Inverse Differential Operators in Time and Space

When one function is defined as a differential operation on another function, it’s often desirable to invert the definition, to effectively “undo” the differentiation. A Green’s function approach is often used to accomplish this, but variations on this theme exist, and we examine a few such variations. The mathematical analysis of is sought in the form if such an inverse operator exists, but physics is defined by both mathematical formula and ontological formalism, as I show for an example based on the Dirac equation. Finally, I contrast these “standard” approaches with a novel exact inverse operator for field equations.

LEIBNIZ′FORMULA OF GENERALIZED DIFFERENCE WITH RESPECT TO A CLASS OF DIFFERENTIAL OPERATORS AND RECURRENCE FORMULA OF THEIR GREEN′S FUNCTION

In this paper, Leibniz' formula of generalized divided difference with respect to a class of differential operators whose basic sets of solutions have power form, is considered. The recurrence formula of Green function about the operators is also given.

DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES

We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D） =^∞∑k=0φkD^k.φk constant numbers an a power of D.Dn, meaning, is there a isomorphism X （from s onto s） such that Xφ（D） = D^nX？. We prove that if φ（D） is equivalent to Dn, then φ（D） is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n = 1.

Generators of Rings of Differential Operators on a Class of k-algebras

In this paper, we find an explicit method to calculate generators of rings of differential operators on a class of k-algebras.

On the Spectra of General Ordinary Quasi-Differential Operators and Their L2w-Solutions

In this paper, we consider the general ordinary quasi-differential expression τ of order n with complex coefficients and its formal adjoint τ+ on the interval [a,b). We shall show in the case of one singular end-point and under suitable conditions that all solutions of a general ordinary quasi-differential equation are in the weighted Hilbert space provided that all solutions of the equations and its adjoint are in . Also, a number of results concerning the location of the point spectra and regularity fields of the operators generated by such expressions may be obtained. Some of these results are extensions or generalizations of those in the symmetric case, while the others are new.

Representations of Inverse Covariances by Differential Operators

In the cost function of three- or four-dimensional variational dataassimilation, each term is weighted by the inverse of its associated error covariance matrix and thebackground error covariance matrix is usually much larger than the other covariance matrices.Although the background error covariances are traditionally normalized and parameterized by simplesmooth homogeneous correlation functions, the covariance matrices constructed from these correlationfunctions are often too large to be inverted or even manipulated. It is thus desirable to finddirect representations of the inverses of background error correlations. This problem is studied inthis paper. In particular, it is shown that the background term can be written into ∫ dx∣Dυ(x)∣~2, that is, a squared 1/2 norm of a vector differential operator D, called theD-operator, applied to the field of analysis increment υ(x). For autoregressive correlationfunctions, the D-operators are of finite orders. For Gaussian correlation functions, the D-operatorsare of infinite order. For practical applications, the Gaussian D-operators must be truncated tofinite orders. The truncation errors are found to be small even when the Gaussian D-operators aretruncated to low orders. With a truncated D-operator, the background term can be easily constructedwith neither inversion nor direct calculation of the covariance matrix. D-operators are also derivedfor non-Gaussian correlations and transformed into non-isotropic forms.

Spectrum of a class of fourth order left-definite differential operators

The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…

ON APPROXIMATION OF SMOOTH FUNCTIONS FROM NULL SPACES OF OPTIMAL LINEAR DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln（ψ） =∑k=0^n akψ（k） where the constant coefficients ak C R may be adapted to f.

Realization of the N（odd）-Dimensional Quantum Euclidean Space by Differential Operators

The quantum Euclidean space is a kind of noncommutative space that is obtained from ordinary Euclidean space by deformation with parameter q. When N is odd, the structure of this space is similar to . Motivated by realization of by differential operators in , we give such realization for and cases and generalize our results to (N odd) in this paper, that is, we show that the algebra of can be realized by differential operators acting on C∞ functions on undeformed space .

Novel Investigation of Stochastic Fractional Differential Equations Measles Model via the White Noise and Global Derivative Operator Depending on Mittag-Leffler Kernel

Because of the features involved with their varied kernels,differential operators relying on convolution formulations have been acknowledged as effective mathematical resources for modeling real-world issues.In this paper,we constructed a stochastic fractional framework of measles spreading mechanisms with dual medication immunization considering the exponential decay and Mittag-Leffler kernels.In this approach,the overall population was separated into five cohorts.Furthermore,the descriptive behavior of the system was investigated,including prerequisites for the positivity of solutions,invariant domain of the solution,presence and stability of equilibrium points,and sensitivity analysis.We included a stochastic element in every cohort and employed linear growth and Lipschitz criteria to show the existence and uniqueness of solutions.Several numerical simulations for various fractional orders and randomization intensities are illustrated.

