time discretization of an integro differential equation of parabolic type the stability and convergence properties of two time discretizations of an integro differential equation of parabolic type are studied the methods reduce to the backward euler and crank nicolson methods if the integral term is absent the integral term is approximated in each case by a quadrature rule with relatively high order truncation error so that a relatively large time step can be used for the quadrature in order to reduce the memory and computational requirements of the method

discontinuous galerkin method for an integro differential equation modeling dynamic fractional order viscoelasticity an integro differential equation modeling dynamic fractional order viscoelasticity with a mittag leffler type convolution kernel is considered a discontinuous galerkin method based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem stability estimates of the discrete problem are proved that are used to prove optimal order a priori error estimates the theory is illustrated by a numerical example

a second order accurate numerical method for a fractional wave equation we study a generalized crank nicolson scheme for the time discretization of a fractional wave equation in combination with a space discretization by linear finite elements the scheme uses a non uniform grid in time to compensate for the singular behaviour of the exact solution at t 0 with appropriate assumptions on the data and assuming that the spatial domain is convex or smooth we show that the error is of order k2 h2 where k and h are the parameters for the time and space meshes respectively

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

lower bounds for quartic anharmonic and double well potentials rigorous and remarkably accurate lower bounds to the lower eigenvalue spectrum of the schrodinger equation with quartic anharmonic and symmetric double well potentials of the form v a b ax2 2 bx4 b 0 are presented this procedure exploits some exactly soluble model potentials and appears to be of quite general utility

a posteriori error estimates for space time finite element approximation of quasistatic hereditary linear viscoelasticity problems abstract we give a space time galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory volterra integral the numerical scheme consists of a continuous galerkin approximation in space based on piecewise polynomials of degree p 0 cg p with a discontinuous galerkin piecewise constant dg 0 or linear dg 1 approximation in time a posteriori galerkin error estimates are derived by exploiting the galerkin framework and optimal stability estimates for a related dual backward problem the a posteriori error estimates are quite flexible strong lp energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth while weak energy norms are used when the data are non smooth in time we also give upper bounds on the dg 0 cg 1 a posteriori error estimates which indicate optimality however a complete analysis is not given

discretization with variable time steps of an evolution equation with a positive type memory term we consider the linearized form of an evolution equation used to model viscoelasticity and heat conduction in materials with memory the equation is discretized in time using several schemes based on the backward euler and crank nicolson methods and incorporating appropriate quadratures for the memory term by permitting variable time steps we remove a restriction present in earlier related work by the first two authors

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

adaptive fast and oblivious convolution in evolution equations with memory to approximate convolutions which occur in evolution equations with memory terms a variable step size algorithm is presented for which advancing n steps requires only o n log n operations and o log n active memory in place of o n 2 operations and o n memory for a direct implementation a basic feature of the fast algorithm is the reduction via contour integral representations to differential equations which are solved numerically with adaptive step sizes rather than the kernel itself its laplace transform is used in the algorithm the algorithm is illustrated on three examples a blowup example originating from a schrodinger equation with concentrated nonlinearity chemical reactions with inhibited diffusion and viscoelasticity with a fractional order constitutive law

on the simultaneous eigenproblem for the x2 λ x2 1 gx2 1 interaction extension of gallas results an ansatz for the eigenfunctions of the x2 x2 1 gx2 1 interaction has been recently introduced by blecher and leach using this ansatz gallas has obtained five solution pairs for 2g 3g 2 gallas results are generalised and extended here

mixed spin ising model on the union jack lattice we show that the mixed spin ising model on the union jack lattice with s 1 2 and s 1 spin operators placed on different sublattices is equivalent to the eight vertex model on a two dimensional manifold in a parameter space the model corresponds to the symmetric eight vertex model and thus is solvable we locate exactly a coexistence surface between two ordered phases of the mixed spin model this surface is bounded by a bicritical line with continuously changing exponents

some observations on the nature of solutions for the interaction v x x2 λx2 1 gx2 the authors study the nature of exact solutions for the nonpolynomial interaction in particular it is shown that all the solutions are supersymmetric it is also shown that this problem admits a dynamical su 2 symmetry so that an arbitrarily large part of the spectrum but not the whole spectrum can be determined exactly

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

entropic order in frustrated antiferromagnets we show that in frustrated spin s ising antiferromagnets the entropy can stabilize a partially ordered ground state in the model on the decorated square lattice partial long range order exists for s or 1 for the model on the union jack lattice the partially ordered ground state exists even for s 1 2 for this lattice the ground state problem is shown to be equivalent in a certain case to the staggered six vertex model we suggest that the model on the triangular lattice for s sc approximately 3 might exhibit a novel kind of partial ordering

ritz volterra projections to finite element spaces and applications to integrodifferential and related equations the object of this paper is to investigate the convergence of finite element approximations to solutions of parabolic and hyperbolic integrodifferential equations and also of equations of sobolev and viscoelasticity type the concept of ritz volterra projection will be seen to unify much of the analysis for the different types of problems optimal order error estimates are obtained in l p for 2 leqq p infty and almost optimal order pointwise results given

reductions of 2 1 dimensional integrable systems via mixed potential eigenfunction constraints new types of reductions of 2 1 dimensional integrable systems which are associated with mixed potential eigenfunction constraints are discussed necessary and sufficient conditions for the admissibility of such constraints are presented classification results for integrable equations will be used for applying these conditions several typical examples are considered including the kp mkp 2 1 dimensional savada kotera kaup kupershmidt harry dym and nizhnik veselov novikov equations