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Mohamed M. S. Nasser
Mohamed M.S. Nasser is a professor of computational complex analysis. He received his B.Sc. degree (Mathematics) from Baghdad University, Iraq, in 1999, M.Sc. degree (Mathematics) and Ph.D. degree (Mathematics) from Universities Teknologi Malaysia, Malaysia, in 2002 and 2005, respectively. From 2005 until 2009, he was an assistant professor at the Department of Mathematics, Faculty of Science, Ibb University (Yemen). In 2009, he joined the Department of Mathematics, Faculty of Science, King Khalid University (Saudi Arabia) as an assistant professor from 2009 to 2011 and as an associate professor from 2011 until August 2015 when he was promoted to full professor. Starting from September 2015, he moved to Qatar University. His research interests include the applied and computational complex analysis, specially, numerical conformal mapping and the Riemann-Hilbert problem.
Supervisors: Ali Hassan Mohd. Murid and Norsarahaida S. Amin
Phone: +974-4403-7532
Address: mms.nasser@qu.edu.qa
د. محمد مسعد سعيد ناصر
قسم الرياضيات والاحصاء والفيزياء
كلية الاداب والعلوم
جامعة قطر
ص. ب. 2713
الدوحة - قطر
Prof. Dr. Mohamed M.S. Nasser
Department of Mathematics, Statistics and Physics
College of Arts and Sciences
Qatar University
P.O. Box: 2713, Doha, Qatar
Supervisors: Ali Hassan Mohd. Murid and Norsarahaida S. Amin
Phone: +974-4403-7532
Address: mms.nasser@qu.edu.qa
د. محمد مسعد سعيد ناصر
قسم الرياضيات والاحصاء والفيزياء
كلية الاداب والعلوم
جامعة قطر
ص. ب. 2713
الدوحة - قطر
Prof. Dr. Mohamed M.S. Nasser
Department of Mathematics, Statistics and Physics
College of Arts and Sciences
Qatar University
P.O. Box: 2713, Doha, Qatar
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This article surveys the important role, in a variety of applied mathematical contexts, played by the so-
called Schottky–Klein (S–K) prime function. While it is a classical special function, introduced by 19th
century investigators, its theoretical significance for applications has only been realized in the last decade
or so, especially with respect to solving problems defined in multiply connected, or ‘holey’, domains. It
is shown here that, in terms of it, many well-known results pertaining only to the simply connected case
(no holes) can be generalized, in a natural way, to the multiply connected case, thereby contextualizing
thosewell-knownresultswithinamoregeneralframeworkofmuchbroaderapplicability.Giventhewide-
ranging usefulness of the S–K prime function it is important to be able to compute it efficiently. Here we
introducebothanewtheoreticalformulationforitscomputation,aswellastwodistinctnumericalmethods
to implement the construction. The combination of these theoretical and computational developments
renders the S–K prime function a powerful new tool in applied mathematics.
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We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For l-times connected domains the method requires solving l boundary integral equations with the Neumann kernel. This can be done in O(l^2n log n) operations, where n is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, nonconvex boundaries, piecewise smooth boundaries, and for domains of high connectivity.
Beside the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's thirty nine canonical slit regions.
We present a fast and accurate numerical method for constructing incompressible, inviscid and irrotational flows in two-dimensional {\it coastal domains}, which are unbounded multiply connected domains above an infinitely long coastline boundary. In the numerical method, we utilize a numerical conformal mapping method based on a boundary integral equation with the generalized Neumann kernel in order to construct conformal mappings from coastal domains onto five of Koebe's canonical domains. The numerical method is fast and accurate, since it just requires $O((m+1)n\ln n)$ operations and it converges with $O(e^{-cn})$ for coastal domains of connectivity $m+1$, where $n$ is the number of nodes in discretizing each smooth boundary component and $c$ is a positive constant. With some examples, we also show that it is applicable to arbitrary coastal domains with high connectivity and complex geometry.
