Narrow Results

In this paper we introduce a class of semiclassical Fourier integral operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schrödinger equations. Our construction is elementary, it is inspired by the joint work of the first author with Yu. Safarov and D. Vasiliev. We consider several simple but basic examples.

We consider the fundamental solution for the Cauchy problem for perturbations of the harmonic oscillator by time dependent potentials which grow at spatial infinity slower than quadratic but faster than linear functions and whose Hessian matrices have a fixed sign. We prove that the fundamental solution at resonant times grows indefinitely at spatial infinity with an algebraic growth rate, which increases indefinitely when the growth rate of perturbations at infinity decreases from the near quadratic to the near linear ones.

We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative partial differential equations (PDEs) based on the spirit of LY algorithm [T.-P. Liu and S.-H. Yu, Dirichlet–Neumann kernel for hyperbolic-dissipative system in half-space, Bull. Inst. Math. Acad. Sin. 7 (2012) 477–543]. The new method can handle more general domains than that of LYs’. We convert the evolutionary PDEs into the elliptic PDEs by the Laplace transformation. Using the Laplace transformation of the fundamental solutions of the evolutionary PDEs and the image method, we can construct Green’s functions for the corresponding elliptic PDEs. Finally, we obtain Green’s functions for the evolutionary PDEs by inverting the Laplace transformation. As a consequence, we establish Green’s functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. On the other hand, the structure of hyperbolic-dissipative PDEs means its fundamental solution is non-symestric and hence the image method does not work. We utilize the idea of Laplace wave train introduced by Liu and Yu in [Navier–Stokes equations in gas dynamics: Greens function, singularity and well-posedness, Comm. Pure Appl. Math. 75(2) (2022) 223–348] to generalize the image method. Combining this with the notions of Rayleigh surface wave operators introduced in [S. J. Deng, W. K. Wang and S.-H. Yu, Green’s functions of wave equations in ${ℝ}_{+}^n\times {ℝ}_{+}$, Arch. Ration. Mech. Anal. 216 (2015) 881–903], we are able to obtain the complete representations of Green’s functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions.

Based on the implicit calculus equation modeling approach, this paper proposes a speculative concept of the potential and wave operators on negative dimensionality. Unlike the standard partial differential equation (PDE) modeling, the implicit calculus modeling approach does not require the explicit expression of the PDE governing equation. Instead the fundamental solution of physical problem is used to implicitly define the differential operator and to implement simulation in conjunction with the appropriate boundary conditions. In this study, we conjecture an extension of the fundamental solution of the standard Laplace and Helmholtz equations to negative dimensionality. And then by using the singular boundary method, a recent boundary discretization technique, we investigate the potential and wave problems using the fundamental solution on negative dimensionality. Numerical experiments reveal that the physics behaviors on negative dimensionality may differ on positive dimensionality. This speculative study might open an unexplored territory in research.

We introduce a fractional stochastic heat equation with second-order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.

We derive eigenfunction expansions for a fundamental solution of Laplace's equation in three-dimensional Euclidean space in 5-cyclidic coordinates. There are three such expansions in terms of internal and external 5-cyclidic harmonics of first, second and third kind. The internal and external 5-cyclidic harmonics are expressed by solutions of a Fuchsian differential equation with five regular singular points.

Let $Z$ denote the holomorphic tangential vector field to the generalized upper-half plane ${𝒟}=\{(z_1,z)\in {ℂ}^2;\rho =\mathrm{\text{Im}} z_1-|z{|}^4>0\}$. In our terminology, $t=\mathrm{\text{Re}} z_1$. Consider the ${\square }_b$ operator on the boundary of ${𝒟}$, $\mathcal{ℒ}=-1/2(Z\overline{Z}+\overline{Z}Z)$; note that $\mathcal{ℒ}$ is nowhere elliptic, but it is subelliptic with step three. The principal result of this paper is the derivation of an explicit fundamental solution $F$ to $\mathcal{ℒ}$. Our approach is based on special functions and their properties.

A new hybrid fundamental-solution-based finite element method (HFS-FEM) is developed for the transient heat conduction analysis of two-dimensional orthotropic materials. By using coordinate transformation and time-stepping approximation, the governing equation of the original problem is converted to the modified Helmholtz equation. In the solution procedure, the homogeneous solutions of the problem are sought by employing HFS-FEM while the particular solutions are approximated with the aid of radial basis function. HFS-FEM has many attractive features compared to the conventional FEM. The computational efficiency and convergence of the present method are examined through several numerical examples.

