Kinds Of Evolution Equations Research Articles

Upper bounds for the blow up time for the Kirchhoff- type equation

In this research, we take into account the Kirchhoff type equation with variable exponent. The Kirchhoff type equation is known as a kind of evolution equations,namely, PDEs, where t is an independent variable. This type problem can be extensively used in many mathematical models of various applied sciences such as flows of electrorheological fluids, thin liquid films, and so on. This research, we investigate the upper bound for blow up time under suitable conditions.

Hyperbolic free boundary problems and applications to wave-structure interactions

Motivated by a new kind of initial boundary value problem (IBVP) with a free boundary arising in wave-structure interaction, we propose here a general approach to one-dimensional IBVP as well as transmission problems. For general strictly hyperbolic 2 × 2 quasilinear hyperbolic systems, we derive new sharp linear estimates with refined dependence on the source term and control on the traces of the solution at the boundary. These new estimates are used to obtain sharp results for quasilinear IBVP and transmission problems, and for fixed, moving, and free boundaries. In the latter case, two kinds of evolution equations are considered. The first one is of kinematic type in the sense that the velocity of the interface has the same regularity as the trace of the solution. Several applications that fall into this category are considered: the interaction of waves with a lateral piston, and a new version of the well-known stability of shocks (classical and undercompressive) that improves the results of the general theory by taking advantage of the specificities of the one-dimensional case. We also consider fully nonlinear evolution equations characterized by the fact that the velocity of the interface is one derivative more singular than the trace of the solution. This configuration is the most challenging; it is motivated by a free boundary problem arising in wave-structure interaction, namely, the evolution of the contact line between a floating object and the water. This problem is solved as an application of the general theory developed here.

Analysis of (3 + 1)-dimensional Boiti–Leon–Manna–Pempinelli equation via Lax pair investigation and group transformation method

Singular manifolds method (SMM) is utilized to detect the Lax pair and Bäcklund transformation of (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation. An extension of group transformation method with three parameters is exploited in transforming the Lax pair directly to a system of ordinary differential equations (ODEs). Two cases of similarity variables are considered resulting in solvable systems of ODEs. New analytical solutions are attained and graphed containing some soliton waves. Some of these solutions are applicable of plasma explosions and fluid dynamics. Both of investigation of Lax pair of (3+1)-dimensional evolution equations and three parameters group method are seldom utilized in this kind of evolution equations.

Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable

The purpose of this article is to study the Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable. The complexification of the spatial variable in these equations is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding (real) solution by replacing the usual sum of variables (translation) by an exponential product (rotation) and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two dimensional spatial variables are generated and their solutions are constructed.

Schrödinger type equations with real-time variable and complex spatial variables

In recent work, heat and Laplace equations, (un)damped wave equations, the Burgers and the Black–Merton–Scholes equations with real-time variable and complex spatial variable were studied. The purpose of this article is to make a similar study for the Schrödinger equation with real-time variable and complex spatial variable. The complexification of the spatial variable in the case of the Schrödinger equation is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding convolution formula by replacing the usual sum of variables (translation) by an exponential product (rotation) and second, one complexifies the spatial variable in the corresponding evolution equation and then one searches for non-analytic and for analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated and their solutions are constructed. It is of interest to note that in the case of the first method, solutions can be studied by employing the powerful theory of groups of linear operators. Then, we show that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. In the case of the higher order Schrödinger equation, the complexification of the spatial variable is made in the corresponding convolution formula.

Wave and telegraph equations with real time variable and complex spatial variables

In two recent papers [C.G. Gal, S.G. Gal and J.A. Goldstein, Evolution equations with real time variable and complex spatial variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753–774; C.G. Gal, S.G. Gal and J.A. Goldstein, Higher order heat and Laplace type equations with real time variable and complex spatial variable, Complex Var. Elliptic Eqns., 55 (2010), pp. 357–373, the classical heat and Laplace equations with real time variable and complex spatial variable are studied. The purpose of this article is to make a similar study for the classical wave and telegraph equations with real time variable and complex spatial variable. The complexification of the spatial variable in the wave and telegraph equations is made by two different methods which produce different equations. By the former method, we complexify the spatial variable in the corresponding formulas by replacing the usual translations x ± ct, c is the speed of propagation, by the rotations ze ±ict and, by the latter, we complexify the spatial variable in the corresponding evolution equation and then we search for analytic and non-analytic solutions. The first method produces solutions that also preserve some geometric properties of the boundary function, such as the univalence, starlikeness, convexity and spirallikeness. Moreover, new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated from both methods and their solutions are constructed. New physical/probabilistic interpretations of the solutions to these equations are also given.

