Semi‐analytical solution of MHD flow through boundary integrals on the pipe wall

A mathematical model is given for the magnetohydrodynamic (MHD) pipe flow as an inner Dirichlet problem in a 2D circular cross section of the pipe, coupled with an outer Dirichlet or Neumann magnetic problem. Inner Dirichlet problem is given as the coupled convection‐diffusion equations for the velocity and the induced current of the fluid coupling also to the outer problem, which is defined with the Laplace equation for the induced magnetic field of the exterior region with either Dirichlet or Neumann boundary condition. Unique solution of inner Dirichlet problem is obtained theoretically reducing it into two boundary integral equations defined on the boundary by using the corresponding fundamental solutions. Exterior solution is also given theoretically on the pipe wall with Poisson integral, and it is unique with Dirichlet boundary condition but exists with an additive constant obtained through coupled boundary and solvability conditions in Neumann wall condition. The collocation method is used to discretize these boundary integrals on the pipe wall. Thus, the proposed procedure is an improved theoretical analysis for combining the solution methods for the interior and exterior regions, which are consolidated numerically showing the flow behavior. The solution is simulated for several values of problem parameters, and the well‐known MHD characteristics are observed inside the pipe for increasing values of Hartmann number maintaining the continuity of induced currents on the pipe wall.     

Dirichlet‐to‐Neumann Map for a Nonlinear Diffusion Equation

A nonlinear diffusive equation with moving boundaries is analyzed by constructing the corresponding Dirichlet‐to‐Neumann map. In particular, the Dirichlet boundary value and the initial condition are used to derive the unknown Neumann boundary value. Then, a contraction‐mapping technique is used to prove existence and uniqueness of the solution for small times.     

Analysis of united boundary‐domain integro‐differential and integral equations for a mixed BVP with variable coefficient

The mixed (Dirichlet–Neumann) boundary‐value problem for the ‘Laplace’ linear differential equation with variable coefficient is reduced to boundary‐domain integro‐differential or integral equations (BDIDEs or BDIEs) based on a specially constructed parametrix. The BDIDEs/BDIEs contain integral operators defined on the domain under consideration as well as potential‐type operators defined on open sub‐manifolds of the boundary and acting on the trace and/or co‐normal derivative of the unknown solution or on an auxiliary function. Some of the considered BDIDEs are to be supplemented by the original boundary conditions, thus constituting boundary‐domain integro‐differential problems (BDIDPs). Solvability, solution uniqueness, and equivalence of the BDIEs/BDIDEs/BDIDPs to the original BVP, as well as invertibility of the associated operators are investigated in appropriate Sobolev spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

Mixed and hybrid formulations for the three‐dimensional magnetostatic problem

We propose mixed and hybrid formulations for the three‐dimensional magnetostatic problem. Such formulations are obtained by coupling finite element method inside the magnetic materials with a boundary element method. We present a formulation where the magnetic field is the state variable and the boundary approach uses a scalar Dirichlet‐Neumann map to describe the exterior domain. Also, we propose a second formulation where the magnetic induction is the state variable and a vectorial Dirichlet‐Neumann map is used to describe the outer field. Numerical discretizations with “edge” and “face” elements are proposed, and for each discrete problem we study an “inf‐sup” condition. © 2002 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 18: 85–104, 2002     

Analytic solution of an exterior Neumann problem in a non‐convex domain

A new method, based on the Kelvin transformation and the Fokas integral method, is employed for solving analytically a potential problem in a non‐convex unbounded domain of ?², assuming the Neumann boundary condition. Taking advantage of the property of the Kelvin transformation to preserve harmonicity, we apply it to the present problem. In this way, the exterior potential problem is transformed to an equivalent one in the interior domain which is the Kelvin image of the original exterior one. An integral representation of the solution of the interior problem is obtained by employing the Kelvin inversion in ?² for the Neumann data and the ‘Neumann to Dirichlet’ map for the Dirichlet data. Applying next the ‘reverse’ Kelvin transformation, we finally obtain an integral representation of the solution of the original exterior Neumann problem. Copyright © 2010 John Wiley & Sons, Ltd.     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Initial‐boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so‐called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well‐posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so‐called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrödinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet‐to‐Neumann map supplied by the defocusing nonlinear Schrödinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial‐boundary value problem on the half‐line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter‐plane space‐time domain.     

Global existence of solutions for non‐small data to non‐linear spherically symmetric thermoviscoelasticity

We consider some initial–boundary value problems for non‐linear equations of thermoviscoelasticity in the three‐dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd.     

