The aim of this paper is twofold. Firstly, we consider the existence of solutions to a type of complex functional-differential equations
(w')nw(n)=awn+1(g)+bw+d
in complex variables. We obtain g is linear when w is a transcendental meromorphic function and a≠0, b, d are constants. In addition, due to the different properties between equations and system of equations, it is meaningful to research systems of equations, this paper is also concerned with a type of system of functional equations, properties of meromorphic solutions are obtained under some proper conditions. Examples are constructed to show that our results are accurate.
Base on the super Lie algebra osp(2/2), we construct the (2+1)-dimensional supersymmetric integrable equations by means of two approaches. One of the technique is in terms of homogeneous spaces of super-Lie algebra, and in the other one, extending the dimension of the system has been used. Moreover, we derive the Bäcklund transformations for the (2+1)-dimensional supersymmetric integrable equations.
Given a homeomorphism of the real line, we define its quasisymmetric exponent, Hölder exponent and algebraic exponent. These exponents capture the local behavior of a homeomorphism and are useful in the study of quasisymmetric maps and quasiconformal maps. In this paper we shall explore the relations among these exponents and give some examples.
Let B be the unit ball in Cn. In this paper, the authors characterize the boundedness of the Bergman type operators Ta, b on the logarithmic weight general function spaces F(p, q, s, k) or from spaces A(p, q, s, k) to spaces L(p, q, s, k).
For harmonic mappings $ f_{i}(z)=h_{i}(z)+\overline{g_{i}(z)}$($ i=1, 2$) defined in the unit disk satisfying the given coefficient conditions, we consider the radii of full convexity and full starlikeness of order $\alpha $ for the convex combination $ (1-t)L^{\epsilon}_{f_{1}}+tL^{\epsilon}_{f_{2}}$, where $ L^{\epsilon}_{f_{i}}=z\frac{\partial f_{i}}{\partial z}-\epsilon\overline{z}\frac{\partial f_{i}}{\partial\overline{z}}(|\epsilon|=1)$ denotes the differential operator of $ f_{i}$. In addition, we obtain the radii of fully convex and full starlikeness of order $\alpha $ for convolution of harmonic mappings under the differential operator. All results are sharp.
A food chain model in the unstirred chemostat with toxins is studied. The stability of the trivial solution and semi-trivial solution is analyzed by means of the stability theory, and a priori estimate of positive solution is given by the maximum principle and the super and sub-solution method. Then, by using the fixed point index theory, the sufficient conditions for the existence of positive solutions are achieved. Finally, the effect of the toxins on the dynamic behavior is discussed by virtue of the perturbation theory and bifurcation theory, and the stability and uniqueness of positive solutions are obtained. The results show that the species can coexist when the growth rates of the microorganisms u and v are larger in the presence of the toxins. Furthermore, if the effect of the toxins is sufficiently large, the system has unique stable positive solution when the growth rate of the microorganism v belongs to a certain range.
In this paper, we study a class of sublinear biharmonic equation with p-Laplacian and Neumann boundary condition. By using a dual approach, we prove the existence and multiplicity of sign-changing solutions for the problem.
In this paper, we consider a class of singularly perturbed boundary value problem with integral boundary condition. Based on the singularly perturbed geometric theory, the existence of step-like contrast structure solution is proved. By virtue of the structure of the solution, we construct the uniformly valid formal asymptotic solution by the boundary layer function method. Finally, an example is given to show the main result.
In this paper, we consider the Cauchy problem of a weakly dissipative modified two-component Dullin-Gottwald-Holm (mDGH2) system. The local well-posedness and the global existence are analyzed, which is to prove that blow-up phenomena cannot happen under the condition $\left(\|y_{0}\|_{L^{2}}^{2}+\|\rho_{0}\|^{2}_{L^{2}}\right)^{\frac{1}{2}}<\frac{4\lambda} {3}.$ We derive the precise blow-up scenario, and then provide several criteria guaranteeing the blow-up of the solutions to the weakly dissipative mDGH2 system. It is worth noting that the solution to the weakly dissipative system is not affected by the weakly dissipative term.
In this paper, we investigate the symmetry property of solutions of the fractional Laplacian with convex nonlinearities. The main result is that all entire weak solutions of the above problem of index 1 on the ball or the annular domain are axially symmetric if the nonlinearity is strictly convex with respect to the solution.
In this paper, we study the bifurcation of positive solutions about parameter $\lambda$ for the quasilinear elliptic equations with Φ-Laplacian operator and concave-convex nonlinearities by using the critical point theory, appropriate truncation and comparison techniques. Furthermore, we obtain the existence of the smallest positive solution and the monotonicity with respect to parameter $\lambda$.
The transient heat conduction equation defined in a three-dimensional semi-infinite cylinder is Considered, in which the nonlinear conditions are imposed on the finite end and the lateral surface of the cylinder. A partial differential inequality for the "energy expression" is established after some constraints are imposed on the boundary conditions, and Phragmén-Lindelöf type results of the heat conduction equation is obtained from the inequality. In the case of decay, it is proved that "total energy" can be controlled by known data.
In this paper, we are concerned with the N(N ≥ 2)-dimensional exterior problem for a class of semilinear wave equations with variable coefficient nonlinearity. We mainly consider the blow up and lifespan of the solutions. Base on [19-20], by utilizing the test function and the harmonic functions of N=2 and N ≥ 3 in the exterior domains with Dirichlet boundary condition, we are able to derive the upper bound of lifespan. In particular, the impact of the variable coefficient nonlinearity on the lifespan has been discussed in detail.
