Generalized Neumann kernel

Author: Mohamed M.S. Nasser

This paper presents an implementation of the integral equations with the generalized Neumann kernel to solve numerically the uniquely and the non-uniquely solvable Riemann-Hilbert problems in Jordan regions with smooth boundaries. The non-uniquely solvable problems are made uniquely solvable by requiring their solutions to satisfy additional constraints. Two type of constraints are presented. Various test numerical examples are presented. The computational efficiency appears significantly excellent.

Authors: Mohamed M.S. Nasser, A.H.M. Murid & N.S. Amin

Based on the recently discovered second kind Fredholm integral equation for the exterior Riemann problem, a boundary integral equation is developed in this paper for the two-dimensional, irrotational, incompressible fluid flow around an airfoil without a cusped trailing edge. The solution of the integral equation contains one arbitrary real constant, which may be determined by imposing the Kutta-Joukowski condition. Comparisons between numerical and analytical values of the pressure coefficient on the surface of the NACA 0009 and NACA 0012 airfoils with zero angle of attack show a very good agreement

Author: Mohamed M.S. Nasser

A Fredholm integral equation of the second kind with the generalized Neumann kernel associated with the Riemann-Hilbert problem on unbounded multiply connected regions will be derived and studied in this paper. The derived integral equation yields a uniquely solvable boundary integral equations for the modified Dirichlet problem on unbounded multiply connected regions.

Formal verification of an operating system kernel manifests absence of errors in the kernel and establishes trust  in  it.  This  paper  evaluates  various  projects on  operating  system  kernel  verification  and  presents  in-depth  survey of  them. The  methodologies  and contributions  of  operating  system verification  projects  have been  discussed  in  the  present  work.  At  the  end,  few  unattended  and  interesting  future  challenges  in operating  system  verification  area  have  been  discussed  and  possible  directions  towards  the  challenge  solution have been described in brief.

Authors: Mohamed M.S. Nasser, Ali H.M. Murid, Ali W.K. Sangawi

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.

Authors:  A.H.M. Murid, Mohamed M.S. Nasser & N.S. Amin

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.

Author: Mohamed M.S. Nasser

This paper presents a new method to solve the interior and the exterior Neumann problems in simply connected regions with smooth boundaries. The method is based on two uniquely solvable Fredholm integral equations of the second kind with
the generalized Neumann kernel. Numerical examples reveal that the present method offers an effective numerical method for the Neumann problems when the boundaries are sufficiently smooth.

A fast method for solving boundary integral equations with the generalized Neumann kernel and the adjoint generalized Neumann kernel is presented. The complexity of the developed method is O((m+1)nlnn) for the integral equation with the generalized Neumann kernel and O((m+1)n) for the integral equation with the adjoint generalized Neumann kernel, where m+1 is the multiplicity of the multiply connected domain and n is the number of nodes in the discretization of each boundary component. Numerical results illustrate that the method gives accurate results even for domains of very high connectivity, domains with piecewise smooth boundaries, domains with close-to-touching boundaries, and domains of real world problems.

We consider a nonlinear integral equation which can be interpreted as a generalization of Theodorsen’s nonlinear integral equation. This equation arises in computing the conformal mapping between simply connected regions. We present a numerical method for solving the integral equation and prove the uniform convergence of the numerical solution to the exact solution. Numerical results are given for illustration.

Authors:Ali W.K. Sangawi, Ali H.M. Murid and M.M.S. Nasser

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.

Authors: A.H.M. Murid & Mohamed M.S. Nasser

Recently, the Riemann problem in the interior domain of a smooth Jordan curve was solved by transforming its boundary condition to a Fredholm integral equation of the second kind with the generalized Neumann kernel. The eigenvalues 1±=λ play an important role in the solvability of these integral equations. In this paper, the necessary and sufficient conditions for 1±=λ to be eigenvalues of the generalized Neumann kernel are given and the corresponding eigenfunctions are derived. Some examples are presented.

Author: Mohamed M S Nasser. We consider a nonlinear integral equation which can be interpreted as a generalization of Theodorsen's nonlinear integral equation. This equation arises in computing the conformal mapping between simply connected regions. We present a numerical method for solving the integral equation and prove the uniform convergence of the numerical solution to the exact solution. Numerical results are given for illustration.

Authors:Ali W.K. Sangawi, Ali H.M. Murid and M.M.S. Nasser

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.

Authors: R. Wegmann, A.H.M. Murid & Mohamed M.S. Nasser

This paper presents two newFredholm integral equations associated to the interior and the exterior Riemann–Hilbert problems in simply connected regions with smooth boundaries.The kernel of these integral equations is the generalized Neumann kernel.The solvability of the integral equations depends on whether
= ±1 are eigenvalues of the kernel which in turn depends on the index of the Riemann–Hilbert problem.The complete discussion of the solvability of the integral equations with the generalized Neumann kernel is presented.The integral equations can be used effectively to solve numerically the Riemann–Hilbert problems.The case of non-uniquely solvable Riemann–Hilbert problems is treated by imposing additional constraints to get a uniquely solvable problem.Fredholm integral equations with generalized Neumann kernels are also derived for the problem of the interior and the exterior harmonic conjugate functions.As applications, we study the problem of conformal mapping to a nearby region and extend Wegmann’s iterative method to general regions.Numerical examples reveal that the present method offers an effective solution technique for the Riemann–Hilbert problems when the boundaries are sufficiently smooth.

Authors: Rudolf Wegmann & Mohamed M.S. Nasser

This paper presents and studies Fredholm integral equations associated with the linear Riemann–Hilbert problems on multiply connected regions with smooth boundary curves. The kernel of these integral equations is the generalized Neumann kernel. The approach is similar to that for simply connected regions (see [R.Wegmann, A.H.M. Murid, M.M.S. Nasser, The Riemann–Hilbert problem and the generalized Neumann kernel, J. Comput. Appl. Math. 182 (2005) 388–415]). There are, however, several characteristic differences, which are mainly due to the fact, that the complement of a multiply connected region has a quite different topological structure. This implies that there is no longer perfect duality between the interior and exterior problems. We investigate the existence and uniqueness of solutions of the integral equations. In particular, we determine the exact number of linearly independent solutions of the integral equations and their adjoints. The latter determine the conditions for solvability. An analytic example on a circular annulus and several numerically calculated examples illustrate the results.

"Author: Mohamed M.S. Nasser

We present a unified boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region G onto the five classical canonical slit domains. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. Using the proposed method, the approximate mapping functions onto the five canonical slit domains can be computed in a unifed way by solving linear systems with a common coe±cient matrix. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The performance of the method is illustrated by several examples for regions with smooth boundaries and with piecewise smooth boundaries."

Author: Mohamed M.S. Nasser

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.