correlation structure of fractional pearson diffusions the stochastic solution to a diffusion equations with polynomial coefficients is called a pearson diffusion if the first time derivative is replaced by a caputo fractional derivative of order less than one the stochastic solution is called a fractional pearson diffusion this paper develops an explicit formula for the covariance function of a fractional pearson diffusion in steady state in terms of mittag leffler functions that formula shows that fractional pearson diffusions are long range dependent with a correlation that falls off like a power law whose exponent equals the order of the fractional derivative

large order perturbation expansions for the charged harmonic oscillator the charged oscillator defined by the hamiltonian h d2 dr2 r2 lambda r in the domain 0 infinity is a particular case of the family of spike oscillators which does not behave as a supersingular hamiltonian this problem is analysed around the three regions lambda to infinity lambda to 0 and lambda to infinity by using rayleigh ritz large order perturbative expansions a path is found to connect the large lambda regions with the small lambda region by means of the renormalization of the series expansions in lambda finally the riccati pade method is used to construct an implicit expansion around lambda to 0 which extends to very large values of mod lambda mod

nonperturbative treatment of singular potentials a nonperturbative but absolutely convergent algorithm is applied to the determination of the eigenfunctions of hamiltonian with a singular potential of the form h d2 dx2 x2 x in the domain 0 which obey dirichlet boundary conditons the formal structure of the algorithm is identical to that of the lanczos algorithm when it is formally extended to self adjoint operators

matrix elements for a generalized spiked harmonic oscillator closed form expressions for the singular potential integrals m x n are obtained with respect to the gol dman and krivchenkov eigenfunctions for the singular potential bx2 a x2 b 0 a 0 the formulas obtained are generalizations of those found earlier by use of the odd solutions of the schrodinger equation with the harmonic oscillator potential aguilera navarro et al j math phys 31 99 1990

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

study of the singular anharmonic potentials by means of the analytic continuation method here we study the singular anharmonic potentials by applying the analytic continuation method of holubec and stauffer 1985 in order to do that we have developed several approximations to the problem because this method cannot be applied when the solution has essential singularities at any point of its domain all the options here shown have the same precision giving us the eigenvalues correct to all the decimal places provided by the computer

solving the fractional nonlinear bloch system using the multi step generalized differential transform method in this paper the multi step differential transform method is employed for the first time to solve a time fractional nonlinear bloch system the nonlinear bloch equation is known to govern the dynamics of an ensemble of spins controlling the basic process of nuclear magnetic resonance nmr this nonlinear bloch equation is formed from a system of nonlinear ordinary differential equations of fractional order the fractional derivatives are described in the caputo sense the results obtained are in good agreement with the ones in the open literature and it is shown that the technique introduced here is robust efficient and easy to implement

a basis for variational calculations in d dimensions in this paper we derive expressions for matrix elements i hj for the hamiltonian h qa q rq in d 2 dimensions the basis functions in each angular momentum subspace are of the form the matrix elements are given in terms of the gamma function for all d the significance of the parameters t and p and scale s are discussed applications to a variety of potentials are presented including potentials with singular repulsive terms of the form r 0 perturbed coulomb potentials d r br ar2 and potentials with weak repulsive terms such as r2 r4 0

singular anharmonicities and the analytic continued fractions the potential v r 2r2 2r 4 is investigated and its bound states are constructed by a generalized hill determinant method it is shown that the binding energy e and another free parameter in the underlying laurent series representation of the wave function may be obtained from a coupled pair of hill determinant equations

discrete direct methods in the fractional calculus of variations finite differences as a subclass of direct methods in the calculus of variations consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem this article generalizes the same idea for fractional variational problems we consider a minimization problem with a lagrangian that depends on the left riemann liouville fractional derivative using the grunwald letnikov definition we approximate the objective functional in an equispaced grid as a multi variable function of the values of the unknown function on mesh points the problem is then transformed to an ordinary static optimization problem the solution to the latter problem gives an approximation to the original fractional problem on mesh points

quantum anharmonic oscillators a new approach the determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy namely the wronskian of two solutions of the schr dinger equation which are regular respectively at the origin and at infinity we show in this paper how to evaluate that wronskian exactly except for numerical rounding errors the procedure is illustrated by application to the gx2 x2n n a positive integer oscillator

spiked harmonic oscillators a complete variational treatment is provided for a family of spiked harmonic oscillator hamiltonians h d2 dx2 bx2 x b 0 0 for arbitrary 0 a compact topological proof is presented that the set s n of known exact solutions for 2 constitutes an orthonormal basis of the hilbert space l2 0 closed form expressions are derived for the matrix elements of h with respect to s these analytical results and the inclusion of a further free parameter facilitate optimized variational estimation of the eigenvalues of h to high accuracy

