rectifiable and unrectifiable oscillations for a class of second order linear differential equations of euler type the main purpose of the paper is to pay attention to a new kind of oscillations so called rectifiable and unrectifiable oscillations which occur in a class of linear second order differential equations including the famous linear euler differential equation it will be shown that this kind of oscillations only depends on a positive real parameter which appears in a singular term of the main equation as well as in a suitable boundary layer condition

local behavior of solutions of quasilinear elliptic equations with general structure this paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form diva x u vu b x u vu where a and b are borel measurable are solutions to the equation diva x u vu b x u vu u for some nonnegative radon measure i among other things it is shown that if u is a holder continuous solution to this equation then the measure u satisfies the growth property j b x r 1 is given by the structure of the differential operator conversely if u is assumed to satisfy this growth condition then it is shown that u satisfies a harnack type inequality thus proving that u is locally bounded under the additional assumption that a is strongly monotonic it is shown that u is holder continuous

rectifiable oscillations of self adjoint and damped linear differential equations of second order abstract asymptotic and oscillatory behaviours near x 0 of all solutions y y x of self adjoint linear differential equation p p q p y q y 0 on 0 t will be studied where p p x and q q x satisfy the so called hartman wintner type condition we show that the oscillatory behaviour near x 0 of p p q is characterised by the nonintegrability of q p on 0 t moreover under this condition we show that the rectifiable resp unrectifiable oscillations near x 0 of p p q are characterised by the integrability resp nonintegrability of q p 3 4 on 0 t next some invariant properties of rectifiable oscillations in respect to the liouville transformation are proved also sturm s comparison type theorem for the rectifiable oscillations is stated furthermore previous results are used to establish such kind of oscillations for damped linear second order differential equation y g x y f x y 0 and especially the bessel type damped linear differential equations are considered finally some open questions are posed for the further study on this subject

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

minkowski bouligand dimension of solutions of the one dimensional p laplacian the upper minkowski bouligand dimension of the graph of continuous solutions denoted by dimmy is studied for a class of nonlinear one dimensional p laplacian some sufficient conditions on the nonlinearities are given such that dimmy takes the prescribed fractional values next a relation between dimmy and the order of growth for singular behaviour of lp norm of derivatives of solutions is given finally the change and stability of dimmy are considered by means of a double parametric problem

numerical solution of the space fractional fokker planck equation the traditional second order fokker planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of brownian motion densities of stable type have been used to describe the probability distribution of these motions the resulting governing equation of these motions is similar to the traditional fokker planck equation except that the order of the highest derivative is fractional in this paper a space fractional fokker planck equation sffpe with instantaneous source is considered a numerical scheme for solving sffpe is presented using the riemann liouville and grunwald letnikov definitions of fractional derivatives the sffpe is transformed into a system of ordinary differential equations ode then the ode system is solved by a method of lines numerical results for sffpe with a constant diffusion coefficient are evaluated for comparison with the known analytical solution the numerical approximation of sffpe with a time dependent diffusion coefficient is also used to simulate levy motion with stable densities we will show that the numerical method of sffpe is able to more accurately model these heavy tailed motions

lp continuity of wave operators in z abstract we recover for discrete schrodinger operators on the lattice z stronger analogues of the results by weder r weder the w k p continuity of the schrodinger wave operators on the line comm math phys 208 1999 507 520 and by d ancona and fanelli p d ancona l fanelli l p boundedness of the wave operator for the one dimensional schrodinger operator comm math phys 268 2006 415 438 on r

