Some numerical methods and comparisons for solving mathematical model of surface decontamination by disinfectant solution

A mathematical model is considered to determine the effectiveness of disinfectant solution for surface decontamination. The decontamination process involved the diffusion of bacteria into disinfectant solution and the reaction of the disinfectant killing effect. The mathematical model is a reaction-diffusion type. Finite difference method and method of lines with fourth-order Runge-Kutta method are utilized to solve the model numerically. To obtain stable solutions, von Neumann stability analysis is employed to evaluate the stability of finite difference method. For stiff problem, Dormand-Prince method is applied as the estimated error of fourth-order Runge-Kutta method. MATLAB programming is selected for the computation of numerical solutions. From the results obtained, fourth-order Runge-Kutta method has a larger stability region and better accuracy of solutions compared to finite difference method when solving the disinfectant solution model. Moreover, a numerical simulation is carried out to investigate the effect of different thickness of disinfectant solution on bacteria reduction. Results show that thick disinfectant solution is able to reduce the dimensionless bacteria concentration more effectively

Circular slit maps of multiply connected regions with application to brain image processing

In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is O((M+ 1) n) , where M+ 1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require O((M+ 1) 3n3) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented

Numerical conformal mapping onto the exterior unit disk with a straight slit and logarithmic spiral slits

This paper presents a fast boundary integral equation method for numerical conformal mapping of unbounded multiply connected regions onto a disk with an infinite straight slit and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method

A Boundary Integral Equation with the Generalized Neumann Kernel for a Certain Class of Mixed Boundary Value Problem

We present a uniquely solvable boundary integral equation with the generalized Neumann kernel for solving two-dimensional Laplace’s equation on multiply connected regions with mixed boundary condition. Two numerical examples are presented to verify the accuracy of the proposed method

Computing the zeros of the Szegö kernel for doubly connected regions using conformal mapping

An explicit formula for the zero of the Szegö kernel for an annulus region is well-known. There exists a transformation formula for the Szegö kernel from a doubly connected region onto an annulus. Based on conformal mapping, we derive an analytical formula for the zeros of the Szegö kernel for a general doubly connected region with smooth boundaries. Special cases are the explicit formulas for the zeros of the Szegö kernel for doubly connected regions bounded by circles, limacons, ellipses, and ovals of Cassini. For a general doubly connected region with smooth boundaries, the zero of the Szegö kernel must be computed numerically. This paper describes the application of conformal mapping via integral equation with the generalized Neumann kernel for computing the zeros of the Szegö kernel for smooth doubly connected regions. Some numerical examples and comparisons are also presented. It is shown that the conformal mapping approach also yields good accuracy for a narrow region or region with boundaries that are close to each other

A Boundary Integral Method For The Planar External Potential Flow Around Airfoils

This Paper Presents A Boundary Integral Equation For The External Potential Flow Problem Around Airfoils Without Cusped Trailing Edge Angle. The Derivation Of The Integral Equation Is Based Upon Reducing The External Potential Flow Problem To An Exterior Riemann Problem. The Solution Technique Is Different From The Known Techniques In The Literature Since It Involves An Application Of The Riemann Problem, Instead Of The Usual Dirichlet Or Neumann Problems. The Solution Of The Integral Equation Contains An Arbitrary Real Constant, Which May Be Determined By Imposing The Kutta-Joukowski Condition. The Integral Equation Is Solved Numerically Using The Nyström Method With Kress Quadrature Rule. Comparisons Between The Calculated And Analytical Values Of The Pressure Coefficient For The Van De Vooren Airfoil And The Karman-Trefftz Airfoil With 15% Thickness Ratio And Different Angles Of Attack Show Very Good Agreement. Numerical Results Of The Pressure Coefficient For NACA0012 Airfoil With Different Angles Of Attack Are Also Presented

An integral equation method for conformal mapping of doubly connected regions involving the Kerzman-Stein kernel

We present an integral equation method for conformal mapping of doubly connected regions onto a unit disc with circular slit of radius m < 1. Our theoretical development is based on the boundary integral equation for conformal mapping of doubly connected region derived by Murid and Razali. In this paper, using the boundary relationship satisfied by the mapping function, a related system of Fredholm integral equation is constructed, provided m is assume known. For numerical experiment, the integral equation is discretized which leads to a system of linear equations. Numerical implementation on a circular annulus is also presented

Analog-q bagi transformasi linear terhadap polinomial Rn

In this paper, a second q-analogue for the Rn polynomial in two variables is given. This second q-analogue is found to be equivalent with the first q-analogue of Rn discussed in the previous work. A special case of this equivalence relation is the linear transformation of the Rn polynomial