This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves by a penetrable, inhomogeneous, Lipschitz obstacle covered with a thin layer of high conductivity. The well posedness of the direct problem is established by the variational method. The inverse problem is also considered in this paper. Under certain assumptions, a uniqueness result is obtained for determining the shape and location of the obstacle and the corresponding surface parameter $\lambda (x)$ from the knowledge of the near field data, assuming that the incident fields are electric dipoles located on a large sphere with polarization $p\in {ℝ}^{\mathrm{3}}$. Our results extend those in the paper by F. Hettlich (1996) to the case of inhomogeneous Lipschitz obstacles.

This paper considers a perturbed Markov-modulated risk model with two-sided jumps, where both the upward and downward jumps follow arbitrary distribution. We first derive a system of differential equations for the Gerber-Shiu function. Furthermore, a numerical result is given based on Chebyshev polynomial approximation. Finally, an example is provided to illustrate the method.

We study the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation $-{{𝒟}_{\mathbf{t}}}^{\mu }x(t)$ = $\lambda f(t,x(t),{{𝒟}_{\mathbf{t}}}^{{\mu }_1}x(t),{{𝒟}_{\mathbf{t}}}^{{\mu }_2}x(t),\dots ,{{𝒟}_{\mathbf{t}}}^{{\mu }_{n-1}}x(t))\dots {{𝒟}_{\mathbf{t}}}^{{\mu }_i}x(0)$ = $0,1\le i\le n-1,{{𝒟}_{\mathbf{t}}}^{{\mu }_{n-1}+1}x(0)=0,{{𝒟}_{\mathbf{t}}}^{{\mu }_{n-1}}x(1)={\sum }_{j=1}^{m-2}{a}_j{{𝒟}_{\mathbf{t}}}^{{\mu }_{n-1}}x({\xi }_j),$ where , $n\ge 3$, , , satisfying , ${{𝒟}_{\mathbf{t}}}^{\mu }$ is the standard Riemann-Liouville derivative, $f\in C((\mathrm{0,1})\times {ℝ}^n,(-\infty ,+\infty ))$, and $f$ is allowed to be changing-sign. By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.

By using the tanh-coth method, we obtained some travelling wave solutions of two well-known nonlinear Sobolev type partial differential equations, namely, the Benney-Luke equation and the higher-order improved Boussinesq equation. We show that the tanh-coth method is a useful, reliable, and concise method to solve these types of equations.

A Godunov-mixed finite element method on changing meshes is presented to simulate the nonlinear Sobolev equations. The convection term of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and the diffusion term by an expanded mixed finite element method. The method can simultaneously approximate the scalar unknown and the vector flux effectively, reducing the continuity of the finite element space. Almost optimal error estimates in $L^2$-norm under very general changes in the mesh can be obtained. Finally, a numerical experiment is given to illustrate the efficiency of the method.

This work is concerned with an initial boundary value problem for a nonlocal porous medium equation with inner absorption and weighted nonlocal boundary condition. We obtain the roles of weight function on whether determining the blowup of nonnegative solutions or not and establish the precise blow-up rate estimates under some suitable condition.

We study the existence and uniqueness of nontrivial solutions for a class of fractional differential system involving the Riemann-Stieltjes integral condition, by using the Leray-Schauder nonlinear alternative and the Banach contraction mapping principle, some sufficient conditions of the existence and uniqueness of a nontrivial solution of a system are obtained.

This paper solves the problem of logistics node space relationship beyond expression based on computer vision technology, proposes internal layout optimization mathematical model of logistics node on the basis of overall consideration of function zone geometry shape, the optimal area utilization rate, and the minimum material handling cost, and then designs a highly mixed genetic simulated annealing algorithm based on multiagent to get layout solution. Through contrasting, the result has shown that the model and algorithms put forward in this paper can realize large-scale internal layout optimization of logistics node under the conditions of complex terrain and multiple constraints.

