An alternating boundary integral based method for a Cauchy problem for the Laplace equation in a quadrant

We study the Cauchy problem for the Laplace equation in a quadrant (quarter-plane) containing a bounded inclusion. Given the values of the solution and its derivative on the edges of the quadrant the solution is reconstructed on the boundary of the inclusion. This is achieved using an alternating iterative method where at each iteration step mixed boundary value problems are being solved. A numerical method is also proposed and investigated for the direct mixed problems reducing these to integral equations over the inclusion. Numerical examples verify the efficiency of the proposed scheme.

Recovering boundary data in planar heat conduction using boundary integral equation method

We consider a Cauchy problem for the heat equation, where the temperature field is to be reconstructed from the temperature and heat flux given on a part of the boundary of the solution domain. We employ a Landweber type method proposed in [2], where a sequence of mixed well-posed problems are solved at each iteration step to obtain a stable approximation to the original Cauchy problem. We develop an efficient boundary integral equation method for the numerical solution of these mixed problems, based on the method of Rothe. Numerical examples are presented both with exact and noisy data, showing the efficiency and stability of the proposed procedure and approximations.

Computer methods for the heat conduction equation

The merits of various numerical methods for the solution of the one and two dimensional heat conduction equation with a radiation boundary condition have been examined from a practical standpoint in order to determine accuracies and efficiencies. It is found that the use of five increments to approximate the space derivatives gives sufficiently accurate results provided the time step is not too large; further, the implicit backward difference method of Liebmann (27) is found to be the most accurate method. On this basis, a new implicit method is proposed for the solution of the three-dimensional heat conduction equation with radiation boundary conditions. The accuracies of the integral and analogue computer methods are also investigated.

Microstructural development in aluminide diffusion coatings on nickel-base superalloy single crystals

An investigation has been made of the microstructural stability of aluminide diffusion coatings during post-coating thermal exposure. This study has employed edge-on transmission electron microscopy to examine high-activity pack aluminised single crystals of a gamma prime strengthened nickel-base superalloy. The influence of exposure temperature, duration and atmosphere as well as the initial coating thickness has been assessed. Two major processes have been found to contribute to microstructural changes in the coating. These are, firstly, the transformation of the coating matrix (β-phase, nominally NiAl) to other Ni-Al based phases, especially γ' (nominally Ni3(Al, Ti)) and, secondly, the precipitation of chromium containing phases. The work has enabled the roles of three processes contributing to γ formation, namely: oxidation of the coating surface, interdiffusion with the substrate and ageing of the coating, to be understood. In addition, the factors leading to the formation of a sequence of chromium-containing phases have been identified.

Comparison of fitting stability of the different soft toric contact lenses

Purpose: To compare lens orientation and rotational recovery of five currently available soft toric lenses. Methods: Twenty subjects were recruited and trialed with each of the study lenses in a random order. Study lenses were PureVision® Toric (B&L), Air Optix® for Astigmatism (Alcon), Biofinity® Toric (CooperVision), Acuvue® Advance for Astigmatism (Vistakon), and Proclear® Toric (CooperVision). Lens orientation in primary position to determine the lens rotation form the vertical position and rotational recovery to primary gaze orientation following a 45° manual misorientation for the different lenses was compared. Results: The Biofinity Toric showed the lowest rotation from the vertical position and the Proclear Toric the highest. Also, the highest and the lowest reorientation speed were related to the Biofinity Toric and the Acuvue Advance for Astigmatism, respectively. The Repeated Measures ANOVA showed a significant difference in the lens rotation (P=. 0.004) and rotational recovery (P<. 0.001) among different contact lenses and the performed multiple comparisons indicated differences in rotation and also in reorientation speed were only seen between the Biofinity Toric when compared to four other lenses (P<. 0.05). Conclusion: Although there was appropriate fitting, based upon lens orientation and reorientation speed, with each of the study lenses it would appear that the optimized ballast technique used in the design of the Biofinity Toric helps reduce lens rotation and improve rotational recovery compared to others.

Optimal Control of a Second Order Parabolic Heat Equation

In this paper, we are concerned with the optimal control boundary control of a second order parabolic heat equation. Using the results in [Evtushenko, 1997] and spatial central finite difference with diagonally implicit Runge-Kutta method (DIRK) is applied to solve the parabolic heat equation. The conjugate gradient method (CGM) is applied to solve the distributed control problem. Numerical results are reported.

Cauchy Problem for Differential Equation with Caputo Derivative

The paper is devoted to the study of the Cauchy problem for a nonlinear differential equation of complex order with the Caputo fractional derivative. The equivalence of this problem and a nonlinear Volterra integral equation in the space of continuously differentiable functions is established. On the basis of this result, the existence and uniqueness of the solution of the considered Cauchy problem is proved. The approximate-iterative method by Dzjadyk is used to obtain the approximate solution of this problem. Two numerical examples are given.

Generalized Strichartz Inequalities for the Wave Equation on the Laguerre Hypergroup

Mathematics Subject Classification: 42B35, 35L35, 35K35

An Abstract Second Kind Fredholm Integral Equation with Degenerated Kernel

Mathematics Subject Classification: 44A40, 45B05

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45

Well-Posedness of the Cauchy Problem for Inhomogeneous Time-Fractional Pseudo-Differential Equations

Mathematics Subject Classification: 26A33, 45K05, 35A05, 35S10, 35S15, 33E12

Sobolev-Morrey Type Inequality for Riesz Potentials, Associated with the Laplace-Bessel Differential Operator

2000 Mathematics Subject Classification: 42B20, 42B25, 42B35

A Fractional Analog of the Duhamel Principle

Mathematics Subject Classification: 35CXX, 26A33, 35S10

On Two Saigo’s Fractional Integral Operators in the Class of Univalent Functions

2000 Mathematics Subject Classification: Primary 26A33, 30C45; Secondary 33A35

Polynomial Expansions for Solutions of Higher-Order Bessel Heat Equation in Quantum Calculus

Mathematics Subject Class.: 33C10,33D60,26D15,33D05,33D15,33D90