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

modeling and numerical analysis of fractional order bloch equations this paper deals with the bloch equations which are a set of macroscopic equations that are used for modeling of nuclear magnetization as a function of time these equations were introduced by felix bloch in 1946 and they are used for a description of the nuclear magnetic resonance nmr this physical phenomenon is used in medicine chemistry physics and engineering to study complex material fractional order generalization of the bloch equations was presented by richard magin et al in 2008 as an opportunity to extend their use to describe a wider range of experimental situations involving heterogeneous porous or composite materials in this paper we describe numerical and simulation models created for matlab simulink of the classical and the fractional order bloch equations the behaviour and stability analysis of the bloch equations are presented as well

a simple numerical method for singular potentials a one parameter family of coordinate transformations is shown to lead to a simple finite difference method which gives highly accurate energies and expectation values for the schrodinger equation in which the potential consists of a smooth term plus a perturbing term which is singular at the origin the method is effective down to very small values of the perturbation parameter and supplements the previously reported perturbation approach which is valuable for large values

fractional bloch equation with delay in this paper we investigate a fractional generalization of the bloch equation that includes both fractional derivatives and time delays the appearance of the fractional derivative on the left side of the bloch equation encodes a degree of system memory in the dynamic model for magnetization the introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation the analysis of this system shows different stability behavior for the t 1 and the t 2 relaxation processes the t 1 decay is stable for the range of delays tested 1 100 ms while the t 2 relaxation in this model exhibited a critical delay typically 6 ms above which the system was unstable delays are expected to appear in nmr systems in both the system model and in the signal excitation and detection processes therefore by including both the fractional derivative and finite time delays in the bloch equation we believe that we have established a more complete and more realistic model for nmr resonance and relaxation

on the eigenvalues of the s state radial equation of a spiked harmonic oscillator eigenvalues of the ground state of the radial schrodinger equation for a spiked harmonic oscillator potential have been evaluated employing two methods numerically via the lanczos grid technique and by means of standard pade approximants constructed from an expansion of large coupling parameter series for the energy numerical results are compared for several values of the parameters characterizing the spiked singular potential

nonsingular spiked harmonic oscillator a perturbative study of a class of nonsingular spiked harmonic oscillators defined by the hamiltonian h d2 dr 2 r 2 r in the domain 0 is carried out in the two extremes of a weak coupling and a strong coupling regimes a path has been found to connect both expansions for near 2

part of the d dimensional spiked harmonic oscillator spectra the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros interdimensional degeneracies emerging from the isomorphism between the angular momentum and the dimensionality of the central force schrodinger equation are used to construct part of the d dimensional spiked harmonic oscillator bound state spectra pslet results are found to compare excellently with those from direct numerical integration and generalized variational methods

on the power series construction of bound states i the energies as zeros of the infinite hill determinants for a broad class of potentials we show that the nonvariational so called hill determinant intuitive identification of binding energies with zeros of certain infinite determinants may be given a rigorous mathematical foundation the essence of the construction lies in an appropriate restriction of the class of the admissible ansatze this eliminates the undesirable confluence of the physical and unphysical components of r that may take place in the general case

energy bounds for a class of singular potentials and some related series perturbation expansions up to third order for the generalized spiked harmonic oscillator hamiltonians h d2 dx2 x2 a x2 x a 0 2 1 1 2 1 4a and small values of the coupling 0 are developed upper and lower bounds for the eigenvalues are computed by the procedure of burrows et al 1987 j phys a math gen 20 889 97 for assessing the accuracy of a truncated perturbation expansion closed form sums for some related perturbation double infinite series then immediately follow as a result of this investigation

study of anharmonic singular potentials a simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a hamiltonian h with singular potentials closed form analytic expressions in n dimensions are obtained for the matrix elements of h with respect to the eigenfunctions of a soluble singular problem with two free parameters a and b the matrix eigenvalues are then optimized with respect to a and b for a given n applications convergence rates and comparisons with earlier work are discussed in detail

eigenvalue bounds for a class of singular potentials in n dimensions the eigenvalue bounds obtained earlier hall r l and saad n 1998 j phys a math gen 31 963 for smooth transformations of the form are extended to n dimensions in particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential and is valid for all discrete eigenvalues arbitrary angular momentum l and spatial dimension n

unified qlm for regular and arbitrary singular potentials a parameterless numerical implementation of the quasilinearization method qlm is constructed and tested to 21 25 digits precision to give quadratically convergent energies e of the klauder effect exhibiting spiked harmonic oscillator with the r or exp r type spikes in a riccati reformulation of the schrodinger equation the radial solution is uniformly quadratic convergent to the same precision as e except in the small minorization interval where the self correcting property of qlm assures geometric convergence like in the picard algorithm to about 12 16 digits sufficient not to affect the convergence of e confirming what is expected on physical grounds it was shown before that for regular potentials immediate onset of quadratic convergence is guaranteed by the initial iteration of the wkb form and that for quadratic convergence of e for power type spikes it suffices to augment this by a nonlinear integration point distribution and by minorization of negative solution values the form of the riccati equation used allows the minorization function to easily be formally defined over the entire interval without the need for a cutoff radius of application and dependence on its scale factor is plateau like and negligible