We study the existence of entire positive solutions for the semilinear elliptic system with quadratic gradient terms, $\mathrm{\Delta }{u}_i+|\nabla u_i{|}^2=p_i(|x|)f_i(u_1,u_2,\dots ,u_d)$ for $i=\mathrm{1,2},\dots ,d$ on $R^N,N\ge 3$ and $d\in \{\mathrm{1,2},3,\dots \}$. We establish the conditions on $p_i$ that ensure the existence of nonnegative radial solutions blowing up at infinity and also the conditions for bounded solutions on the entire space. The condition on $f_i$ is simple and different to the Keller-Osserman condition.

This paper presents a combined mathematical treatment for a special automatic music transcription system. This system is specially made for computer-synthesized music. The combined mathematical treatment includes harmonic selection, matrix analysis, and probability analysis method. The algorithm reduces dimension by PCA and selects candidates first by human auditory model and harmonic structures of notes. It changes the multiple-F0 estimation question into a mathematical problem and solves it in a mathematical way. It can be shown in this paper that the experimental results indicate that this method has very good recognition results.

We generalize a fixed point theorem in partially ordered complete metric spaces in the study of A. Amini-Harandi and H. Emami (2010). We also give an application on the existence and uniqueness of the positive solution of a multipoint boundary value problem with fractional derivatives.

We study a fractional differential equation dynamics model arising from the analysis of real estate asset securitization by using the generalized fixed point theorem for weakly contractive mappings in partially ordered sets. Based on the analysis for the existence and uniqueness of the solution and scientific numerical calculation of the solution, in further study, some optimization schemes for traditional risk control process will be obtained, and then the main results of this paper can be applied to the forefront of research of real estate asset securitization.

This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.

The bifurcations near a primary homoclinic orbit to a saddle-center are investigated in a 4-dimensional reversible system. By establishing a new kind of local moving frame along the primary homoclinic orbit and using the Melnikov functions, the existence and nonexistence of 1-homoclinic orbit and 1-periodic orbit, including symmetric 1-homoclinic orbit and 1-periodic orbit, and their corresponding codimension 1 or codimension 3 surfaces, are obtained.

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the space $C([0,\infty )$ × $\text{R})$ $\cap L^{\infty }([0,\infty )$; $H^1(\text{R}))$ provided that its initial value $u_0(x)$ belongs to the space $H^1(R)$. A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

We investigate the positive solutions of a class of second-order nonlinear singular differential equations with multi-point boundary value conditions on an infinite interval in Banach spaces. The tools we used are the cone theory and Mönch fixed point theorem and a monotone iterative technique. An example is also given to demonstrate the applications of our results, which include and extend some existing results.

We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system $-{{𝒟}_{\mathbf{t}}}^{\alpha }x(t)=f(t,x(t),x\text{'}(t),x”(t),\dots ,x^{(n-2)}(t))$, , $x(0)=x\text{'}(0)=\cdots =x^{(n-2)}(0)=0$, $x^{(n-2)}(1)={\int }_0^1{x}^{(n-2)}(s)dA(s)$, where , $n\in \mathbb{N}$ and $n\ge 2$, ${{𝒟}_{\mathbf{t}}}^{\alpha }$ is the standard Riemann-Liouville derivative, ${\int }_0^{1}x(s)dA(s)$ is linear functionals given by Riemann-Stieltjes integrals, $A$ is a function of bounded variation, and $dA$ can be a changing-sign measure. The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal integral boundary value problem for fractional differential equation are obtained. Our analysis relies on Schauder's fixed-point theorem and upper and lower solution method.

We consider the 3D fluid system with the linear Soret effect. We obtain a logarithmically improved regularity criterion in the BMO space.

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid size $H$. The nonlinearities are expanded about the coarse grid solution on a fine gird of size $h$, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with the $H^1$-norm $O(h+H^2)$ which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfy $h=O(H^2)$. An example is also given to illustrate the theoretical results.

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense: $U_{ad}=\{u\in X;{\displaystyle \int {\mathrm{\Omega }}_{U}u⩾0,t\in [0,T]}\}$. Then the a posteriori error estimates in $L^{\infty }(0,T;H^1(\Omega ))$-norm and $L^2(0,T;L^2(\Omega ))$-norm for both the state and the control approximation are given.