Formal Differential Operator Research Articles

Efficient pure qP-wave modeling and reverse time migration in tilted transversely isotropic media calculated by a finite-difference approach

Subsurface anisotropy is commonly induced by shale layers, aligned cracks, and fine bedding and has a significant impact on seismic wave propagation. Ignoring anisotropic effects during seismic migration degrades image quality. Therefore, we derive a pure qP-wave equation with high accuracy for modeling seismic wave propagation in tilted transversely isotropic (TTI) media. However, the derived pure qP-wave equation requires a computationally expensive spectral-based method for performing numerical simulations. This is unsuitable for large-scale industrial applications, particularly 3D applications. For numerical efficiency, we first decompose the newly derived wave equation into some fractional differential operators and spatial derivatives. The spatial derivatives can be directly solved by conventional finite-difference (FD) approaches. Then, we use an asymptotic approximation to find an equivalent form of fractional differential operators, obtaining scalar operators that we can discretize with the FD method. Numerical tests show that our TTI pure qP-wave equation with an FD discretization can produce accurate and highly efficient wavefield simulations in TTI media. We also use our TTI pure qP-wave equation with an FD discretization as a forward engine to implement TTI reverse time migration (RTM). Synthetic examples and a field data test demonstrate that our TTI RTM can effectively correct the anisotropic effects, providing high-quality imaging results while maintaining good computational efficiency.

Uniqueness of second-order elliptic operators with unbounded and degenerate coefficients in L 1-spaces

Let . Let and . Consider the formal second-order differential operator in . We show that the closure of is quasi-m-accretive under certain conditions on the coefficients.

Nonlinear vibroacoustic response and sound transmission loss analysis of functionally graded doubly curved shallow shells

This paper presents the semi-analytical nonlinear vibroacoustic and sound transmission loss behaviors of functionally graded doubly curved shallow. The model is developed by considering the simple power law scheme of material distribution in the thickness direction, under thermal load and incident oblique plane sound wave, as well as Donnell’s nonlinear shallow shell theory. By defining the stress function in terms of the force resultants and applying differential operators to these resultants, the coupled compatibility equation and nonlinear equation of motion are obtained only with regard to the stress function and displacement components. The methods of inverse differential operator and average form approach are then utilized to solve the particular and homogeneous solutions of the stress function associated with different boundary conditions. Based on this solution and the use of the Galerkin method with trigonometric mode shape functions, the nonlinear partial differential equation of motion is turned into the Duffing equation. Next, the method of multiple scales is implemented to capture the frequency of the shell corresponding to transverse motion. Finally, the nonlinear vibration and acoustic responses of shells are studied by considering the variation of important parameters such as aspect ratio, dimensionless amplitude, volume fraction power of functionally graded material, phase portrait, sound transmission loss, velocity, average mean square velocity of drive point, and the sound power level of the shells.

A shallow physics-informed neural network for solving partial differential equations on static and evolving surfaces

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network (PINN) for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of a level set function, the surface normal and mean curvature used in the surface differential expressions can be computed easily. So, instead of imposing the normal extension constraints used in literature, we write the surface differential operators in the form of traditional Cartesian differential operators and use them in the loss function directly. We demonstrate a series of performance study for the present methodology by solving Laplace–Beltrami equations and surface diffusion equations on complex static surfaces. With just a moderate number of neurons used in the hidden layer, we are able to attain satisfactory prediction results. We then extend the present methodology to solve the advection–diffusion equation on an evolving surface with a given velocity. To track the deforming surface, we additionally introduce a network, in which a prescribed hidden layer is employed to enforce the topological structure of the surface and learn the homeomorphism between the surface and the prescribed topology. The proposed network structure is designed to track the surface and solve the equation simultaneously. Again, the numerical results show comparable accuracy as the static cases. As an application, we simulate surfactant transportation on a droplet surface under shear flow and obtain some physically plausible results.

