Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.     

X-ray crystal and molecular structure of upper rim monoformylated calix[4]arene system

Selective formylation under controlled conditions leads to the formation of a monoformylated compound 11-formyl-25,27-bis(ethoxycarbonylmethoxy)-calix[4]arene (1). The spectroscopic results indicated that the calix[4]arene is present in a cone conformation. The crystal and molecular structure of (1) has been determined by X-ray diffraction methods. The crystal data are monoclinic, space group P2 ₁/c, M= 624.66, a= 10.341(1), b=15.176(1), c= 20.121(1) Å, = 91.90°(1), V= 3156.0(4) ų, Z= 4, D _c = 1.315 g cm^–3. The structure confirms the cone conformation for the molecule in the solid state as well.     

Studies of Erythrina alkaloids: VII—13C NMR spectral studies of some Erythina alkaloids

A variety of alkaloids have been isolated from the seeds of a range of Erythrina species, including dienoid, alkenoid and lactonic derivatives. The ¹³C NMR spectra of these compounds have been assigned by internal comparisons within this series of related compounds and also by the use of model compounds.     

Higher isoprenoids—XIII : Partial syntheses from cycloartenol,cyclolaudenol—part 5: transformation of cyclolaudenol to 31-norcyclolaudenone

A new general procedure for the functionalization of 4α-Me group in triterpenoids, reported earlier, has been utilized for the conversion of cyclolaudenol to 31-norcyclolaudenone a triterpene isolated from Musa sapientum. A convenient method for the cleavage of CC bond in enamines with RuO₄ is reported.     

Optimal quadratures for analytic functions

For integrals _–1 ¹ w(x)f(x)dx with and with analytic integrands, we consider the determination of optimal abscissasx _i ^o and weightsA _i ^o , for a fixedn, which minimize the errorE _n (f)= _–1 ¹ w(x)f(x)dx – _i ⁼¹ⁿ A _i f(x _i ) over an appropriate Hilbert spaceH ²(E ; w(z)) of analytic functions. Simultaneously, we consider the simpler problem of determining intermediate-optimal weightsA _i *, corresponding to (preassigned) Gaussian abscissasx _i ^G , which minimize the quadrature error. For eachw(x), the intermediate-optimal weightsA _i * are obtained explicitly, and these come out proportional to the corresponding Gaussian weightsA _i ^G . In each case,A _i ^G =A _i *+O( ^–4n ), . For , a complete explicit solution for optimal abscissas and weights is given; in fact, the set {x _i ^G ,A _i *;i=1,...,n} to provides the optimal abscissas and weights. For otherw(x), we study the closeness of the set {x _i ^G ,A _i *;i=1,...,n} to the optimal solution {x _i ^o ,A _i ^o ;i=1,...,n} in terms of _n (), the maximum absolute remainder in the second set ofn normal equations. In each case, _n () is, at least, of the order of ^–4n for large.     

Digital simulation of counterflow regenerator two-dimensional model by two relatively new techniques

Two relatively new finite-difference explicit methods, Alternating Direction Approximation and the Dufort and Frankel scheme, have been compared for digital simulation of counterflow thermal regenerators. The methods are applied to evaluate the thermal performance of regenerators. The temperature histories in the solid are considered as functions of two space dimensions. The solutions are programmed to include the effect of gas radiation and variable specific heats of gases. Of the two methods compared, the Alternating Direction Approximation is found to be simpler, easy to apply, and self-starting.