Analysis of the immersed boundary method for a finite element Stokes problem

Convergence results are presented for the immersed boundary (IB) method applied to a model Stokes problem. As a discretization method, we use the finite element method. First, the immersed force field is approximated using a regularized delta function. Its error in the W^{?1, p} norm is examined for 1 ≤ p < n/(n ? 1), with n representing the space dimension. Subsequently, we consider IB discretization of the Stokes problem and examine the regularization and discretization errors separately. Consequently, error estimate of order h^{1 ? α} in the W^{1, 1} × L¹ norm for the velocity and pressure is derived, where α is an arbitrary small positive number. The validity of those theoretical results is confirmed from numerical examples.     

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.     

Two-point and three-point boundary value problems for n th-order nonlinear differential equations

In this article, we are concerned with existence and uniqueness of solutions of four kinds of two-point boundary value problems for nth-order nonlinear differential equations by “Shooting” method, and studied existence and uniqueness of solutions of a kind of three-point boundary value problems for nth-order nonlinear differential equations by “Matching” method.     

Existence and uniqueness of positive solutions for some singular boundary value problems with linear functional boundary conditions

By using the upper and lower solution method and fixed point theory, we investigate some nonlinear singular second-order differential equations with linear functional boundary conditions. The nonlinear term f(t, u) is nonincreasing with respect to u, and only possesses some integrability. We obtain the existence and uniqueness of the C[0,1] positive solutions as well as the C ¹[0, 1] positive solutions.     

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u_tt?u_xx=b(u,Du)u_xx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem 1.1 is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span 1.8 in the general case and 1.10 in the special case. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonlinear partial differential equations of fourth order under mixed boundary conditions

Solutions to nonlinear partial differential equations of fourth order are studied. Boundary regularity is proved for solutions that satisfy mixed boundary conditions. Various geometric situations including so called triple points are considered. Regularity is measured in Sobolev-Slobodeckii spaces and the results are sharp in this scale. The approach is based on the use of a first order difference quotient method.     

Non - Classical Mixed Interface Problems for Anisotropic Bodies

The paper deals with the three-dimensional mathematical problems of the elasticity theory of anisotropic piece-wise homogeneous bodies. Non - classical mixed type boundary value problems are studied when on one part (S₁) of the interface the rigid contact conditions (jumps of displacement and stress vectors) are given, while conditions of the non - classical interface Problem H or Problem G are imposed on the remaining part (S₂) of the interface, i. e., in both cases jumps of the normal components of displacement and stress vectors are known on S₂ and, in addition, in the first one (Problem H) the tangent components of the displacement vector and in the second one (Problem G) the tangent components of the stress vector are given from the both sides on S₂. The investigation is carried out by the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

Regularity for Fully Nonlinear Elliptic Equations with Neumann Boundary Data

We obtain local C ^α, C ^1,α, and C ^2,α regularity results up to the boundary for viscosity solutions of fully nonlinear uniformly elliptic second order equations with Neumann boundary conditions.     

Finite difference schemes for the fourth-order parabolic equations with different boundary value conditions

In this paper, the fourth-order parabolic equations with different boundary value conditions are studied. Six kinds of boundary value conditions are proposed. Several numerical differential formulae for the fourth-order derivative are established by the quartic interpolation polynomials and their truncation errors are given with the aid of the Taylor expansion with the integral remainders. Effective difference schemes are presented for the third Dirichlet boundary value problem, the first Neumann boundary value problem and the third Neumann boundary value problem, respectively. Some new embedding inequalities on the discrete function spaces are presented and proved. With the method of energy analysis, the unique solvability, unconditional stability and unconditional convergence of the difference schemes are proved. The convergence orders of derived difference schemes are all O(τ² + h²) in appropriate norms. Finally, some numerical examples are provided to confirm the theoretical results.     

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.     