Linearization of Nonlinear Affi ne Control Systems with Non-Involutive Distributions by the Introduction of Linearizing Controls

In the article an algorithm for finding linear equivalents (exact linearization) for noninvolutive distributions of control vector fields is considered. In contrast to the common approach to solving this problem — the use of dynamic linearization (the introduction of integrators), which leads to an expansion of the state space — an algorithm for obtaining involutive distributions and ensuring local controllability based on linearizing controls is proposed. The essence of the algorithm: choose such a control and find an explicit expression for it that a controlled vector field associated with this control, when attached to a drift vector field, will provide local controllability and involution of the corresponding distributions. To check the involution of distributions and find the decomposition functions of vector fields on the basis of the current distribution, and on them directly the conditions imposed on the linearization controls, the author has developed an algorithm and a program in the Maple package for finding these functions. For the convenience of presentation and maximum clarity of the proposed approach, in the article is using notation not generally accepted in applied differential geometry. This applies primarily to the representation of vector fields in coordinate form or in the form of differential operators, which is often not specified, but it is assumed that the shape of the vector field is determined from the context. In the article, these forms are clearly separated and their specific use is shown. An example is considered — a nonlinear affine control system of the fifth order with three controls, in which all stages of synthesis are reflected in detail.

Couple stress-based nonlinear primary resonant dynamics of FGM composite truncated conical microshells integrated with magnetostrictive layers

The size-dependent geometrically nonlinear harmonically soft excited oscillation of composite truncated conical microshells (CTCMs) made of functionally graded materials (FGMs) integrated with magnetostrictive layers is analyzed. It is supposed that the FGM CTCMs are subjected to mechanical soft excitations together with external magnetic fields. An analytical framework is created by a microstructure-dependent shell model having the 3rd-order distribution of shear deformation based on the modified couple stress (MCS) continuum elasticity. With the aid of the discretized form of differential operators developed via the generalized differential quadrature technique, a numerical solution methodology is introduced for obtaining the couple stress-based amplitude and frequency responses related to the primary resonant dynamics of the FGM CTCMs. Jump phenomena due to the loss of the first stability branch and falling down to the lower stable branch can be seen in the nonlinear primary resonance of the FGM CTCMs. It is demonstrated that the hardening type of nonlinearity results in bending the frequency response to the right side, and the MCS type of size effect weakens this pattern. Moreover, for higher material gradient indexes, the hardening type of nonlinearity is enhanced, and the MCS-based frequency response bends more considerably to the right side.

Stochastic Filtering in Electromagnetics

This article presents the estimation of electric and magnetic fields using the Kalman filter (KF). The electric and magnetic fields in the entire space have been estimated using the scalar and vector potential. For this estimation, the measurements at a sparse discrete set of spatial pixels have been used. To implement the KF, the state space model has been obtained using the wave equations with sources satisfied by the scalar and vector potential. The proposed method has been implemented on a Hertzian dipole antenna. The fields estimated using KF have been compared with the recursive least squares (RLS) method. The KF presents better estimation than RLS, as it is an optimal estimator. This work uses the Kronecker product for compact representation of discretized fields in the form of vectors and partial differential operators in the form of matrices.

Three-dimensional nonlinear primary resonance of functionally graded rectangular small-scale plates based on strain gradeint elasticity theory

Abstract In this paper, a size-dependent three-dimensional (3D) nonlinear weak formulation is provided to examine the nonlinear primary resonance problem for functionally graded rectangular small-scale plates. The small-scale factors are taken into formulation by choosing the Mindlin's strain gradeint theory (SGT). According to the variational differential quadrature (VDQ) method, first, the displacement field, nonlinear strain-displacement and constitutive relations as well as the potential and kinetic energies are expressed as the vector and matrix forms. Then, by applying the discretized form of differential operators obtained via the generalized differential quadrature (GDQ) method, the discretized form of aforementioned relations is achieved. Finally, Hamilton's principle is employed to access the weak form of 3D nonlinear governing equations of thick rectangular small-scale plates. The achieved formulation is solved via a multi-step numerical technique to address the size-dependent nonlinear primary resonance of considered system under the harmonic lateral force. In addition to reducing the run time, computational effort and CPU usage, the feature of proposed weak form formulation is that one can employ it in other solution approaches such as finite element method. Also, the use of this formulation provides the possibility of recovering models on the basis of other types of size-dependent theories such as modified strain gradient and modified couple stress theories (MSGT and MCST). In the numerical results, the effects of boundary conditions, small-scale parameter, material index and geometry are examined.