We prove a weak form of the soliton resolution conjecture for — $H^2$ uniformly bounded in time — solutions of semilinear fourth-order Schrödinger equations, in dimensions $n\ge 5$, and with a mass supercritical-energy subcritical power type nonlinearity, by using a strategy developed by T. Tao in 2007. More precisely, we prove that the solutions are decomposed into a sum of two terms: a free solution and a nonradiative term that approaches asymptotically an object that has similar properties to those of a finite sum of solitons. The asymptotic behavior of the nonradiative term is derived from its asymptotic frequency localization and its asymptotic spatial localization. There are two main differences between the present contribution and T. Tao’s earlier work. The first one appears when we prove the asymptotic frequency localization: we fill a gap of regularity by using the better dispersive properties of the high frequency pieces of the free solution. The second one appears when we prove the asymptotic spatial localization. A key estimate depends on the fundamental solution that does not have an explicit form. We overcome the difficulty by introducing a modified fundamental solution and exploiting the symmetries of the characters of the phases.

This paper presents an advanced formulation of the time-domain two-dimensional (2D) boundary element method (BEM) for an elastic, homogeneous unsaturated soil subjected to dynamic loadings. Unlike the usual time-domain BEM, the present formulation applies a convolution quadrature which requires only the Laplace-domain instead of the time-domain fundamental solutions. The coupled equations governing the dynamic behavior of unsaturated soils ignoring contributions of the inertia effects of the fluids (water and air) are derived based on the poromechanics theory within the framework of a suction-based mathematical model. In this formulation, the solid skeleton displacements $u_i$, water pressure $p_w$ and air pressure $p_a$ are presumed to be independent variables. The fundamental solutions in Laplace transformed-domain for such a dynamic $u-p_w-p_a$ theory have been obtained previously by authors. Then, the BE formulation in time is derived after regularization by partial integrations and time and spatial discretizations. Thereafter, the BE formulation is implemented in a 2D boundary element code (PORO-BEM) for the numerical solution. To verify the accuracy of this implementation, the displacement response obtained by the boundary element formulation is verified by comparison with the elastodynamics problem.

A hybrid finite element method is proposed for the heat conduction analysis with variable thermal conductivities. A linear combination of fundamental solutions is employed to approximate the intra-element temperature field while standard one-dimensional shape functions are utilized to independently define the frame temperature field along the element boundary. The influence of variable thermal conductivities embeds in the intra-element temperature field via the fundamental solution. A hybrid variational functional, which involves integrals along the element boundary only, is developed to link the two assumed fields to produce the thermal stiffness equation. The advantage of the proposed method lies that the changes in the thermal conductivity are captured inside the element domain. Numerical examples demonstrate the accuracy and efficiency of the proposed method and also the insensitivity to mesh distortion.

The objective of this paper is to present some sufficient conditions for approximate controllability of semilinear fractional stochastic control system with delay. The results hold when the nonlinear function is Lipschitz continuous. Sufficient conditions are obtained by separating the given fractional semilinear stochastic system into two systems namely a semilinear fractional system and a fractional linear stochastic system. To prove our results, the Schauder fixed point theorem is applied. At the end, one example is given to illustrate the results.

Fundamental solutions are constructed for the equation

We describe the partial hypoellipticity of differential operators with constant coefficients in terms of fundamental solutions.

These remarks are devoted to a basic properties of the double-complex Laplace operator , where z, w are complex variables and U(z, w) is a double-complex value function. A decomposition into two complex operators is given. An initial problem for the double-complex Laplace operator on the whole space ℂ(1, j) is solved with a D'Alember's type formula. An initial problem on the cartesian product of two unit squares in ℂ for the zero eigenvalue problem for the double-complex Laplace operator is treated with the Fourier method of separating of the variables in double-complex context. The corresponding system of two equations of second order and the corresponding operator of fourth order with real coefficients are obtained.

This short communication deals with several properties of the double-complex Laplace operator, namely it is constructed a fundamental distribution solution, it is proved the partial C^∞ hypoellipticity and it is proved a theorem on microlocal propagation of singularities of the solutions along the leaves (two dimensional) of the real and imaginary parts of the Hamiltonian vector field of the symbol.