This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a boundary integral equation with the adjoint generalized Neumann kernel for the derivative of the boundary correspondence function $\theta'$. Only the right-hand side of the integral equation is different from a canonical region to another. The function $\theta'$ is integrated to obtain the boundary correspondence function $\theta$. The integration constants as well as the parameters of the canonical region are computed using the same uniquely solvable integral equation.
A numerical example is presented to illustrate the accuracy of the proposed method.
This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions. For each canonical region, two integral equations are solved before one can approximate the boundary values of the mapping function. Cauchy’s type integrals are used for computing the mapping function and its inverse for interior points. This method also works for regions with piecewise smooth boundaries. Three examples are given to illustrate the effectiveness of the proposed method."
This article surveys the important role, in a variety of applied mathematical contexts, played by the so-
called Schottky–Klein (S–K) prime function. While it is a classical special function, introduced by 19th
century investigators, its theoretical significance for applications has only been realized in the last decade
or so, especially with respect to solving problems defined in multiply connected, or ‘holey’, domains. It
is shown here that, in terms of it, many well-known results pertaining only to the simply connected case
(no holes) can be generalized, in a natural way, to the multiply connected case, thereby contextualizing
thosewell-knownresultswithinamoregeneralframeworkofmuchbroaderapplicability.Giventhewide-
ranging usefulness of the S–K prime function it is important to be able to compute it efficiently. Here we
introducebothanewtheoreticalformulationforitscomputation,aswellastwodistinctnumericalmethods
to implement the construction. The combination of these theoretical and computational developments
renders the S–K prime function a powerful new tool in applied mathematics.
-----------------------------------------------------------------------------------------
We present a numerical method for the computation of the conformal map from unbounded multiply-connected domains onto lemniscatic domains. For l-times connected domains the method requires solving l boundary integral equations with the Neumann kernel. This can be done in O(l^2n log n) operations, where n is the number of nodes in the discretization of each boundary component of the multiply connected domain. As demonstrated by numerical examples, the method works for domains with close-to-touching boundaries, nonconvex boundaries, piecewise smooth boundaries, and for domains of high connectivity.
Beside the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's thirty nine canonical slit regions.
We present a fast and accurate numerical method for constructing incompressible, inviscid and irrotational flows in two-dimensional {\it coastal domains}, which are unbounded multiply connected domains above an infinitely long coastline boundary. In the numerical method, we utilize a numerical conformal mapping method based on a boundary integral equation with the generalized Neumann kernel in order to construct conformal mappings from coastal domains onto five of Koebe's canonical domains. The numerical method is fast and accurate, since it just requires $O((m+1)n\ln n)$ operations and it converges with $O(e^{-cn})$ for coastal domains of connectivity $m+1$, where $n$ is the number of nodes in discretizing each smooth boundary component and $c$ is a positive constant. With some examples, we also show that it is applicable to arbitrary coastal domains with high connectivity and complex geometry.
This paper presents a new uniquely solvable boundary integral equation for computing the conformal mapping, its derivative and its inverse from bounded multiply connected regions onto the five classical canonical slit regions. The integral equation is derived by reformulating the conformal mapping as an adjoint Riemann-Hilbert problem. From the adjoint Riemann-Hilbert problem, we derive a boundary integral equation with the adjoint generalized Neumann kernel for the derivative of the boundary correspondence function $\theta'$. Only the right-hand side of the integral equation is different from a canonical region to another. The function $\theta'$ is integrated to obtain the boundary correspondence function $\theta$. The integration constants as well as the parameters of the canonical region are computed using the same uniquely solvable integral equation.
A numerical example is presented to illustrate the accuracy of the proposed method.
This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for computing conformal mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions. For each canonical region, two integral equations are solved before one can approximate the boundary values of the mapping function. Cauchy’s type integrals are used for computing the mapping function and its inverse for interior points. This method also works for regions with piecewise smooth boundaries. Three examples are given to illustrate the effectiveness of the proposed method."
A boundary integral equation method for numerical evaluation of the conformal mapping and its inverse from unbounded multiply connected regions onto five canonical slit regions is presented in this paper. This method is based on a uniquely solvable boundary integral equation with the adjoint generalized Neumann kernel. This method is accurate and reliable. Some numerical examples are presented to illustrate the effectiveness of this method."