A Coupled Thermomechanical Crack Propagation Behavior of Brittle Materials by Peridynamic Differential Operator

This study proposes a comprehensive,coupled thermomechanical model that replaces local spatial derivatives in classical differential thermomechanical equations with nonlocal integral forms derived from the peridynamic differential operator(PDDO),eliminating the need for calibration procedures.The model employs a multi-rate explicit time integration scheme to handle varying time scales in multi-physics systems.Through simulations conducted on granite and ceramic materials,this model demonstrates its effectiveness.It successfully simulates thermal damage behavior in granite arising from incompatible mineral expansion and accurately calculates thermal crack propagation in ceramic slabs during quenching.To account for material heterogeneity,the model utilizes the Shuffle algorithm andWeibull distribution,yielding results that align with numerical simulations and experimental observations.This coupled thermomechanical model shows great promise for analyzing intricate thermomechanical phenomena in brittle materials.

Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Transient heat conduction problems widely exist in engineering.In previous work on the peridynamic differential operator(PDDO)method for solving such problems,both time and spatial derivatives were discretized using the PDDO method,resulting in increased complexity and programming difficulty.In this work,the forward difference formula,the backward difference formula,and the centered difference formula are used to discretize the time derivative,while the PDDO method is used to discretize the spatial derivative.Three new schemes for solving transient heat conduction equations have been developed,namely,the forward-in-time and PDDO in space(FT-PDDO)scheme,the backward-in-time and PDDO in space(BT-PDDO)scheme,and the central-in-time and PDDO in space(CT-PDDO)scheme.The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem.Results show that the FT-PDDO scheme is conditionally stable,whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable.The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method.The convergence rate in space for these three methods is two.These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems.The accuracy and validity of the schemes are verified by comparison with analytical solutions.

Euler’s First-Order Explicit Method–Peridynamic Differential Operator for Solving Population Balance Equations of the Crystallization Process

Using Euler’s first-order explicit(EE)method and the peridynamic differential operator(PDDO)to discretize the time and internal crystal-size derivatives,respectively,the Euler’s first-order explicit method–peridynamic differential operator(EE–PDDO)was obtained for solving the one-dimensional population balance equation in crystallization.Four different conditions during crystallization were studied:size-independent growth,sizedependent growth in a batch process,nucleation and size-independent growth,and nucleation and size-dependent growth in a continuous process.The high accuracy of the EE–PDDO method was confirmed by comparing it with the numerical results obtained using the second-order upwind and HR-van methods.The method is characterized by non-oscillation and high accuracy,especially in the discontinuous and sharp crystal size distribution.The stability of the EE–PDDO method,choice of weight function in the PDDO method,and optimal time step are also discussed.

ON SELF-ADJOINTNESS OF THE PRODUCT OF TWO n-ORDER DIFFERENTIAL OPERATORS ON[a,b]

Self adjointness of the product of differential operators on is studiedhere, using the construction theory of self adjoint operators, we give a sufficient and necessary condition of self adjointness problem.

Weighted variation inequalities for differential operators and singular integrals in higher dimensions

We prove weighted q-variation inequalities with 2<q<∞for sharp truncations of singular integral operators in higher dimensions.The vector-valued extensions of these inequalities are also given.Parallel results are proven for differential operators.

On the Self-adjointness of the Product Operators of Two mth-Order Differential Operators on [0,+∞)

In the present paper,the self-adjointness of the product of two ruth-order differential operators on [0,+∞)is studied.By means of the construction theory of self-adjoint operators and matrix computation,we obtain a sufficient and necessary condition to ensure that the product operator is self-adjoint,which extends the results in the second order case.

On the Spectrum of a Class of Differential Operators and Embedding Theorems

The author considers the embedding problem of weighted Sobolev spaces Hnp in weighted Ls spaces Ls,r,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.

On Self-Adjointness of the Product of Two Second-Order Differential Operators

In this paper, the problem of self-adjointness of the product of two differential operators is considered. A number of results concerning self-adjointness of the product L2L1 of two second-order self-adjoint differential operators are obtained by using the general construction theory of self-adjoint extensions of ordinary differential operators.

On computing Grbner bases in rings of differential operators

Insa and Pauer presented a basic theory of Grobner basis for differential operators with coefficients in a commutative ring in 1998, and a criterion was proposed to determine if a set of differential operators is a GrSbner basis. In this paper, we will give a new criterion such that Insa and Pauer＇s criterion could be concluded as a special case and one could compute the Grobner basis more efficiently by this new criterion.

Hermitizable, isospectral complex second-order differential operators

The first aim of the paper is to study the Hermitizability of secondorder differential operators,and then the corresponding isospectral operators.The explicit criteria for the Hermitizable or isospectral properties are presented.The second aim of the paper is to study a non-Hermitian model,which is now well known.In a regular sense,the model does not belong to the class of Hermitizable operators studied in this paper,but we will use the theory developed in the past years,to present an alternative and illustrated proof of the discreteness of its spectrum.The harmonic function plays a critical role in the study of spectrum.Two constructions of the function are presented.The required conclusion for the discrete spectrum is proved by some comparison technique.