Generalized Dirichlet‐to‐Neumann Map in Time‐Dependent Domains

We study the heat, linear Schrödinger (LS), and linear KdV equations in the domain l(t) < x < ∞ , 0 < t < T , with prescribed initial and boundary conditions and with l(t) a given differentiable function. For the first two equations, we show that the unknown Neumann or Dirichlet boundary value can be computed as the solution of a linear Volterra integral equation with an explicit weakly singular kernel. This integral equation can be derived from the formal Fourier integral representation of the solution. For the linear KdV equation we show that the two unknown boundary values can be computed as the solution of a system of linear Volterra integral equations with explicit weakly singular kernels. The derivation in this case makes crucial use of analyticity and certain invariance properties in the complex spectral plane. The above Volterra equations are shown to admit a unique solution.     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

Mixed FEM and BEM coupling for the three‐dimensional magnetostatic problem

We present two new mixed finite element methods coupled with a boundary method for the three dimensional magnetostatic problem. Such formulations are obtained by coupling a finite element method inside a bounded domain with a boundary integral method involving either the Calderon equations or the inverse of Dirichlet Neumann operator to treat the exterior domain. First, we present the formulations and then prove that our mixed formulations are well posed and that they lead to a convergent Galerkin method. Finally, we give numerical results for a sphere immersed in a homogeneous (source) field in the two formulations. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 443–462, 2003     

Boundary layers for cellular flows at high Péclet numbers

We analyze the behavior of solutions of steady advection‐diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ? 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(?^1/2), ? = Pe^?1, around the flow separatrices. We construct an ?‐dependent approximate “water pipe problem” purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ?‐independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ?‐independent problem on this graph. Finally, we show that the Dirichlet‐to‐Neumann map of the water pipe problem approximates the Dirichlet‐to‐Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers. © 2004 Wiley Periodicals, Inc.     

Integral equations of the first kind in the theory of oscillating plates

A modified matrix of fundamental solutions is used to derive and solve first-kind integral equations for the problem of high-frequency harmonic oscillations of an infinite elastic plate with a hole when Dirichlet or Neumann conditions are prescribed on the boundary curve.     

Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions

This paper presents a new boundary integral method for the solution of Laplace’s equation on both bounded and unbounded multiply connected regions, with either the Dirichlet boundary condition or the Neumann boundary condition. The method is based on two uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. Numerical results are presented to illustrate the efficiency of the proposed method.     

Least‐squares mixed Galerkin formulation for variable‐coefficient fractional differential equations with D‐N boundary condition

We propose a least‐squares mixed variational formulation for variable‐coefficient fractional differential equations (FDEs) subject to general Dirichlet‐Neumann boundary condition by splitting the FDE as a system of variable‐coefficient integer‐order equation and constant‐coefficient FDE. The main contributions of this article are to establish a new regularity theory of the solution expressed in terms of the smoothness of the right‐hand side only and to develop a decoupled and optimally convergent finite element procedure for the unknown and intermediate variables. Numerical analysis and experiments are conducted to verify these findings.     

Mathematical problems of the theory of elasticity of chiral materials for Lipschitz domains

By the potential method, we investigate the Dirichlet and Neumann boundary value problems of the elasticity theory of hemitropic (chiral) materials in the case of Lipschitz domains. We study properties of the single‐ and double‐layer potentials and of certain, generated by them, boundary integral operators. These results are applied to reduce the boundary value problems to the equivalent first and the second kind integral equations and the uniqueness and existence theorems are proved in various function spaces. Copyright © 2005 John Wiley & Sons, Ltd.     

Modified boundary integral formulations for the Helmholtz equation

In this paper we describe some modified regularized boundary integral equations to solve the exterior boundary value problem for the Helmholtz equation with either Dirichlet or Neumann boundary conditions. We formulate combined boundary integral equations which are uniquely solvable for all wave numbers even for Lipschitz boundaries Γ=∂Ω. This approach extends and unifies existing regularized combined boundary integral formulations.     

Analysis of segregated boundary‐domain integral equations for variable‐coefficient problems with cracks

Segregated direct boundary‐domain integral equation (BDIE) systems associated with mixed, Dirichlet and Neumann boundary value problems (BVPs) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed for domains with interior cuts (cracks). The main results established in the paper are the BDIE equivalence to the original BVPs and invertibility of the BDIE operators in the corresponding Sobolev spaces. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Finite energy solutions of self‐adjoint elliptic mixed boundary value problems

This paper describes existence, uniqueness and special eigenfunction representations of H¹‐solutions of second order, self‐adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H¹(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H¹(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.