A three-dimensional analytical solution of transport problem of convection-drying is accomplished using Green's function method (GFM). Mass conservation, momentum and energy equations must be solved to obtain convective heat and mass transfer coefficients. In most papers, these equations are solved using numerical methods alike. The mass transfer coefficients are calculated using analogy between the thermal and concentration boundary layers. The results of previous studies are used for these coefficients values. Green's function method has been used as a new simple method to solve equations of transport problem. The heat and mass transfer equations are solved using Green's function method and taking diffusion equation-related adjoint differential operator equal to Delta function. Of course, these equations are coupled with thermal diffusivity, because this parameter is a function of temperature and is used in mass transfer equation. These two coupled equations were solved with a good approximation which is a definition of a weighted function for moisture distribution. Thus, temperature and moisture distributions within the body obtained as functions of x, y, z and an independent time variable t. After these steps it is seen that there is a good consistency between the results and the existing solutions. The prominent advantages of the proposed solution are techniques which are less time and money consuming compared to numerical and experimental methods.

Adjoint n-point Boundary Value Problem for the Linear Differential Equation and Green's Function

The n-point problem with linear boundary conditions of general type is studied in this work. We have found the boundary conditions for the adjoint differential operator. The Green's function has been constructed where we have used solutions of the adjoint differential equation and studied its new properties. Through the Green's function and saltus of its derivatives, we have solved the nonhomogeneous n-point boundary value problem for the linear differential equation with variable coefficients.

Sobolev-Trenogin Principle in Construction of the Boundary Value Problem Adjoint to the Linear Multipoint Problem

The research has been done in two directions. In a linear case, the adjoint boundary problem have been built. We have managed to do it in a classical continuous case without resort to such terms of functional analysis as an adjoint space, adjoint operator, etc. In a non-linear case, we have considered the problem with a small parameter and discussed an issue of applicability of some aspects of a theory of bifurcation of the nonlinear equations' solutions (Trenogin et al., 1991). We have built a boundary problem adjoint to the linear multipoint problem. We have studied unicity of its linear and adjoint differential operators with multipoint boundary conditions and generalized it for the m-point problem.

Global Compensated Compactness Theorem for General Differential Operators of First Order

Let A1(x, D) and A2(x, D) be differential operators of the first order acting on l-vector functions \({u= (u_1, \ldots, u_l)}\) in a bounded domain \({\Omega \subset \mathbb{R}^{n}}\) with the smooth boundary \({\partial\Omega}\). We assume that the H1-norm \({\|u\|_{H^{1}(\Omega)}}\) is equivalent to \({\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_1u\|_{H^{\frac{1}{2}}(\partial\Omega)}}\) and \({\sum_{i=1}^2\|A_iu\|_{L^2(\Omega)} + \|B_2u\|_{H^{\frac{1}{2}}(\partial\Omega)}}\), where Bi = Bi(x, ν) is the trace operator onto \({\partial\Omega}\) associated with Ai(x, D) for i = 1, 2 which is determined by the Stokes integral formula (ν: unit outer normal to \({\partial\Omega}\)). Furthermore, we impose on A1 and A2 a cancellation property such as \({A_1A_2^{\prime}=0}\) and \({A_2A_1^{\prime}=0}\), where \({A^{\prime}_i}\) is the formal adjoint differential operator of Ai(i = 1, 2). Suppose that \({\{u_m\}_{m=1}^{\infty}}\) and \({\{v_m\}_{m=1}^{\infty}}\) converge to u and v weakly in \({L^2(\Omega)}\), respectively. Assume also that \({\{A_{1}u_m\}_{m=1}^{\infty}}\) and \({\{A_{2}v_{m}\}_{m=1}^{\infty}}\) are bounded in \({L^{2}(\Omega)}\). If either \({\{B_{1}u_m\}_{m=1}^{\infty}}\) or \({\{B_{2}v_m\}_{m=1}^{\infty}}\) is bounded in \({H^{\frac{1}{2}}(\partial\Omega)}\), then it holds that \({\int_{\Omega}u_m\cdot v_m \,{\rm d}x \to \int_{\Omega}u\cdot v \,{\rm d}x}\). We also discuss a corresponding result on compact Riemannian manifolds with boundary.

