Helicon waves in a flaring magnetic field

1. A Preview of Applications and Techniques. 2. Fourier Series. Supplement on Convergence. 3. Partial Differential Equations in Rectangular Coordinates. 4. Partial Differential Equations in Polar and Cylindrical Coordinates. Supplement on Bessel Functions. 5. Partial Differential Equations in Spherical Coordinates. Supplement on Legendre Functions. 6. Sturm-Liouville Theory with Engineering Applications. 7. The Fourier Transform and its Applications. 8. The Laplace and Hankel Transforms with Applications. 9. Finite Difference Numerical Methods. 10. Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations. 11. An Introduction to Quantum Mechanics. Supplement on Orthogonal Polynomials. Appendix A: Ordinary Differential Equations: Review of Concepts and Methods. Appendix B: Tables of Transforms. References. Answers to Selected Exercises. Index.

4 - Introductory PDE model

This paper develops and validates through a series of presentable examples, a comprehensive high-precision, and ultrafast computing concept for solving nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) with cellular neural networks (CNN). The core of this concept is a straightforward scheme that we call "nonlinear adaptive optimization (NAOP),” which is used for a precise template calculation for solving nonlinear ODEs and PDEs through CNN processors. One of the key contributions of this work is to demonstrate the possibility of transforming different types of nonlinearities displayed by various classical and well-known nonlinear equations (e.g., van der Pol-, Rayleigh-, Duffing-, Rössler-, Lorenz-, and Jerk-equations, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in the case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultrafast solver of nonlinear ODEs and/or PDEs. This clearly enables a CNN-based, real-time, ultraprecise, and low-cost computational engineering. As proof of concept, two examples of well-known ODEs are considered namely a second-order linear ODE and a second order nonlinear ODE of the van der Pol type. For each of these ODEs, the corresponding precise CNN templates are derived and are used to deduce the expected solutions. An implementation of the concept developed is possible even on embedded digital platforms (e.g., field programmable gate array (FPGA), digital signal processor (DSP), graphics processing unit (GPU), etc.). This opens a broad range of applications. Ongoing works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting ODEs and PDEs equation models such as Lorenz-, Rössler-, Navier Stokes-, Schrödinger-, Maxwell-, etc.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found

We introduce a partial differential equation whose structure, in many respects, mimics that of the second Painlevé equation. For this equation we give auto-Bäcklund transformations of both ordinary differential equation and partial differential equation types. Our discussion is sufficiently generally phrased to allow us to draw some general conclusions about the connection between auto-Bäcklund transformations and the integrability of ordinary and partial differential equations.

Surface-wave instability in helicon wave propagation in layered structures

In this paper, numerical solution of partial differential equations (PDEs) is considered by multivariate padé approximations. We applied these method to two examples. First, PDE has been converted to power series by two‐dimensional differential transformation, Then the numerical solution of equation was put into multivariate padé series form. Thus, we obtained numerical solution of PDE. Copyright © 2010 John Wiley & Sons, Ltd.

A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.

Chapter 10 - Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables

Layered superconductors represent unusual materials that invoke the interest of many researchers. To describe electrodynamics in layered superconductors, well-known coupled sine-Gordon equations are widely used. The field components of the waves localized on layered superconductors in the presence of dc magnetic field can be found in WKB (quasiclassical) approximation. However, this approximation does not allow studying dispersion curves in important regions, namely near the light line and at the frequencies close to the Josephson plasma frequency. In this work, we show that the theoretical study of plasma waves localized on a layered superconductor can be performed in more accurate form using Legendre functions.

Experimental results have shown that anti-cancer therapies, such as radiotherapy and chemotherapy, can modulate the cell cycle and generate cell cycle phase-dependent responses. As a result, obtaining a detailed understanding of the cell cycle is one possible path towards improving the efficacy of many of these therapies. Here, we consider a basic structured partial differential equation (PDE) model for cell progression through the cell cycle, and derive expressions for key quantities, such as the population growth rate and cell phase proportions. These quantities are shown to be periodic and, as such, we compare the PDE model to a corresponding ordinary differential equation (ODE) model in which the parameters are linked by ensuring that the long-term ODE behaviour agrees with the average PDE behaviour. By design, we find that the ODE model does an excellent job of representing the mean dynamics of the PDE model within just a few cell cycles. However, by probing the parameter space we find cases in which this mean behaviour is not a good measure of the PDE population growth. Our analytical comparison of two caricature models (one PDE and one ODE system) provides insight into cases in which the simple ODE model is an appropriate approximation to the PDE model.

ABSTRACTThe problem of feedback control of bandwidth allocation for Internet links is analysed in this article. It is shown that the procedure for numerical solution of a fluid flow partial differential equation (PDE) which describes the bandwidth allocation on a link results into a set of ordinary differential equations (ODEs) and into an associated state-space equations model. For the local subsystems, into which the flow PDE is decomposed, it becomes possible to apply boundary-based feedback control. The solution of the control problem proceeds by showing that the state-space model of the flow PDE stands for a differentially flat system. Next, for each subsystem which is related to an ODE, a virtual control input is computed, which can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the data-flow PDE is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the data flow PDE of the communication network so as to assure that all its state variables will converge to the desirable set points.

The method of Lyapunov functions is used to investigate the stability of systems with distributed and lumped parameters, described by linear partial and ordinary differential equations. The original high-order partial differential equations are represented by a first-order system of partial differential evolution equations and constraint equations; to do that, we introduce additional variables. Passing to the first-order system of partial differential equations and representing ordinary differential equations in the normal Cauchy form, we obtain a possibility to construct the Lyapunov function as a sum of integral and classical quadratic forms and develop general methods of the investigation of the stability of a broad class of systems with distributed and lumped parameters. For example, we consider the stability of the work of the wind-driven lift pump and take into account the elasticity of the shaft transmitting the torque from the wind engine to the pump.

Adaptive event-triggered tracking control for strict-feedback nonlinear ODE cascaded n+1 coupled hyperbolic PDE systems.

The present paper contains the low‐frequency expansions of solutions of a large class of exterior boundary value problems involving second‐order elliptic equations in two dimensions. The differential equations must coincide with the Helmholtz equation in a neighbourhood of infinity, however, they may depart radically from the Helmholtz equation in any bounded region provided they retain ellipticity. In some cases the asymptotic expansion has the form of a power series with respect to k2 and k2 (ln k + a)−1, where k is the wave number and a is a constant. In other cases it has the form of a power series with respect to k2, coefficients of which depend polynomially on In k. The procedure for determining the full low‐frequency expansion of solutions of the exterior Dirichlet and Neumann problems for the Helmholtz equation is included as a special case of the results presented here.