In the article, we provide a necessary and sufficient condition such that the complete p-elliptic integral of the second kind is concave with respect to Hölder mean, which is a generalization of the previously known result on the complete elliptic integral of the second kind.
Existing works on approximate controllability of fractional differential equations often assume that the nonlinear item is uniformly bounded and the corresponding fractional linear system is approximate controllable, which is, however, too constrained. In this paper, we omit these two assumptions and investigate the approximate controllability of Hilfer fractional integro-differential equations using sequence method.
The present paper is concerned with the Weyl classification of second order singular Sturm-Liouville equations with nonlocal point potential. We give the Weyl classification for these equations and give sufficient and necessary conditions for the division of these two kinds. Furthermore, the most important part is the situation of square integrable solutions for λ on the real axis, which has essential differences with the classical equation, and corresponding sufficient and necessary conditions are also obtained.
In this paper, we propose a deformed Boussinesq-type integrable hierarchy of nonlinear evolution equations associated with a 3×3 matrix spectral problem by using the zero-curvature equation. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first two members in the hierarchy. Some explicit solutions to the first deformed Boussinesq-type equation are given by utilizing the Darboux transformation.
In this paper, we first derive some blow-up results for two ordinary differential inequalities with variable coefficients, which are the generalizations of Theorem 3.1 in Li and Zhou[3]. Second, as an application of the improved ordinary differential inequality, we consider the Cauchy problem for the semilinear wave equation with scale-invariant damping and deduce the upper bound of the lifespan for the case $\mu>1$ and $ 1 < p < 1+\frac{2}{n}$ under some suitable assumptions for the initial data. The method for the latter result is due to Lai and Zhou[11].
In this paper, an efficient reproducing kernel method(RKM) is proposed for solving the piecewise smooth BVP. By defining an operator $ {\cal L}:{W_{2}^{3}[0, 1]}\rightarrow {L_{2}[0, 1]}$ and applying the reproducing kernel property, we have skillfully solved the complex piecewise smooth BVP. The theorems show the accuracy of the algorithm. Furthermore, we get that the approximate solution $u_{n}(x)$ convergence to the exact one with order $O(h^2)$. That is, $u_{n}(x)$ has second order convergence in the sense of norm $\left\|\displaystyle. \right\|_{W_2^{3}}$. Finally, some numerical experiments are given to illustrate the algorithm is accuracy, simple and efficient.
In this paper, we consider the complex Ginzburg-Landau equations with hereditary effects and the nonlinear term satisfying the polynomial growth of arbitrary $p-1$ $(p>2)$ order. We analyze the well-posedness of solutions and prove the existence of the pullback attractors in $C_{L^{2}(\Omega)}$ by applying the contractive functions method.
This paper takes advantages of the Discontinuous Galerkin (DG) method and Lagrangian scheme to present a second-order Runge-Kutta(RK) DG method for solving Lagrangian compressible Euler equations on unstructured triangular meshes. The method is more succinct than other fully Lagrangian schemes with the Jacobian matrix associated with the map between Lagrangian and Eulerian spaces, the solver of vertex velocity in the method has good adaptability for many problems. Numerical examples are presented to illustrate the robustness and second-order accuracy of the scheme.
In this paper, a fully discrete viscosity-splitting finite element method is developed and studied for the fluid-fluid interaction model. This method applies decomposition technique of viscosity in time and mixed finite element method in space, where the temporal term includes two steps. In the first step, a backward Euler scheme is utilized for the temporal discretization, semi-implicit scheme is applied for the nonlinearity term and the geometric averaging method is used to deal with the fluid interface. Then, in the second step, we only solve a linear Stokes problem without spatial iteration per time step for each individual domain. Hence, the viscosity-splitting finite element method splits nonlinearity and incompressibility. Moreover, the stability and convergence of the method are established by rigorous analysis. Finally, numerical experiments are presented to show the performance of the proposed method.
In order to give the algorithm's ability to distinguish general parameters, this paper proposes the concept of algorithm parameter resolution. This paper combines the idea of clustering to give the definition and calculation method of parameter resolution. The least squares estimation and the total least absolute deviations estimation method are used to analyze the parameter resolution of unary linear regression model. Experimental results show that both algorithms have properties:as the SNR increases, the accuracy of the parameter resolution is higher; the local parameter resolution is consistent with the overall parameter resolution; the standard deviation of noise and parameter resolution of least squares satisfy a linear relationship which has been proved by using interval estimation theory. Finally, the least squares and the total least absolute deviations are used to estimate the parameter resolution of two similar audio signals. The experimental results illustrate the rationality and effectiveness of the definition of parameter resolution. Parameter resolution is a criterion for measuring whether two similar signals can be separated, it is also an effective indicator for evaluating the accuracy of models and algorithms.
Unified formulation of charge-conserving current assignment in the Electromagnetic Particle-in-Cell simulation in two and three dimensional Cartesian geometry is proposed. This formulation is adapt for macroparticle with constant charge distribution. From the law of charge conservation, it is revealed that the involving current densities caused by the movement of a macroparticle with certain charge distribution satisfy a linear algebraic system which is usually underdetermined. By solving the linear system, all the possible current assignments could be obtained as the general solution of the system and any charge-conserving current assignment scheme is a special solution of the system. When a macroparticle with constant shape function moving in two dimensional Cartesian geometry three possible situations are presented, and when the macroparticle moving in three dimensional Cartesian geometry, the general and simplest situation is picked. The corresponding linear algebraic systems are listed and solved, and the concrete solution expressions are given. Three prevailing charge-conserving methods as three special solutions of these expressions are listed and analysed to give the corresponding parameters.
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