On the Inhomogeneous Conservative Wiener–Hopf Equation

We prove the existence of a solution to the inhomogeneous Wiener–Hopf equation whose kernel is a probability distribution generating a random walk drifting to ?∞. Asymptotic properties of a solution are found depending on the corresponding properties of the free term and the kernel of the equation.     

Extremum Problem for the Wiener–Hopf Equation

The extremum problem for the Wiener–Hopf equation obtained by replacing the condition u(x) = 0, x < 0, by the condition of the minimum of the quadratic functional of the function u(x)exp(–x), – < x < , is solved in closed form.     

On the Volterra Factorization of the Wiener–Hopf Integral Operator

Mathematical Notes - The problem of the factorization of the Wiener–Hopf integral operator in the form of the product of the upper and lower Volterra operators is considered. Conditions for...     

Exchange the Order of a Successive Integral with Bochner-Martinelli Kernel

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A Class of Singular Integral Equation of Convolution Type with csc(τ-θ) Kernel

In this paper, we propose and discuss a class of singular integral equation of convolution type with csc(τ- θ) kernel in class L2[-π, π]. Using discrete Fourier transform and the lemma, this kind of equations is transformed to discrete system of equations, and then we obtain the solvable conditions and the explicit solutions in class L2[-π, π].     

Symmetric Exponential Integrators with an Application to the Cubic Schrödinger Equation

In this article, we derive and study symmetric exponential integrators. Numerical experiments are performed for the cubic Schrödinger equation and comparisons with classical exponential integrators and other geometric methods are also given. Some of the proposed methods preserve the L ²-norm and/or the energy of the system.     

Nonlinear Evolutionary Schrödinger Equation in the Supercritical Case

Theoretical and Mathematical Physics - We prove that for some initial data, solutions of the Cauchy problem for the nonlinear Schrödinger equation in the supercritical case are destroyed after...     

Geodesics in the Engel Group with a Sub-Lorentzian Metric—— the Space-Like Case

Let $E$ be the Engel group and $D$ be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quasi-pendulum equation by Jacobi functions, and then gets the space-like Hamiltonian geodesics in the Engel group with a sub-Lorentzian metric.     

Geodesics in the Engel Group with a Sub-Lorentzian Metric——the Space-Like Case

Let E be the Engel group and D be a bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, the author constructs a parametrization of a quasi-pendulum equation by Jacobi functions, and then gets the space-like Hamiltonian geodesics in the Engel group with a sub-Lorentzian metric.     

Stationary Spherically Symmetric Solutions of the Vlasov–Poisson System in the Three-Dimensional Case

Doklady Mathematics - We consider the three-dimensional stationary Vlasov–Poisson system of equations with respect to the distribution function of the gravitating matter $$f =...     

An Integral Representation Result for the Γ-Limit of Functionals with Non-standard Growth Conditions in the Case of Elasticity

An integral representation result on regular functions is proved for the o -limit of a sequence of integral functionals defined in the vectorial case and modelled on elasticity theory functional Z z f (( x , e ( u )) dx where convex lagrangians satisfy a non-standard estimate $$ -c_{0} + c_{1} | \xi|^{\alpha }\leq f ( (x,\xi ) \leq c_{0} + c_{2} | \xi|^{\beta },\quad 1 \lt \alpha \leq \beta \lt \frac {n\alpha }{n-\alpha },\enskip c_{0}\geq 0,\enskip c_{1},c_{2} \gt 0. $$ When the limit integrand does not show Lavrent'ev phenomenon the representation result is also true on the whole space W 1, f ( z ; R n ).     

A Cauchy Type Problem for a Degenerate Equation with the Riemann–Liouville Derivative in the Sectorial Case

Siberian Mathematical Journal - Under study is the unique solvability of a Cauchy type problem and a generalized Schowalter–Sidorov type problem for a class of linear inhomogeneous equations...     

On the Wiener–Hopf factorization for Lévy processes with bounded positive jumps

We study the Wiener–Hopf factorization for Lévy processes with bounded positive jumps and arbitrary negative jumps. We prove that the positive Wiener–Hopf factor can be expressed as an infinite product involving solutions to the equation ψ(z)=q$\psi \left(z\right)=q$, where ψ$\psi$ is the Laplace exponent. Under additional regularity assumptions on the Lévy measure we obtain an asymptotic expression for these solutions. When the process is spectrally negative with bounded jumps, we derive a series representation for the scale function. In order to illustrate possible applications, we discuss the implementation of numerical algorithms and present the results of several numerical experiments.     

A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator

A new efficient compact difference scheme is proposed for solving a space fractional nonlinear Schrödinger equation with wave operator. The scheme is proved to conserve the total mass and total energy in a discrete sense. Using the energy method, the proposed scheme is proved to be unconditionally stable and its convergence order is shown to be of $ \mathcal{O}( h^6 + \tau^2) $ in the discrete $ L_2 $ norm with mesh size $ h $ and the time step $ \tau $. Moreover, a fast difference solver is developed to speed up the numerical computation of the scheme. Numerical experiments are given to support the theoretical analysis and to verify the efficiency, accuracy, and discrete conservation laws.     

A Fast, Direct Algorithm for the Lippmann–Schwinger Integral Equation in Two Dimensions

The state-of-the-art, large-scale numerical simulations of the scattering problem for the Helmholtz equation in two dimensions rely on iterative solvers for the Lippmann–Schwinger integral equation, with an optimal CPU time O(m ³ log(m)) for an m-by-m wavelength problem. We present a method to solve the same problem directly, as opposed to iteratively, with the obvious advantage in efficiency for multiple right-hand sides corresponding to distinct incident waves. Analytically, this direct method is a hierarchical, recursive scheme consisting of the so-called splitting and merging processes. Algebraically, it amounts to a recursive matrix decomposition, for a cost of O(m ³), of the discretized Lippmann–Schwinger operator. With this matrix decomposition, each back substitution requires only O(m ² log(m)); therefore, a scattering problem with m incident waves can be solved, altogether, in O(m ³ log(m)) flops.     

Optimal Control for the Convective Cahn–Hilliard Equation in 2D Case

In this paper, for 2D convective Cahn–Hilliard equation, the optimal control problem is considered, the existence of optimal solution is proved and the optimality system is established.