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

stability of the line soliton of the kp ii equation under periodic transverse perturbations we prove the nonlinear stability of the kdv solitary waves considered as solutions of the kp ii equation with respect to periodic transverse perturbations our proof uses a miura transform which sends the solutions of an mkp ii equation to solutions of the kp ii equation

fractal oscillations of self adjoint and damped linear differential equations of second order abstract for a prescribed real number s 1 2 we give some sufficient conditions on the coefficients p x and q x such that every solution y y x y c 2 0 t of the linear differential equation p x y q x y 0 on 0 t is bounded and fractal oscillatory near x 0 with the fractal dimension equal to s this means that y oscillates near x 0 and the fractal box counting dimension of the graph y of y is equal to s as well as the s dimensional upper minkowski content generalized length of y is finite and strictly positive it verifies that y admits similar kind of the fractal geometric asymptotic behaviour near x 0 like the chirp function y ch x a x s x which often occurs in the time frequency analysis and its various applications furthermore this kind of oscillations is established for the bessel chirp and other types of damped linear differential equations given in the form y x y g x y 0 x 0 t in order to prove the main results we state a new criterion for fractal oscillations near x 0 of real continuous functions which essentially improves related one presented in 1

on asymptotic stability in 3d of kinks for the model we add to a kink which is a 1 dimensional structure two transversal directions we then check its asymptotic stability with respect to compactly supported perturbations in 3d and a time evolution under a nonlinear wave equation nlw the problem is inspired by work by jack xin on asymptotic stability in dimension larger than 1 of fronts for reaction diffusion equations the proof involves a separation of variables the transversal variables are treated as in work on nonlinear klein gordon equation nlkg originating from klainerman and from shatah in a particular elaboration due to delort et al the longitudinal variable is treated by means of a result by weder on dispersion for schrodinger operators in id

the w k p continuity of the schrodinger wave operators on the line we prove that the wave operators for the schrodinger equation on the line are continuous on the sobolev spaces w k p 1 p moreover if the potential is exceptional and where f 1 x 0 is a jost solution at zero energy the wave operators are continuous on w k 1 and on w k

on asymptotic stability of moving kink for relativistic ginzburg landau equation we prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the ginzburg landau type in one space dimension for any odd initial condition in a small neighborhood of the kink the solution asymptotically in time is the sum of the kink and dispersive part described by the free klein gordon equation the remainder converges to zero in a global norm

rectifiable oscillations in second order linear differential equations abstract we study the linear differential equation p y x f x y x 0 on i 0 1 where the coefficient f x is strictly positive and continuous on i and satisfies the hartman wintner condition at x 0 the four main results of the paper are i a criterion for rectifiable oscillations of p characterized by the integrability of f x 4 on i ii a stability result for rectifiable and unrectifiable oscillations of p in terms of a perturbation on f x iii the s dimensional fractal oscillations for which we assume also f x c x when x 0 2 and s max 1 3 2 2 and iv the co existence of rectifiable and unrectifiable oscillations in the absence of the hartman wintner condition on f x explicit examples related to the above results are given

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

coupled continuous time random walk approach to the rachev ruschendorf model for financial data abstract in this paper we expand the rachev ruschendorf asset pricing model introducing a coupled continuous time random walk ctrw like form of the random number of price changes such a form results from the concept of the random clustering procedure that resembles the coarse graining methods of statistical physics and on the other hand indicates applicability of the ctrw idea widely used in physics to model anomalous diffusion for describing financial markets in the framework of the proposed model we derive the limiting distributions of log returns and the corresponding pricing formulas for european call option in order to illustrate the obtained theoretical results we present their fitting with several sets of financial data

the unit condition and global existence for a class of nonlinear klein gordon equations abstract global existence for a special class of nonlinear klein gordon equations in two space dimensions is proved in particular it is shown that for small initial data the equation tt u u u u 2 has a global solution

reprint of boundary conditions for fractional diffusion abstract this paper derives physically meaningful boundary conditions for fractional diffusion equations using a mass balance approach numerical solutions are presented and theoretical properties are reviewed including well posedness and steady state solutions absorbing and reflecting boundary conditions are considered and illustrated through several examples reflecting boundary conditions involve fractional derivatives the caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion since the resulting boundary value problem is not positivity preserving

