An integral equation for conformal mapping of multiply connected regions onto a circular region

Abstract. An integral equation is presented for the conformal mapping of multiply connected regions of connectivity m+1 onto a circular region. The circular region is bounded by a unit circle, with centre at the origin, and m number of circles inside the unit circle. The development of theoretical part is based on the boundary integral equation related to a non-homogeneous boundary relationship. An example for verification purpose is given in this paper for the conformal mapping from an annulus onto a doubly connected circular region with centres and radii are assumed to be known

Conformal mapping of unbounded multiply connected regions onto canonical slit regions

We present a boundary integral equation method for conformal mapping of unbounded multiply connected regions onto five types of canonical slit regions. For each canonical region, three linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unboundedmultiply connected region. The integral equations are uniquely solvable. The kernels involved in these integral equations are the modified Neumann kernels and the adjoint generalized Neumann kernels

Nonlinear partial differential equations model related to oxidation pond treatment system: a case study of mPHO at Taman Timor Oxidation Pond, Johor Bahru

This study presents a mathematical model examining wastewater pollutant removal through an oxidation pond treatment system. This model was developed to describe the reaction between microbe-based product mPHO (comprising Phototrophic bacteria (PSB)), dissolved oxygen (DO) and pollutant namely chemical oxygen demand (COD). It consists of coupled advection-diusion-reaction equations for the microorganism (PSB), DO and pollutant (COD) concentrations, respectively. The coupling of these equations occurred due to the reactions between PSB, DO and COD to produce harmless compounds. Since the model is nonlinear partial dierential equations (PDEs), coupled, and dynamic, computational algorithm with a specic numerical method, which is implicit Crank-Nicolson method, was employed to simulate the dynamical behaviour of the system. Furthermore, numerical results revealed that the proposed model demonstrated high accuracy when compared to the experimental data

Solving Robin problems in bounded doubly connected regions via an integral equation with the eneralized Neumann Kernel

This paper presents a boundary integral equation method for finding the solution of Robin problems in bounded multiply connected regions. The Robin problems are formulated as a Riemann-Hilbert problems which lead to systems of integral equations and the related differential equations are also constructed that give rise to unique solutions are shown. Numerical results on several test regions are presented to illustrate that the approximate solution when using this method for the Robin problems when the boundaries are sufficiently smooth are accurate

Integral equation for the Ahlfors map on multiply connected regions

This paper presents a new boundary integral equation with ɵ’ the adjoint Neumann kernel associated with where ɵ’ is the boundary correspondence function of Ahlfors map of a bounded multiply connected region onto a unit disk. The proposed boundary integral equation is constructed from a boundary relationship satisfied by the Ahlfors map of a multiply connected region. The integral equation is solved numerically for ɵ’ using combination of Nystrom method, GMRES method, and fast multiple method. From the computed values of ɵ’ we solve for the boundary correspondence function ɵ’ which then gives the Ahlfors map. The numerical examples presented here prove the effectiveness of the proposed metho

Conformal mapping and periodic cubic spline interpolation

Previous studies have shown computation of conformal mapping in which the exact parameterization of the boundary of the region is assumed known. However there are regions whose boundaries have no known exact parameterization. Periodic cubic spline interpolation had been introduced to approximate and obtain the parameterization. We present a numerical procedure to generate periodic cubic spline from the boundary of a 2-dimensional object by using Mathematica software. First we obtain Cartesian coordinates points from the boundary of this 2-dimensional object. Then we convert them into polar coordinates form. Finally the cubic spline is generated based on this polar coordinate points. Some results of our numerical experiments are presented

An integral equation method for solving neumann problems on simply and multiply connected regions with smooth boundaries

This research presents several new boundary integral equations for the solution of Laplace’s equation with the Neumann boundary condition on both bounded and unbounded multiply connected regions. The integral equations are uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. The complete discussion of the solvability of the integral equations is also presented. Numerical results obtained show the efficiency of the proposed method when the boundaries of the regions are sufficiently smooth

Integral equation approach for computing green’s function on doubly connected regions via the generalized Neumann kernel

This research is about computing the Green’s function on doubly connected regions by using the method of boundary integral equation. The method depends on solving a Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nystrӧm method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gauss elimination method. Mathematica plots of Green’s functions for several test regions are also presente

Modeling of microbial approach in wastewater treatment process: a case study of mPHO in Taman Timor oxidation pond, Johor, Malaysia

In this study, we consider the application of biological based product mPHO that contains Phototrophic bacteria (PSB) for the degradation of bacteria Coliform (pollutant) in Taman Timor Oxidation Pond, Johor, Malaysia. A mathematical model is developed to describe the reaction between microorganism and pollutant. The model facilitates the determination of mPHO optimum amount for achieving the maximum pollutant decontamination in the oxidation pond. A partial differential equation model with coupled equation is developed, and the parameters of the model are estimated using the real data collected from the oxidation pond under study. The numerical simulations are also presented to illustrate the performance of proposed model

Analytical and numerical methods for the Riemann problem

In this survey we consider the classical and new methods for computing the analytical and numerical solutions to the Riemann problem, a class of boundary value problems for analytic functions, in a simply connected region Ω+ with smooth boundary Γ = ∂Ω+ in the complex plane. The classical methods for solving this problem based on reducing the Riemann problem to the Dirichlet problem or to the Hilbert problem where it is required the availability of a suitable conformal mapping from Ω+ onto the unit disk D. Recently, the authors introduce a new method for solving the Riemann problem by transforming its boundary condition to a Fredholm integral equation of the second kind with the generalized Neumann kernel. This method has several advantages in terms of numerical operations as well as ease in programming. This paper sketches these classical and new methods and shows the advantages of our method for solving the Riemann problem using Fredholm integral equations