The regularized trace of a self adjoint differential operator of higher order with unbounded operator coefficient

Let L 0 and L be operators which are formed by the differential expressions. ℓ 0 ( y ) = ( - 1 ) m y ( 2 m ) ( x ) + Ay ( x ) and ℓ ( y ) = ( - 1 ) m y ( 2 m ) ( x ) + Ay ( x ) + Q ( x ) y ( x ) respectively, in the space H 1 = L 2(0, π; > H), with same boundary condition y (2 i−1) (0) = y (2 i−1) ( π) = 0, ( i = 1, 2, … , m) where H is an infinite dimensional separable Hilbert space. Here, A is an unbounded self adjoint operator in H and, for every x ∈ [0, π], Q( x) is a self-adjoint trace class operator in H. Assuming the operator A and the operator function Q( x) satisfy some additional conditions, the following formula has been found. lim p → ∞ m q = 1 n p λ q - μ q - 1 π ∫ 0 π ( Q ( x ) φ j q , φ j q ) dx = 1 4 [ trQ ( 0 ) + trQ ( π ) ] - 1 2 π ∫ 0 π trQ ( x ) dx for the regularized trace of L. Here, n 1 < n 2 < ⋯and j 1, j 2, …are sequences of natural numbers with a particular property. Furthermore, μ 1 ⩽ μ 2 ⩽ ⋯and λ 1 ⩽ λ 2 ⩽ ⋯are the eigenvalues of the operators L 0 and L, respectively; and φ 1, φ 2, …is a complete orthonormal sequence consisting of eigenvectors of the operator A.

On Computing of Eigenvalues of Differential Equations Q = λP with Eigenparameter in Boundary Conditions

Problem statement: Our purpose of this study is to use sinc methods to compute approximately the eigenval-ues of second-order operator pencil of the form Q-λP. Approach: Where Q is second order self adjoint differential operator and P is a first order and λ∈C is an eigenvalue parameter. Results: The eigenparameter appears in the boundary conditions linearly. Using computable error boundswe obtain eigenvalue enclosures in a simple way. Conclusion/Recommendations: We give some numerical examples and makecompanions with existing results.

The second regularized trace of a self adjoint differential operator given in a finite interval with bounded operator coefficient

In this work a formula for the second regularized trace of a self adjoint differential operator which is given in a finite interval and with bounded operator coefficient was found.

Generalized Variational Problems and Euler–Lagrange equations

This paper introduces three new operators and presents some of their properties. It defines a new class of variational problems (called Generalized Variational Problems, or GVPs) in terms of these operators and derives Euler–Lagrange equations for this class of problems. It is demonstrated that the left and the right fractional Riemann–Liouville integrals, and the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of these operators, and they are obtained by considering a special kernel. Further, the Euler–Lagrange equations developed for functional defined in terms of the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of the Euler–Lagrange equations developed here. Examples are considered to demonstrate the applications of the new operators and the new Euler–Lagrange equations. The concepts of adjoint differential operators and adjoint differential equations defined in terms of the new operators are introduced. A new class of generalized Lagrangian, Hamiltonian, and action principles are presented. In special cases, these formulations lead to fractional adjoint differential operators and adjoint differential equations, and fractional Lagrangian, Hamiltonian, and action principle. Thus, the new operators introduce a generalized approach to many problems in classical mechanics in general and variational calculus in particular. Possible extensions of the subject and the concepts discussed here are also outlined.

Adjoint vector fields and differential operators on representation spaces

Let $G$ be a semisimple algebraic group with Lie algebra $\g$. In 1979, J. Dixmier proved that any vector field annihilating all $G$-invariant polynomials on $\g$ lies in the $\bbk[\g]$-module generated by the "adjoint vector fields", i.e., vector fields $\varsigma$ of the form $\varsigma(y)(x)=[x,y]$, $x,y\in\g$. A substantial generalisation of Dixmier's theorem was found by Levasseur and Stafford. They explicitly described the centraliser of $\bbk[\g]^G$ in the algebra of differential operators on $\g$. On the level of vector fields, their result reduces to Dixmier's theorem. The purpose of this paper is to explore similar problems in the general context of affine algebraic groups and their rational representations.