Normally ordered forms of powers of differential operators and their combinatorics

We investigate the combinatorics of the general formulas for the powers of the operator h∂d, where ∂ is a differential operator on an arbitrary noncommutative ring in which h is central. New formulas for the generalized Stirling numbers are obtained, as well as results on the divisibility by primes of the coefficients which occur in the normally ordered form of h∂d. All of the above applies to the theory of formal differential operator rings.

A pseudo dispersive, nonlinear equation and its corresponding linear envelope equation

A general dispersive and nonlinear equation is considered. The equation is given in the form of pseudo differential operators. Some examples of dispersion and nonlinearity types are discussed. Taking a special case of the dispersion and nonlinear terms, the equation leads to a well-known Korteweg - de Vries (KdV) equation. A derivation of the linear wave packet evolution is also discussed. This will put a fundamental step for deriving a Nonlinear Schrödinger (NLS) equation for more general dispersive and nonlinear terms. NLS equations are in the interest of applied mathematics, theoretical physics as well as engineering.

The Powers Sums, Bernoulli Numbers, Bernoulli Polynomials Rethinked

Utilizing translation operators we get the powers sums on arithmetic progressions and the Bernoulli polynomials of order munder the form of differential operators acting on monomials. It follows that (d/dn-d/dz) applied on a power sum has a meaning and is exactly equal to the Bernoulli polynomial of the same order. From this new property we get the formula giving powers sums in term of sums of successive derivatives of Bernoulli polynomial multiplied withprimitives of the same order of n. Then by changing the two arguments z,n into Z=z(z-1), λ where λ designed the 1st order power sums and proving that Bernoulli polynomials of odd order vanish for arguments equal to 0, 1/2, 1, we obtain easily the Faulhaber formula for powers sums in term of polynomials in λ having coefficients depending on Z. These coefficients are found to be derivatives of odd powers sums on integers expressed in Z. By the way we obtain the link between Faulhaber formulae for powers sums on integers and on arithmetic progressions. To complete the work we propose tables for calculating in easiest manners possibly the Bernoulli numbers, the Bernoulli polynomials, the powers sums and the Faulhaber formula for powers sums.

Feller Generators with Measure-valued Drifts

We construct a L p -strong Feller process associated with the formal differential operator − Δ + σ ⋅∇ on $\mathbb R^{d}$ , $d \geqslant 3$ , with drift σ in a wide class of measures (e.g. the sum of a measure having density in weak L d space and a Kato class measure), by exploiting a quantitative dependence of the smoothness of the domain of an operator realization of − Δ + σ ⋅∇ generating a holomorphic C 0-semigroup on $L^{p}(\mathbb R^{d})$ , p > d − 1, on the value of the relative bound of σ.

Mexican hat wavelet transform of distributions

ABSTRACTTheory of Weierstrass transform is exploited to derive many interesting new properties of the Mexican hat wavelet transform. A real inversion formula in the differential operator form for the Mexican hat wavelet transform is established. Mexican hat wavelet transform of distributions is defined and its properties are studied. An approximation property of the distributional wavelet transform is investigated which is supported by a nice example.