Collocation Trefftz methods for the stokes equations with singularity

This study explores the boundary methods for the two‐dimensional homogeneous Stokes equations and investigates the particular solutions (PS) satisfying the Stokes equations. Smooth solutions for the Stokes equations are provided by explicit fundamental solutions (FS) and PS in this study, and singular corner solutions are also provided from linear elastostatics given in Li et al. ([Eng. Anal. Bound. Elem. 34 (2009), 533‐648, 2009). A new singularity model with an interior crack is proposed and solved by the collocation Trefftz method (CTM). The proposed method achieves highly accurate solutions with the first leading coefficient having 10 significant digits. These solutions may be used as a benchmark for testing results obtained by other numerical methods. Error bounds are derived for the CTM solutions using the PS. For a general corner, the exponent ν_k in r can only be obtained by numerical solutions of a system of nonlinear algebraic equations. Therefore, the combined method using many FS plus a few singular solutions is inevitable in most applications. For singularity problems, combining a few singular solutions with the FS is an advanced topic and is successfully implemented in Lee et al. (Eng. Anal. Bound. Elem. 24 (2010), 632–654); however, combining a few singular solutions with the smooth PS fails to converge in the first leading coefficient. As a result, the aforementioned method is not applicable to the singularity problems addressed in this article. With the help of particular and singular solutions, the hybrid Trefftz method with Lagrange multipliers can be developed for the Stokes equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Positive solutions of non‐positone Dirichlet boundary value problems with singularities in the phase variables

The paper presents existence results for positive solutions of the differential equations x ″ + μh (x) = 0 and x ″ + μf (t, x) = 0 satisfying the Dirichlet boundary conditions. Here μ is a positive parameter and h and f are singular functions of non‐positone type. Examples are given to illustrate the main results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary element method and wave equation

In this paper the boundary integral expression for a one-dimensional wave equation with homogeneous boundary conditions is developed. This is done using the time dependent fundamental solution of the corresponding hyperbolic partial differential equation. The integral expression developed is a generalized function with the same form as the well-known D'Alembert formula. The derivatives of the solution and some useful invariants on the characteristics of the partial differential equation are also calculated.The boundary element method is applied to find the numerical solution. The results show excellent agreement with analytical solutions.A multi-step procedure for large time steps which can be used in the boundary element method is also described.In addition, the way in which boundary conditions are introduced during the time dependent process is explained in detail. In the Appendix the main properties of Dirac's delta function and the Heaviside unit step function are described.     

On first-order discrete boundary value problems

This article analyzes a nonlinear system of first-order difference equations with periodic and non-periodic boundary conditions. Some sufficient conditions are presented under which: potential solutions to the equations will satisfy certain a priori bounds; and the equations will admit at least one solution. The methods involve new dynamic inequalities and use of Brouwer degree theory. The new results are compared with those featuring in the theory of solutions to boundary value problems for differential equations.     

The exact number of solutions for the second order nonlinear boundary value problem

A second order nonlinear differential equation with homogeneous Dirichlet boundary conditions is considered. An explicit expression for the root functions for an autonomous nonlinear boundary value problem is obtained using the results of the paper [SOMORA, P.: The lower bound of the number of solutions for the second order nonlinear boundary value problem via the root functions method, Math. Slovaca 57 (2007), 141–156]. Other assumptions are supposed to prove the monotonicity of root functions and to get the exact number of solutions. The existence of infinitely many solutions of the boundary value problem with strong nonlinearity is obtained by the root function method as well. The paper was supported by the Grant VEGA No. 2/7140/27, Bratislava.     

Abstract wave equations with acoustic boundary conditions

We define an abstract setting to treat wave equations equipped with time‐dependent acoustic boundary conditions on bounded domains of R ⁿ . We prove a well‐posedness result and develop a spectral theory which also allows to prove a conjecture proposed in [13]. Concrete problems are also discussed. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solvability,asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω⁺) is embedded in an unbounded fluid domain (Ω^?). The corresponding physical process is described by the boundary‐transmission problem for second‐order partial differential equations. In particular, in the bounded domain Ω⁺, we have a 4×4 dimensional matrix strongly elliptic second‐order partial differential equation, while in the unbounded complement domain Ω^?, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Interaction of acoustic waves and piezoelectric structures

In the paper, we investigate the basic transmission problems arising in the model of fluid‐solid acoustic interaction when a piezo‐ceramic elastic body ( Ω ⁺ ) is embedded in an unbounded fluid domain ( Ω ^? ). The corresponding physical process is described by boundary‐transmission problems for second order partial differential equations. In particular, in the bounded domain Ω ⁺ , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω ^? , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.     

Oscillations of higher order differential equations of neutral type

In this paper, sufficient conditions have been obtained for oscillation of solutions of a class of nth order linear neutral delay-differential equations. Some of these results have been used to study oscillatory behaviour of solutions of a class of boundary value problems for neutral hyperbolic partial differential equations.