Nonlinear waves in a fluid-filled thin viscoelastic tube

In the present paper the propagation property of nonlinear waves in a thin viscoelastic tube filled with incompressible inviscid fluid is studied. The tube is considered to be made of an incompressible isotropic viscoelastic material described by Kelvin—Voigt model. Using the mass conservation and the momentum theorem of the fluid and radial dynamic equilibrium of an element of the tube wall, a set of nonlinear partial differential equations governing the propagation of nonlinear pressure wave in the solid—liquid coupled system is obtained. In the long-wave approximation the nonlinear far-field equations can be derived employing the reductive perturbation technique (RPT). Selecting the exponent α of the perturbation parameter in Gardner—Morikawa transformation according to the order of viscous coefficient η, three kinds of evolution equations with soliton solution, i.e. Korteweg—de Vries (KdV)—Burgers, KdV and Burgers equations are deduced. By means of the method of traveling-wave solution and numerical calculation, the propagation properties of solitary waves corresponding with these evolution equations are analysed in detail. Finally, as a example of practical application, the propagation of pressure pulses in large blood vessels is discussed.

Dynamic information theory and information description of dynamic systems

In this paper, we develop dynamic statistical information theory established by the author. Starting from the ideas that the state variable evolution equations of stochastic dynamic systems, classical and quantum nonequilibrium statistical physical systems and special electromagnetic field systems can be regarded as their information symbol evolution equations and the definitions of dynamic information and dynamic entropy, we derive the evolution equations of dynamic information and dynamic entropy that describe the evolution laws of dynamic information. These four kinds of evolution equations are of the same mathematical type. They show in unison when information transmits in coordinate space outside the systems that the time rate of change of dynamic information densities originates from their drift, diffusion and dissipation in state variable space inside the systems and coordinate space in the transmission processes, and that the time rate of change of dynamic entropy densities is caused by their drift, diffusion and production in state variable space inside the systems and coordinate space in the transmission processes. When space noise can be neglected, an information wave will appear. If we only consider the information change inside the systems, dynamic information evolution equations reduce to information equations corresponding to the dynamic equations which describe evolution laws of the above dynamic systems. This reveals that the evolution laws of respective dynamic systems can be described by information equations in a unified fashion. Hence, the evolution processes of these dynamic systems can be abstracted as the evolution processes of information. Furthermore, we present the formulas for information flow, information dissipation rate, and entropy production rate. We prove that the information production probably emerges in a dynamic system with internal attractive interaction between the elements, and derive a formula for this information production rate. Thereby, we obtain an expression for the time rate of total information change of a dynamic system, which is equal to the algebraic sum of the formulas for information dissipation rate, information production rate and information inflow rate, and which exhibits a unified information description for the evolution including degradation and self-organizing evolution of dynamic systems.

Evolution equations with real time variable and complex spatial variables

It is known that if the time variable in the heat and wave equations is complex and belongs to a sector in ℂ, then the theory of analytic semigroups becomes a powerful tool of study. Also, it is known that if both variables, time and spatial, are complex, then e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. The purpose of this article is, in a sense, complementary: to study the complex versions of the classical heat and Laplace equations, obtained by ‘complexifying’ now the spatial variable only (and keeping the time variable real). This ‘complexification’ is studied by two different methods, which produce different equations: first, one complexifies the spatial variable in the corresponding semigroups of operators and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. It is of interest to note that in the case of the first method, besides the fact that the solutions can be studied by using the theory of semigroups of linear operators, also these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. Also, by both methods new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated and their solutions are constructed.

Solutions of nonlinear evolution inclusions

In this paper, the solution of the nonlinear evolution inclusion problem of the form u ′ ( t ) + B ( t , u ( t ) ) ∋ f ( t ) is studied. In this problem, the operators are of type ( M ) or type ( S + ) , which are different from those of pseudo-monotone operators that had been studied by many authors. At the same time, we study the perturbation problem. In fact, many kinds of evolution equations can be generalized by this problem. The former results are improved and generalized by our conclusions, and we will give more applications.

Aq-Schrödinger equation based on a Hopfq-deformation of the Witt algebra

In an earlier paper a q-Schrodinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q-deformation of the Witt algebra. This q-deformation is a deformation in the category of Lie algebras and hence the corresponding q-Witt algebra has a trivial Hopf algebra structure. In this paper, we extend the above algebra by the addition of a set of shift-type generators, which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics. The latter gives rise to a new kind of evolution equation and it is shown that in the limit q1 a specific class of Schrodinger equations is obtained from it. This specification of a particular class is a new phenomenon, because in earlier references, where a different q-deformation has been implemented or no deformation has been used at all, such a class could not be determined uniquely. The extended algebra used here has a nontrivial Hopf structure. The appearance of the shift-type generator in the q-deformed picture hence leads to a selection of a particular type of dynamics and delivers in the limit q1 new information for the characterization of the corresponding dynamics in the undeformed situation.

Usage de l'algebre de lie su(n) dans l'etude des systems à n etats. VI. Equations linéaires d'evolution positive

AbstractThe n‐level quantum system density‐operators evolution is not usually given by the Liouville equation, but by a more general one. This equation must keep density‐operators trace, hermiticity, and positiveness. These three conditions restrict the available kinds of evolution equations. In this paper we investigate linear equations for systems without memory effects. By using the first two conditions and the formalism we introduced in earlier papers, the evolution equation takes the form of a first order differential equation concerning the n2 – 1 dimension “coherence‐vector.” The third one is the essential object of this paper. Moreover, we obtain a cononical splitting of this equation into four parts that may be separately studied.