Numerical solution of Navier–Stokes–Korteweg systems by Local Discontinuous Galerkin methods in multiple space dimensions

Compressible liquid–vapor flow with phase transitions can be described by systems of Navier–Stokes–Korteweg type. They extend the Navier–Stokes equations by nonlinear higher-grade terms which take the form of either differential or nonlocal integral operators. A numerical approximation method on the basis of the Local Discontinuous Galerkin method in multiple space dimensions is suggested for isothermal flows. It relies on a specific discretization of a non-conservative formulation. To enhance the performance of the overall scheme two techniques are used: (i) local spatial adaptivity based on gradient indicators for the density and (ii) parallelism based on domain decomposition.The paper concludes with numerical experiments in two and three space dimensions. They show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena.

Integral Equations for Electromagnetic Scattering at Multi-Screens

In (X. Claeys and R. Hiptmair, Integral equations on multi-screens. Integral Equ Oper Theory, 77(2):167–197, 2013) we developed a framework for the analysis of boundary integral equations for acoustic scattering at so-called multi-screens, which are arbitrary arrangements of thin panels made of impenetrable material. In this article we extend these considerations to boundary integral equations for electromagnetic scattering. We view tangential multi-traces of vector fields from the perspective of quotient spaces and introduce the notion of single-traces and spaces of jumps. We also derive representation formulas and establish key properties of the involved potentials and related boundary operators. Their coercivity will be proved using a splitting of jump fields. Another new aspect emerges in the form of surface differential operators linking various trace spaces.

Formal Pseudodifferential Operators in One and Several Variables, Central Extensions, and Integrable Systems

We review some aspects of the theory of Lie algebras of (twisted and untwisted) formal pseudodifferential operators in one and several variables in a general algebraic context. We focus mainly on the construction and classification of nontrivial central extensions. As applications, we construct hierarchies of centrally extended Lie algebras of formal differential operators in one and several variables, Manin triples and hierarchies of nonlinear equations in Lax and zero curvature form.

New numerical technique for solving the fractional Huxley equation

Purpose – The purpose of this paper is to investigate the possibility of extension to the variational iteration and the Adomian decomposition methods for solving nonlinear Huxley equation with time-fractional derivative. Design/methodology/approach – Objectives achieved the main methods: the fractional derivative of f (x) in the Caputo sense is first stated. Second, the time-fractional Huxley equation is written in a differential operator form where the differential operator is in Caputo sense. After acting on both sides by the inverse operator of the fractional differential operator in Caputo sense, the Adomian's decomposition is then used to get the power series solution of the resulted time-fractional Huxley equation. Also, a second objective is achieved by applying the variational iteration method to get approximate solutions for the time-fractional Huxley equation. Findings – There are some important findings to state and summarize here. First, the variational iteration method and the decomposition method provide the solutions in terms of convergent series with easily computable components for this considered problem. Second, it seems that the approximate solution of time-fractional Huxley equation using the decomposition method converges faster than the approximate solution using the variational iteration method. Third, the variational iteration method handles nonlinear equations without any need for the so-called Adomian polynomials. However, Adomian decomposition method provides the components of the exact solution, where these components should follow the summation given in Equation (21). Originality/value – This paper presents new materials in terms of employing the variational iteration and the Adomian decomposition methods to solve the problem under consideration. It is expected that the results will give some insightful conclusions for the used techniques to handle similar fractional differential equations. This emphasizes the fact that the two methods are applicable to a broad class of nonlinear problems in fractional differential equations.

Gln+1 ALGEBRA OF MATRIX DIFFERENTIAL OPERATORS AND MATRIX QUASI-EXACTLY-SOLVABLE PROBLEMS

The generators of the algebra <em>gl<sub>n+1</sub></em> in the form of differential operators of the first order acting on <strong>R</strong><sup><em>n</em></sup> with matrix coefficients are explicitly written. The algebraic Hamiltonians for matrix generalization of 3−body Calogero and Sutherland models are presented.

Simple computation of reaction–diffusion processes on point clouds

The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulk-surface coupling are provided for a variety of complex point cloud surfaces.

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.