transient chaos in fractional bloch equations the bloch equation provides the fundamental description of nuclear magnetic resonance nmr and relaxation t 1 and t 2 this equation is the basis for both nmr spectroscopy and magnetic resonance imaging mri the fractional order bloch equation is a generalization of the integer order equation that interrelates the precession of the x y and z components of magnetization with time and space dependent relaxation in this paper we examine transient chaos in a non linear version of the bloch equation that includes both fractional derivatives and a model of radiation damping recent studies of spin turbulence in the integer order bloch equation suggest that perturbations of the magnetization may involve a fading power law form of system memory which is concisely embedded in the order of the fractional derivative numerical analysis of this system shows different patterns in the stability behavior for a near 1 00 in general when a is near 1 00 the system is chaotic while for 0 98 a 0 94 the system shows transient chaos as the value of a decreases further the duration of the transient chaos diminishes and periodic sinusoidal oscillations emerge these results are consistent with studies of the stability of both the integer and the fractional order bloch equation they provide a more complete model of the dynamic behavior of the nmr system when non linear feedback of magnetization via radiation damping is present

spiked oscillators exact solution a procedure to obtain the eigenenergies and eigenfunctions of a quantum spiked oscillator is presented the originality of the method lies in an adequate use of asymptotic expansions of wronskians of algebraic solutions of the schrodinger equation the procedure is applied to three familiar examples of spiked oscillators

eigenvalue bounds for a class of singular potentials we study smooth transformations of the solvable potentials eigenvalue approximation formulae are obtained which provide lower or upper energy bounds for all the discrete energy eigenvalues accordingly as the transformation functions g and f are both convex or both concave detailed results are presented for the special case of two term singular potentials of the form and also for the potentials and for

large and entire large solutions for a class of nonlinear problems abstract in this paper our main purpose is to establish the existence of entire large positive solutions to the quasilinear equation p u x h u we give also necessary conditions for the existence and nonexistence of such solutions we distinguish the cases when is radial and when it is nonradial

chaotic attractors in incommensurate fractional order systems in this paper based on the stability theorems in fractional differential equations a necessary condition is given to check the existence of 1 scroll 2 scroll or multi scroll chaotic attractors in a fractional order system this condition is proposed for incommensurate order systems in general but in the special case it converts to the condition given in the previous works for the commensurate fractional order systems though the presented condition is only a necessary and not sufficient condition for the existence of chaos it can be used as a powerful tool to distinguish for what parameters and orders of a given fractional order system chaotic attractors can not be observed and for what parameters and orders the system may generate chaos it can also be used as a tool to confirm or reject results of a numerical simulation some of the numerical results reported in the previous literature are confirmed by this tool

sustaining stable dynamics of a fractional order chaotic financial system by parameter switching in this paper a simple parameter switching ps methodology is proposed for sustaining the stable dynamics of a fractional order chaotic financial system this is achieved by switching a controllable parameter of the system within a chosen set of values and for relatively short periods of time the effectiveness of the method is confirmed from a computer aided approach and its applications to chaos control and anti control are demonstrated in order to obtain a numerical solution of the fractional order financial system a variant of the grunwald letnikov scheme is used extensive simulation results show that the resulting chaotic attractor well represents a numerical approximation of the underlying chaotic attractor which is obtained by applying the average of the switched values moreover it is illustrated that this approach is also applicable to the integer order financial system

connection factors in the schrodinger equation with a polynomial potential given a second order differential equation with two singular points namely the origin and infinity the connection factors allow to split a power series solution into formal solutions with known asymptotic behavior a procedure is suggested to obtain those factors as quotients of wronskians of the mentioned solutions in the case of a schrodinger equation with a polynomial potential application of the procedure to particular cases whose connection factors are already known allows us to obtain new relations for quotients and products of gamma functions

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

existence non existence and asymptotic behavior of blow up entire solutions of semilinear elliptic equations abstract we are mainly concerned with existence non existence and the behavior at infinity of non negative blow up entire solutions of the equation u x f u in r n no monotonicity condition is assumed upon f and in fact we obtain solutions with a prescribed behavior both at infinity and at the origin the method used to get existence is based upon lower and upper solutions techniques while for non existence we explore radial symmetry estimates on an associated integral equation and the keller osserman condition

chaos in fractional ordered liu system abstract present paper deals with fractional version of a dynamical system introduced by c liu l liu and t liu c liu l liu t liu a novel three dimensional autonomous chaos system chaos solitons fractals 39 4 2009 1950 1958 numerical investigations on the dynamics of this system have been carried out properties of the system have been analyzed by means of lyapunov exponents furthermore the minimum effective dimensions have been identified for chaos to exist in commensurate and incommensurate orders it is noteworthy that the results obtained are consistent with the analytical conditions given in the literature