Galerkin/Least-Squares Finite Element Processes for BVP in h, p, k Mathematical Framework

This paper presents an investigation of the details of mathematical and computational aspects of the Galerkin/least-squares and the Galerkin/weak form least-squares finite element process within the mathematical and computational framework [1, 2, 3] based on h, p, k as independent computational parameters and requiring that the integral forms be variationally consistent (VC). Higher-order global differentiability of order (k−1) defined by the order k of the approximation space is essential for incorporating correct physics of the processes in the computations and that k is an independent parameter in addition to h and p in all finite element computations. In this paper the attributes of the Galerkin method, the Galerkin method with weak form, and least-squares processes in h, p, k framework with variationally consistent (VC) or variationally inconsistent (VIC) integral forms are utilized to investigate the mathematical features of the Galerkin/least-squares processes (GAL/LSP) and Galerkin/weak form least-squares process (GAL/WF/LSP) to establish (1) when such processes have a sound mathematical basis (2) role of functionals resulting from the Galerkin method, the Galerkin method with weak form and least-squares processes in GAL/LSP and GAL/WF/LSP (3) Importance of minimally conforming spaces and role of higher order spaces in GAL/LSP and GAL/WF/LSP. It is concluded that GAL/LSP only have sound mathematical basis for self adjoint differential operators when viewed within h,p,k framework with the requirement that the integral forms be variationally consistent. For non-self adjoint and non-linear differential operators GAL/LSP and GAL/WF/LSP do not have a sound mathematical basis within the proposed framework. Currently used finite element processes based on GAL/WF/LSP utilizing local approximations of class C 0 violate basic mathematical principles, are in violation with physics and hence do not provide a valid and mathematically sound approach. Investigation of the GAL/LSP and GAL/WF/LSP when the local approximations are in minimally conforming spaces or in spaces of order higher than minimally conforming spaces, shows that the functionals from GAL and GAL/WF processes only have detrimental affect on the LSP process. Numerical studies are presented to demonstrate various aspects and features.

Structure of Binary Darboux-Type Transformations for Hermitian Adjoint Differential Operators

For Hermitian adjoint differential operators, we consider the structure of Darboux–Backlund-type transformations in the class of parametrically dependent Hilbert spaces. By using the proposed new method, we obtain the corresponding integro-differential symbols of the operators of transformations in explicit form and consider the problem of their application to the construction of two-dimensional Lax-integrable nonlinear evolution equations and their Darboux–Backlund-type transformations.

Structure of Binary Transformations of Darboux Type and Their Application to Soliton Theory

On the basis of generalized Lagrange identity for pairs of formally adjoint multidimensional differential operators and a special differential geometric structure associated with this identity, we propose a general scheme of the construction of corresponding transformation operators that are described by nontrivial topological characteristics. We construct explicitly the corresponding integro-differential symbols of transformation operators, which are used in the construction of Lax-integrable nonlinear two-dimensional evolutionary equations and their Darboux–Backlund-type transformations.

On a spectrum of blow–up patterns for a higher–order semilinear parabolic equation

We describe a spectrum of blow–up patterns for the 2 m th–order semilinear parabolic equation, u t = ( Δ ) m u + | u | p , x R N , t > 0 ; m > 1 , p > 1. This problem is well understood in the second–order case m = 1 (the semilinear heat equation), where for p ⩽ p s = ( N + 2)/( N − 2) + the inner space–time structure of blow–up patterns is locally governed by separable Hermite polynomials of arbitrary finite order as eigenfunctions of a linear self–adjoint differential operator. We consider m > 1 and describe a principle idea of constructing of a spectrum of blow–up patterns which in the inner self–similar region are composed of a countable subset of separable Kummer9s polynomials, which are eigenfunctions of a non–self–adjoint linear differential operator. A matching procedure extends such linearized structures to the nonlinear intermediate region where the asymptotic blow–up behaviour is described by a unique self–similar solution of the first–order Hamilton–Jacobi equation.

Strongly Harmonic Forms for Representations in the Discrete Series

LetGbe a semisimple connected Lie group and letKbe a maximal compact subgroup. Assume that rankG=rankK, and letT⊂Kbe a Cartan subgroup ofG. The quotientG/Tcarries an indefiniteG-invariant hermitian form. The standard ∂ Dolbeault operator has a formal adjoint differential operator ∂*invwith respect to the invariant hermitian form. Letsdenote the complex dimension ofK/T. We form the indefinite harmonic space Hs(G/T,Lχ+2ρ)={(0,s)+Lχ+2ρ×valued forms in Ker∂∩Ker∂*inv}. In this paper we show that under some positivity conditions onχthe cohomology space Hs(G/T,Lχ) contains a copy of the representation in the discrete series ofGwith parameterχ.