This paper presents a uniquely solvable boundary integral equation with the generalized Neumann kernel for the Ahlfors map of bounded multiply connected regions.
This paper presents a fast boundary integral equation method for approximating the conformal mapping from bounded and unbounded multiply connected regions of connectivity $m+1$ onto the canonical region obtained by removing $m$ rectilinear slits from a strip. The method is based on a combination of a uniquely solvable boundary integral equation with generalized Neumann kernel and the Fast Multipole Method. The presented method requires $O((m+1)n\ln n)$ operations where $n$ is the number of nodes in the discretization of each boundary component. The numerical results of some test calculations illustrate that our method has the ability to handle regions with complex geometry and very high connectivity.
Beside the above canonical region, the presented fast method also can be used to compute the numerical conformal mapping from multiply connected regions onto Koebe's thirty nine canonical slit regions."
Abstract: In this paper we propose a new method for solving the mixed boundary value problem for the Laplace equation in unbounded multiply connected regions. All simple closed curves making up the boundary are divided into two sets. The Dirichlet condition is given for one set and the Neumann condition is given for the other set. The mixed problem is reformulated in the form of a Riemann-Hilbert (RH) problem which leads to a uniquely solvable Fredholm integral equation of the second kind. Three numerical examples are presented to show the effectiveness of the proposed method."
Abstract
This paper presents a boundary integral method for approximating the conformal mapping from bounded multiply connected regions onto the fifth category of Koebe's classical canonical slit regions. The method is based on a uniquely solvable boundary integral equation with generalized Neumann kernel. The results of some test calculations illustrate the performance of the presented method."
Abstract.
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."
In this paper we present a boundary integral equation method for the numerical conformal mapping of a bounded multiply connected region onto a radial slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method."
In this paper 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 rela-
tionship satisfied by an analytic function on an unbounded multiply
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.
In this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a circular slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.
Abstract. This paper presents a boundary integral equation method for the
numerical conformal mapping of bounded multiply connected region onto an
annulus with circular slits. The method is based on some uniquely solvable linear integral equations with classical, adjoint and generalized Neumann kernels.
These boundary integral equations are constructed from a boundary relationship that relates the mapping function $f$ on a multiply connected region with $f'$, $\theta'$ and $|f|$, where $\theta$ is the boundary correspondence function. Some numerical examples are presented to illustrate the efficiency of the presented method.
In this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a parallel slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.
In this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a disk with circular slits. The method is based on some uniquely solvable boundary integral equations with classical adjoint and generalized Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to
illustrate the efficiency of the presented method.
Two uniquely solvable boundary integral equations for calculating the incompressible potential flow past multiple aerofoils with smooth boundaries are presented. The kernels of the integral equations are the Neumann kernel and the adjoint Neumann kernel. Numerical examples reveal that the present method offers an effective solution technique for the potential flow problem.
This paper presents a boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region $G$ onto the second, third and fourth categories of Koebe's canonical slit domains. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The method is an extension of the author's method for the first category of Koebe's canonical slit domains (see [M.M.S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, \emph{SIAM J. Sci. Comput.}~31 (2009) 1695--1715]). Three numerical examples are presented to illustrate the performance of the proposed method.
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This paper presents a boundary integral equation method for conformal mapping of an unbounded multiply connected region onto a radial slits region. Two linear boundary integral equations are constructed from a boundary relationship satisfied by an analytic function on an unbounded multiply connected region. These integral equations are uniquely solvable. The kernels involved in these integral equations are the adjoint generalized Neumann kernels. Two numerical examples are presented to show the effectiveness of the proposed method.
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In this paper, we solve the mixed boundary value problem on bounded multiply connected region by using the method of boundary integral equation. Our approach in this paper is to reformulate the mixed boundary value problem into the form of Riemann-Hilbert problem. The Riemann-Hilbert 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 adjoint generalized Neumann kernel. As an examination of the proposed method, some numerical examples for some different test regions are presented.