Convolution-Based Model-Solving Method for Three-Dimensional, Unsteady, Partial Differential Equations

Neural networks are increasingly used widely in the solution of partial differential equations (PDEs). This letter proposes 3D-PDE-Net to solve the three-dimensional PDE. We give a mathematical derivation of a three-dimensional convolution kernel that can approximate any order differential operator within the range of expressing ability and then conduct 3D-PDE-Net based on this theory. An optimum network is obtained by minimizing the normalized mean square error (NMSE) of training data, and L-BFGS is the optimized algorithm of second-order precision. Numerical experimental results show that 3D-PDE-Net can achieve the solution with good accuracy using few training samples, and it is of highly significant in solving linear and nonlinear unsteady PDEs.

On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

On Sobolev bilinear forms and polynomial solutions of second-order differential equations

Given a linear second-order differential operator $${\mathcal {L}}\equiv \phi \,D^2+\psi \,D$$ L ≡ ϕ D 2 + ψ D with non zero polynomial coefficients of degree at most 2, a sequence of real numbers $$\lambda _n$$ λ n , $$n\geqslant 0$$ n ⩾ 0 , and a Sobolev bilinear form $$\begin{aligned} {\mathcal {B}}(p,q)\,=\,\sum _{k=0}^N\left\langle {{\mathbf {u}}_k,\,p^{(k)}\,q^{(k)}}\right\rangle , \quad N\geqslant 0, \end{aligned}$$ B ( p , q ) = ∑ k = 0 N u k , p ( k ) q ( k ) , N ⩾ 0 , where $${\mathbf {u}}_k$$ u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are linear functionals defined on polynomials, we study the orthogonality of the polynomial solutions of the differential equation $${\mathcal {L}}[y]=\lambda _n\,y$$ L [ y ] = λ n y with respect to $${\mathcal {B}}$$ B . We show that such polynomials are orthogonal with respect to $${\mathcal {B}}$$ B if the Pearson equations $$D(\phi \,{\mathbf {u}}_k)=(\psi +k\,\phi ')\,{\mathbf {u}}_k$$ D ( ϕ u k ) = ( ψ + k ϕ ′ ) u k , $$0\leqslant k \leqslant N$$ 0 ⩽ k ⩽ N , are satisfied by the linear functionals in the bilinear form. Moreover, we use our results as a general method to deduce the Sobolev orthogonality for polynomial solutions of differential equations associated with classical orthogonal polynomials with negative integer parameters.

The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

Closed-Form Solutions of Linear Ordinary Differential Equations with General Boundary Conditions

This paper deals with the solution of boundary value problems for ordinary differential equations with general boundary conditions. We obtain closed-form solutions in a symbolic form of problems with the general n-th order differential operator, as well as the composition of linear operators. The method is based on the theory of the extensions of linear operators in Banach spaces.

Scalar Casimir effect for a conducting cylinder in a Lorentz-violating background

Following a field-theoretical approach, we study the scalar Casimir effect upon a perfectly conducting cylindrical shell in the presence of spontaneous Lorentz symmetry breaking. The scalar field is modeled by a Lorentz-breaking extension of the theory for a real scalar quantum field in the bulk regions. The corresponding Green’s functions satisfying Dirichlet boundary conditions on the cylindrical shell are derived explicitly. We express the Casimir pressure (i.e. the vacuum expectation value of the normal–normal component of the stress–energy tensor) as a suitable second-order differential operator acting on the corresponding Green’s functions at coincident arguments. The divergences are regulated by making use of zeta function techniques, and our results are successfully compared with the Lorentz invariant case. Numerical calculations are carried out for the Casimir pressure as a function of the Lorentz-violating coefficient, and an approximate analytical expression for the force is presented as well. It turns out that the Casimir pressure strongly depends on the Lorentz-violating coefficient and it tends to diminish the force.

Asymptotics of the Spectrum of a Periodic Boundary ValueProblem for an Odd-Order Differential Operator

The spectrum of a differential operator of high odd order with periodic boundary conditions is studied. The asymptotics of the fundamental system of solutions of the differential equation defining the operator are obtained by the method of successive Picard approximations. With the help of this fundamental system of solutions the periodic boundary conditions are studied. As a result, the equation for the eigenvalues of the differential operator under study is obtained, which is a quasi-polynomial. The indicator diagram of this equation, which is a regular polygon, is investigated. In each of the sectors of the complex plane, defined by the indicator diagram, the asymptotics of the eigenvalues of the operator under study is found. An equation for the eigenvalues of the differential operator under study is derived. The indicator diagram of this equation has been studied. The asymptotics of the eigenvalues of the studied operator in different sectors of the indicator diagram is found.

A Second Regularized Trace Formula for a Fourth Order Differential Operator

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.