Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

An optimal two-step quadratic spline collocation method for the Dirichlet biharmonic problem

Numerical Algorithms - A two-step quadratic spline collocation method is formulated for the solution of the Dirichlet biharmonic problem on the unit square rewritten as a coupled system of two...     

Optimal superconvergent one step quadratic spline collocation methods

We formulate new optimal quadratic spline collocation methods for the solution of various elliptic boundary value problems in the unit square. These methods are constructed so that the collocation equations can be solved using a matrix decomposition algorithm. The results of numerical experiments exhibit the expected optimal global accuracy as well as superconvergence phenomena. AMS subject classification (2000)  65N35, 65N22     

Retrieving the heat transfer coefficient for jet impingement from transient temperature measurements

Algorithm of retrieving the heat transfer coefficient (HTC) from transient temperature measurements is presented. The unknown distributions of two types of boundary conditions: the temperature and heat flux are parameterized using a small number of user defined functions. The solutions of the direct heat conduction problems with known boundary temperature and flux are expressed as a superposition of auxiliary temperature fields multiplied by unknown parameters. Inverse problem is formulated as a least squares fit of calculated and measured temperatures and is cast in a form of a sum of two objective functions. The first results originates from an inverse problem for retrieving the boundary temperature the second comes from the inverse problem for reproducing the boundary heat flux. The final form of the objective function is obtained by enforcing constant in time value of the heat transfer coefficient. This approach leads to substantial regularization of the results, when compared with the standard technique, where HTC is calculated from separately reconstructed temperature and heat flux on the boundary. The validation of the numerical procedure is carried out by reconstructing a known distribution of the HTC using simulated measurements laden by stochastic error. The proposed approach is also used to reconstruct the distribution of the HTC in a physical experiment of heating a cylindrical sample using an impinging jet.     

Numerical solutions of the orbital equations for diatomic molecules

A basis of Hermite splines is used in conjunction with the collocation method to solve the orbital equations for diatomic molecules. Accurate solutions of the Hartree-Fock equations are obtained using iterative methods over most regions of space, while solving the equations by Gaussian elimination near the nuclear centres. In order to improve the speed and accuracy of our iterative scheme, a new self-adjoint form of the Hartree-Fock equation is derived. Using this new equation, our iterative subroutines solve the Hartree-Fock equations to one part in 10⁶. The Gaussian elimination routines are accurate to better than one part in 10⁸.     

Modified nodal cubic spline collocation for elliptic equations

We consider the modified nodal cubic spline collocation method for a general, variable coefficient, second order partial differential equation in the unit square with the solution subject to the homogeneous Dirichlet boundary conditions. The bicubic spline approximate solution satisfies both the Dirichlet boundary conditions and a perturbed partial differential equation at the nodes of a uniform partition of the square. We prove existence and uniqueness of the approximate solution and derive an optimal fourth order maximum norm error bound. The resulting linear system is solved efficiently by a preconditioned iterative method. Numerical results confirm the expected convergence rates. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Stereoselective Synthesis of 2,4,6-Trimethylcyclohex-4-ene-1,3-diol Derivatives and of polypropionate fragments with four contiguous stereogenic centers

The Diels-Alder adduct (±)- 3 of 2,4-dimethylfuran and 1-cyanovinyl acetate was converted stereoselectively into benzyl 6-(4-chlorophenylsulfonyl)-1,3-exo,5-trimethyl-7-oxabicyclo[2.2.1]hept-5-en-2-exo-yl ( 26 ) and -2-endo-yl ether ( 36 ). Addition of LiAlH₄ to the latter led to the 3-O-benzyl derivatives 28 and 37 of (1RS,2SR,3SR,6SR)- and (1RS,2SR,3RS,6SR)-5-(4-chlorophenylsulfonyl)-2,4,6-trimethylcyclohex-4-ene-1,3-diol, respectively. Methylenation of 6-exo-(4-chlorophenylthio)-1-methyl-5-methylidene-7-oxabicyclo[2.2.1]heptan-2-one ( 16 ), obtained by reaction of (±)- 3 with 4-Cl-C₆H₄SCl and saponification gave, 6-exo-(4-chlorophenylthio)-1-methyl-3,5-dimethylidene-7-oxabicyclo [2.2.1]heptan-2-one ( 43 ), the reduction of which with K-Selectride afforded 6-exo-(4-chlorophenylthio)-1,3-endo-dimethyl-5-methylidene-7-oxabicyclo[2.2.1]heptan-2-endo-ol ( 44 ). The 3-O-benzyl derivative 48 of (1RS,2RS,3RS,6SR)-5-(4-chlorophenylsulfonyl)- 2,4,6-trimethylcyclohex-4-ene-1,3-diol was derived from 44 via based-induced oxa-ring opening of benzyl 6-endo-(4-chlorophenylsulfonyl)-1,3-endo-5-endo-trimethyl-7-oxabicyclo[2.2.1]hept-2-endo-yl ether ( 49 ). Benzylation of 28 , followed by reductive desulfonylation and oxidative cleavage of the cyclohexene moiety afforded (2RS,3SR,4RS,5RS)-3,5-bis(benzyloxy)-2,4-dimethyl-6-oxoheptanal ( 32 ).     

A modified sinc quadrature rule for functions with poles near the arc of integration

Sinc function approach is used to obtain a quadrature rule for estimating integrals of functions with poles near the are of integration. Special treatment is given to integration over the intervals (–, ), (0, ), and (–1, 1). It is shown that the error of the quadrature rule converges to zero at the rateO(exp(–cN)) asN , whereN is the number of nodes used, and wherec is a positive constant which is independent ofN.