fractal oscillations for a class of second order linear differential equations of euler type abstract the s dimensional fractal oscillations for continuous and smooth functions defined on an open bounded interval are introduced and studied the main purpose of the paper is to establish this kind of oscillations for solutions of a class of second order linear differential equations of euler type next it will be shown that the dimensional number s only depends on a positive real parameter appearing in a singular term of the main equation it continues some recent results on the rectifiable and unrectifiable oscillations given in pasic m pasic rectifiable and unrectifiable oscillations for a class of second order linear differential equations of euler type j math anal appl 335 2007 724 738 and wong j s w wong on rectifiable oscillation of euler type second order linear differential equations electron j qual theory differ equ 20 2007 1 12

on the variational structure of breather solutions i sine gordon equation abstract in this paper we describe stability properties of the sine gordon breather solution these properties are first described by suitable variational elliptic equations which also implies that the stability problem reduces in some sense to i the study of the spectrum of explicit linear systems and ii the understanding of how bad directions if any can be controlled using low regularity conservation laws then we discuss spectral properties of a fourth order linear matrix system using numerical methods we confirm that all spectral assumptions leading to the h 2 h 1 stability of sg breathers are numerically satisfied even in the ultra relativistic singular regime

multidimensional stability of planar travelling waves the multidimensional stability of planar travelling waves for systems of reaction diffusion equations is considered in the case that the diffusion matrix is the identity it is shown that if the wave is exponentially orbitally stable in one space dimension then it is stable for x rn n 2 furthermore it is shown that the perturbation of the wave decays like t n 1 4 as t the result is proved via an application of linear semigroup theory

stability theory of solitary waves in the presence of symmetry ii abstract consider an abstract hamiltonian system which is invariant under a group of operators we continue to study the effect of the group invariance on the stability of solitary waves applications are given to bound states and traveling wave solutions of nonlinear wave equations

modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian the efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics previous wave equations which use time domain fractional operators require the storage of the complete pressure field at previous time steps such operators are convolution based this makes them unsuitable for many three dimensional problems of interest here a wave equation that utilizes two lossy derivative operators based on the fractional laplacian is derived these operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into fourier based pseudospectral and k space methods without the increase in memory required by their time domain fractional counterparts a framework for encoding the developed wave equation using three coupled first order constitutive equations is discussed and the model is demonstrated through several one two and three dimensional simulations

global existence of small amplitude solutions to nonlinear klein gordon equations in four space time dimensions consider the following class of nonlinear wave equations square text u u text text f u u u n k g where t 2 1 2 2 2 3 2 is the d alembertian of the 4 dimensional minkowski space time and f a smooth function of u u t x and its first and second partial derivates vanishing together with its first derivatives at u u u 0 in other words f is at least quadratic in u u u in a neighborhood of the trivial solution u 0 we subject u to the pure initial value problem text u mkern 1mu varepsilon text f x u t varepsilon g x at t 0 i v p with f g c 0 3 and e 0 a small parameter

reflected spectrally negative stable processes and their governing equations this paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one reflected at its infimum first we derive the forward equation using the theory of sun dual semigroups the resulting forward equation is a boundary value problem on the positive half line that involves a negative riemann liouville fractional derivative in space and a fractional reflecting boundary condition at the origin then we apply numerical methods to explicitly compute the transition density of this space inhomogeneous markov process for any starting point to any desired degree of accuracy finally we discuss an application to fractional cauchy problems which involve a positive caputo fractional derivative in time

global existence of small solutions for quadratic quasilinear klein gordon systems in two space dimensions consider a quasi linear system of two klein gordon equations with masses m1 m2 we prove that when m1a2m2 and m2a2m1 such a system has global solutions for small smooth compactly supported cauchy data this extends a result proved by sunagawa j differential equations 192 2 2003 308 in the semi linear case moreover we show that global existence holds true also when m1 2m2 and a convenient null condition is satisfied by the nonlinearities

global existence of small solutions to nonlinear evolution equations abstract we study the global existence and asymptotic behaviour of small solutions of a large class of nonlinear partial differential equations if the nonlinear terms are of high degree the solutions will be asymptotic to solutions of the linear equation

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems