Special Initial Value Problems Research Articles

COMPOSITIONS OF FRACTIONAL DERIVATIVES AS DZHRBASHYAN - NERSESYAN DERIVATIVE

The issues of unique solvability for a special initial value problem to two classes of a linear inhomogeneous equation with a derivative of Dzhrbashyan - Nersesyan are investigated. In one of the classes, the operator at the unknown function is bounded, in the other it is sectorial. The representation of the composition of any number of Gerasimov - Caputo and/or Riemann - Liouville differential operators in the form of a Dzhrbashyan - Nersesyan derivative is also proved.

Asymptotic analysis of Sturm–Liouville problem with nonlocal integral-type boundary condition

In this study, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the one-dimensional Sturm–Liouville equation with one classical-type Dirichlet boundary condition and integral-type nonlocal boundary condition. We investigate solutions of special initial value problem and find asymptotic formulas of arbitrary order. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic formulas of arbitrary order. We apply the obtained results to the problem with integral-type nonlocal boundary condition.

English

A set of new discrete models have been created using Nonstandard methods. A new renormalized denominator function has been derived. This has been applied to some special initial value problems in ordinary differential equation. Result from a suitable numerical solver created from these schemes showed that , the schemes are suitable and they carry along the dynamics of the original equation. I. INTRODUCTION The concept of numerical instability as analyzed in the works of Micken's 1994 has thrown more light on the fact that a lot of standard methods exhibit various form of instability primarily due to the denominator h. In particular a lot of schemes derived using these standard methods fail to preserve the dynamics of the original phenomena as represented by the original differential equation. Micken's has suggested a five point rule that can help numerical modelers to avoid numerical instability and produce schemes that carry along the dynamics of the original differential equation. These rules are both advisory and instructional as some merely point to pitfalls that must be avoided while two of the rules basically required that the schemes be created to satisfy certain conditions. This paper is concerned with deriving schemes that obey the dynamics of the original differential equation.

Buckling and deformation of Hollomon's power-law tubes

A long, thin, inextensible cylindrical tube made of Hollomon's power-law material acted upon by a uniform normal pressure is considered. The nonlinear boundary value problem that governs the equilibrium states of such a tube is formulated as a differential system of equations. Perturbation solutions are obtained for the cases of small pressure values in the neighborhood of the critical buckling pressures. Numerical solutions based on a special initial value problem Matlab solver, Newton's and shooting methods are obtained. The results show that a high strength material tube deforms similarly to an elastic material tube for values of strain hardening exponent.

The complex dynamics of the cardiovascular system

In our paper, we consider the dynamics of blood pressure, initiated in [M.U. Akhmet, G.A. Bekmukhambetova, A prototype compartmental model of blood pressure distribution, Nonlinear Anal. RWA (in press)], concentrating on the interaction between systemic arterial pressure and periphery blood pressure. A system of impulsive differential equations is applied as a model. The main result of the present paper is the existence of Devaney’s chaos ingredients: sensitivity of solutions, transitivity, and existence of infinitely many periodic solutions, in the case, when the moments of discontinuity are defined as a special initial value problem. The method of creating a chaos [M.U. Akhmet, Dynamical synthesis of the quasi-minimal set, Internat. J. Bifur. Chaos (in press); M.U. Akhmet, Devaney’s chaos in a relay system, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1486–1493; M.U. Akhmet, Li–Yorke chaos in the system with impacts, J. Math. Anal. Appl. 351 (2009) 804–810] is applied. Appropriate examples are provided, including a simulation of the chaotic attractor.

Li–Yorke chaos in the system with impacts

The analogue of Li–Yorke chaos [T.Y. Li, J. Yorke, Period three implies chaos, Amer. Math. Monthly 87 (1975) 985–992] for a special initial value problem of a non-autonomous impulsive differential equation is developed.

Section determinants of unitary operators and the operator-valued Riccati differential equation

A special initial value problem for the operator-valued Riccati differential equation (RDE) is used for studying section determinants of unitary operators. A solvability criterion for this initial value problem is derived by providing an a-priori estimate of the solution. Alternatively, the same estimate is derived by a method that is based entirely upon linear algebra.

Evolutionary processes in protoplanetary accretion disks: the propagation of axisymmetric shock waves

In this paper we investigate both the global and the local hydrodynamics of axisymmetric accretion disks around young stellar objects under the simultaneous action of viscosity, self-gravity and pressure forces. For simplicity, we take for the global model a polytropic equation of state, make the infinitely thin disk approximation and characterize the surface density and temperature profiles in the disk as power laws in the radial distance r from the protostar. We solve the problem of the general density profile of a Keplerian disk showing that self-gravity could not be an important factor for the fast formation of the rocky cores of giant gaseous planets in our solar system. Under the hypothesis that the unperturbed rotation curve of the disk is nearly Keplerian throughout the radial extent, we can estimate with our polytropic model a lower limit for the resulting masses M d( r) of stable disks up to 100 AU. These masses are in the range of the so-called minimum mass solar nebular ( d M s ≈ 0.01–0.02 ). By adopting a simplified viscosity model, where the height-integrated turbulent dynamical viscosity ν is a function of the surface density σ like η ∝ σ Γ , we derive in the local shearing sheet model linearized evolution equations for small density perturbations describing both a diffusion process and the propagation of acoustic density waves. We solve a special initial value problem and calculate the appropriate Green's function. The analytical solutions so obtained describe in the case Γ < 0 the successive formation of quasi-stationary ring-shaped density structures in a disk with a definite mode of maximum instability, whereas in the case Γ > Γ c the density wave equation describes the propagation of an “overstable” ring-shaped acoustic density wavelet to the outer ranges of the accretion disk. Whereas the group velocity of the wave packet is subsonic, the phase velocities of individual wave crests in the wave packet are supersonic. The mode of maximum instability, the growth rate and the number of growing waves in the wavelet are controlled by Γ and α. Our present knowledge concerning turbulent viscosity in protoplanetary disks is not sufficient to decide whether or not the case Γ > Γ c is realized. The suggested structuring processes in the linear theory should initiate in the non-linear regime the formation of narrow ring-shaped density shock waves moving through the protoplanetary disk. These non-linear waves could produce extremely spatially and temporally heterogeneous temperature regions in the disk. We speculate that ring-shaped density waves, excited by inner boundary conditions and which have dominated the disk's evolution at early times, are responsible both for the fast growth of dust to planetesimals and at least for the rapid accretion of the rocky cores of giant gaseous planets in the protoplanetary accretion disk (shock wave trigger hypothesis). We derive provisional scaling rules for planetary systems regarding the spacing of orbits as a function of the mass ratio of the protoplanetary disk to the protostar. However, further analytical work and linear as well as nonlinear numerical simulations of density waves excited by inner boundary conditions are needed to consolidate the results and speculations of our linear wave mechanics in the future.

Semiconductor Device Noise Computation Based on the Deterministic Solution of the Poisson and Boltzmann Transport Equations

Numerical simulation results of noise due to current fluctuations along an n+−n−n+ submicron structure are presented. The mathematical framework is based on the interpretation of the equations describing electron transport in the semiclassical transport model as stochastic differential equations (SDE). According to this formalism the key computations for the spectral density describing the noise process are reduced to a special initial value problem for the Boltzmann transport equation (BTE). The algorithm employed in the computation of the space dependent noise autocovariance function involves two main processes: the stationary self-consistent solution of the Boltzmann and Poisson equations, and a transient solution of the BTE with special initial conditions. The solution method for the BTE is based on the Legendre polynomial method. Noise due to acoustic and optical scattering and the effects of nonparabolicity are considered in the physical model.

A generalised pompeiu formula related to differential equations in Cn with higher order

This paper deals with a generalization of the Pompeiu formula for partial differential equation with higher order in the n dimensional complex space. In the centre of the investigation lies a construction of a fundamental solution. The problem of constructing such a fundamental solution can be reduced to solving a special initial value problem. This solution will be turned out to be a generalized Riemann-Vekua-function. With the help of the fundamental solution an integral representation for functions being smooth enough can be found. Functions, which are generalizations to higher dimensions of the polyanalytic or polyharmonic functions in the complex plane, can be represented by the boundary values of the function and some of its derivatives.

Linear partial differential operators with holomorphic coefficients

This paper deals with a generalization of the Pompeiu formula for partial differential equation with higher order in the complex plane. In the centre of the investigation lies the construction of a fundamental solution. The problem of constructing such a fundamental solution can be reduced to a special initial value problem for ordinary differential equation. With the help of the fundamental solution an integral representation for functions being smooth enough can be found. Further investigations of this integral formula under function theoretic view points lead to a conclusion for ordinary differential equation, namely every solution can be obtained by a solution of the special initial value problem, mentioned above This result can be used to prove an algebraic statement about the zeroes of a polynomial.

Global extrapolation of numerical methods for initial value problems

Two-, three- and four-grid extrapolation procedures are considered for the special initial value problem of arbitrary order q > 1. A number of new extrapolation formulas are developed for convergent numerical methods of arbitrary order p≧2. The global extrapolations of three existing P-stable numerical methods are considered and tested on two second order initial value problems from the literature. Global extrapolation procedures for general initial value problems are also considered.

Buckling Loads of Columns with Varying Cross Sections

A new numerical method is presented for evaluating the buckling loads of columns with varying cross sections. By this method, the traditional eigenvalue problem is transformed into a special initial value problem, and the buckling loads of the columns can be evaluated by an iteration procedure, which is to satisfy the boundary conditions specified in each case using numerical integrations.. Two numerical examples, a built-up column and a two-component column, are provided for discussion. It is found that the proposed method can meet the precision requirements very well and is a convenient tool for structural designers.

Global extrapolation on three and four grids for special initial value problems

Three- and four-grid extrapolation procedures are considered for special initial value problems of arbitrary order q > 1.

Evaluation of natural frequencies of non-uniform beams by numerical integration

A new numerical method is presented for evaluating the natural frequencies of non-uniform beams. By this method, the traditional eigenvalue problem is transformed into a special initial value problem. All the natural frequencies of the beams can be evaluated by satisfying the boundary conditions specified in each case using numerical integrations.

The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion—Nonisothermal case

where a and b are the characteristic constants of the fluid. Thus when the temperature is sufficiently low, p ' (v) is positive, and thus the system (I.1) is elliptic on a certain interval (c~, fl), while p ' (v) is negative on (b, ~) (liquid phase) as well as on (/~, oo) (vapor phase) on which (1.1) is hyperbolic. The Riemann problem for (I.1) is a special initial-value problem in which the initial data are of the form

Un metodo di approssimazione bilaterale per equazioni differenziali ordinarie

In this paper we describe a method, based on predictor-corrector formulas, for the bilateral approximation of the solution of special initial value problems for ordinary differential equations.