Asymptotic numerical analysis of the diffusion‐discrete absorption equation

The diffusion‐discrete absorption (DDA) equation is considered. This equation contains the standard diffusion term and the discrete sorption expressed by a sum of large number of δ‐functions with the support at a non‐uniform mesh multiplied by the unknown function (concentration). The main result of the paper is the homogenization (continualization) of this equation when the small parameter is the characteristic step h of the mesh. The error estimates are proved for the difference of the exact solution of the DDA equation and the solution of the homogenized differential equation. Copyright © 2012 John Wiley & Sons, Ltd.     

On the numerical solution of certain boundary control problems for vibrating media in one space-dimension

This paper is concerned with the numerical solution of control problems which consist of minimizing certain quadratic functionals depending on control functions in L ₂[0,1] for some given time T > 0 and bounded with respect to the maximum norm. These control functions act upon the boundary conditions of a vibrating system in one space-dimension which is governed by a wave equation of spatial order 2n They are to be chosen in such a way that a given initial state of vibration at time zero is transferred into the state of rest. This requirement can be expressed by an infinite system of moment equations to be satisfied by the control functions

The control problem is approximated by replacing this infinite system by finitely many, say N, equations (truncation) and by choosing piecewise constant functions as controls (discretization). The resulting problem is a quadratic optimization problem which is solved very efficiently by a multiplier method

Convergence of the solutions of the approximating problems to the solution of the control problem, as N tends to infinity and the discretization is infinitely refined, is shown under mild assumptions. Numerical results are presented for a vibrating beam     

On differential equations satisfied by modular forms

We use the theory of modular functions to give a new proof of a result of P. F. Stiller, which asserts that, if t is a non-constant meromorphic modular function of weight 0 and F is a meromorphic modular form of weight k with respect to a discrete subgroup of SL ₂ () commensurable with SL ₂ (), then F, as a function of t, satisfies a (k+1)-st order linear differential equation with algebraic functions of t as coefficients. Furthermore, we show that the Schwarzian differential equation for the modular function t can be extracted from any given (k+1)-st order linear differential equation of this type. One advantage of our approach is that every coefficient in the differential equations can be relatively easily determined. Mathematics Subject Classification (2000):11F03, 11F11.     

Difference of a Hauptmodul for Γ0(N ) and certain Gross-Zagier type CM value formulas

In this work, we show that the difference of a Hauptmodul for a genus zero group Γ₀(N) as a modular function on Y₀(N) × Y₀(N) is a Borcherds lift of type(2, 2). As applications, we derive the monster denominator formula like product expansions for these modular functions and certain Gross-Zagier type CM value formulas.     

A two-space dimensional semilinear heat equation perturbed by (Gaussian) white noise

A two-space dimensional heat equation perturbed by a white noise in a bounded volume is considered. The equation is perturbed by a non-linearity of the type λ : f(AU) :, where :: means Wick (re)ordering with respect to the free solution;λ, A are small parameters, U denotes a solution, f is the Fourier transform of a complex measure with compact support. Existence and uniqueness of the solution in a class of Colombeau-Oberguggenberger generalized functions is proven. An explicit construction of the solution is given and it is shown that each term of the expansion in a power series in λ is associated with an L ²-valued measure when A is a small enough. Received: 20 July 1997 / Revised version: 1 February 2001 / Published online: 9 October 2001     

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L²[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.     

Convergence of periodically forced rank-type equations

Consider a difference equation which takes the k-th largest output of m functions of the previous m terms of the sequence. If the functions are also allowed to change periodically as the difference equation evolves, this is analogous to a differential equation with periodic forcing. A large class of such non-autonomous difference equations are shown to converge to a periodic limit, which is independent of the initial condition. The period of the limit does not depend on how far back each term is allowed to look back in the sequence, and is in fact equal to the period of the forcing.     

Determine a Space-Dependent Source Term in a Time Fractional Diffusion-Wave Equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time fractional diffusion-wave equation from a part of noisy boundary data. Based on the series expression of solution for the direct problem, we improve the regularity of the weak solution for the direct problem under strong conditions. And we obtain the uniqueness of inverse space-dependent source term problem by the Titchmarsh convolution theorem and the Duhamel principle. Further, we use a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation to find a stable source term. Numerical examples are provided to show the effectiveness of the proposed method.

     

Parameterizations of the Chazy Equation

The Chazy equation y?= 2yy″? 3y′² is derived from the automorphic properties of Schwarz triangle functions S(α, β, γ; z) . It is shown that solutions y which are analytic in the fundamental domain of these triangle functions, only correspond to certain values of α, β, γ . The solutions are then systematically constructed. These analytic solutions provide all known and one new parameterization of the Eisenstein series P, Q, R introduced by Ramanujan in his modular theories of signature 2, 3, 4, and 6.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

A Class of Quadratic Integral-Differential Equations

A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform.     

p-Adic Analogue of the Wave Equation

In this paper, a p-adic analogue of the wave equation with Lipschitz source is considered. Since it is hard to arrive the solution of the problem, we propose a regularized method to solve the problem from a modified p-adic integral equation. Moreover, we give an iterative scheme for numerical computation of the regularlized solution.

     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

Polynomial solutions of the classical equations of Hermite,Legendre, and Chebyshev

The classical differential equations of Hermite, Legendre, and Chebyshev are well known for their polynomial solutions. These polynomials occur in the solutions to numerous problems in applied mathematics, physics, and engineering. However, since these equations are of second order, they also have second linearly independent solutions that are not polynomials. These solutions usually cannot be expressed in terms of elementary functions alone. In this paper, the classical differential equations of Hermite, Legendre, and Chebyshev are studied when they have a forcing term x ^M on the right-hand side. It will be shown that for each equation, choosing a certain initial condition is a necessary and sufficient condition for ensuring a polynomial solution. Once this initial condition is determined, the exact form of the polynomial solution is presented.     

On dual integral equations in the semiconservative case

A dual integral equations on the whole real axis with an unknown function f is considered. It is supposed that the kernel functions of the equations are even and representable as superposition of exponents. The equation is transferred to a system of integral equations on the positive semi-axis with two unknown functions: f ₁(x) = f(x) and f ₂(x) = f(−x). Applying a factorization method and using the solution of Ambartsumian equation, a system of Laplace transforms α ₁, α ₂ of functions f ₁, f ₂ is obtained. Under some conditions on the free term, the existence and uniqueness of the solution of that system is proved in the semi-conservative case. A construction of the functions α ₁, α ₂ is given by means of successive approximations, and a construction method of the solution (f ₁, f ₂) by α ₁, α ₂ is described.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.     

Functions of ultraexponential and infralogarithm types and general solution of the Abel functional equation

We propose generalized forms of ultraexponential and infralogarithm functions introduced and studied earlier by the author and present two classes of special functions, namely, ultraexponential and infralogarithm f -type functions. As a result of this investigation, we obtain a general solution of the Abel equation α(f(x)) = α (x) + 1 under some conditions on a real function f and prove a new completely different uniqueness theorem for the Abel equation stating that an infralogarithm f -type function is its unique solution. We also show that an infralogarithm f -type function is an essentially unique solution of the Abel equation. Similar theorems are proved for ultraexponential f -type functions and their functional equation β(x) = f(β(x − 1)), which can be considered as dual to the Abel equation. We also solve a certain problem unsolved before and study some properties of two considered functional equations and some relations between them.