Dirac method for nonlinear and non-homogenous boundary value problems of plates

The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.

Wavelet Multi-Resolution Interpolation Galerkin Method for Linear Singularly Perturbed Boundary Value Problems

In this study,a wavelet multi-resolution interpolation Galerkin method(WMIGM)is proposed to solve linear singularly perturbed boundary value problems.Unlike conventional wavelet schemes,the proposed algorithm can be readily extended to special node generation techniques,such as the Shishkin node.Such a wavelet method allows a high degree of local refinement of the nodal distribution to efficiently capture localized steep gradients.All the shape functions possess the Kronecker delta property,making the imposition of boundary conditions as easy as that in the finite element method.Four numerical examples are studied to demonstrate the validity and accuracy of the proposedwavelet method.The results showthat the use ofmodified Shishkin nodes can significantly reduce numerical oscillation near the boundary layer.Compared with many other methods,the proposed method possesses satisfactory accuracy and efficiency.The theoretical and numerical results demonstrate that the order of theε-uniform convergence of this wavelet method can reach 5.

THE NONLINEAR BOUNDARY VALUE PROBLEM FOR k HOLOMORPHIC FUNCTIONS IN C^(2)

k holomorphic functions are a type of generation of holomorphic functions.In this paper,a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in C^(2).The existence of the solution for the problem is studied in detail with the help of the boundary properties of Cauchy type singular integral operators with a k holomorphic kernel.Furthermore,the integral representation for the solution is obtained.

A Class of Strongly Nonlinear Singular Perturbed Boundary Value Problems

In this paper, a class of strongly nonlinear singular perturbed boundary value problems are coasidered by the theory of differential inequalities and the correction of boundary layer, under which the existence of solution is proved and the uniformly valid asymptotic expansions is obtained as well.

A CLASS OF SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM

The singularly perturbed boundary value problem for the nonlinear boundary conditions is considered.Under suitable conditions,the asymptotic behavior of solution for the original problems is studied by using theory of differential inequalities.

POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 < t < 1, au(0) - betau ' (0) = 0, gammau(1) + deltau ' (1) = 0 under the conditions 0 less than or equal to f(0)(+) < M-1, m(1) < f(infinity)(-)less than or equal to infinity or 0 less than or equal to f(infinity)(+)< M-1, m(1) < f (-)(0)less than or equal to infinity where f(0)(+) = lim(u -->0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.

Singular Perturbations for a Class of Fourth-order Nonlinear Boundary Value Problem

This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y（4） =f（t,y＇,y＇＇,y＇＇＇）,0t1,0ε1 with the nonlinear boundary conditions y（0）=y＇（1）=0,p（y＇＇（0）,y＇＇＇（0））=0,q（y＇＇（1）,y＇＇＇（1））=0 where f：[0,1]×R3→R is continuous,p,q：R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities.

ON THE EXISTENCE AND NODAL CHARACTER OF SOLUTIONS OF SINGULAR NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper we prove the existence of k-node solution of singular nonlinear boundary value problems for each k is-an-element-of N union {0}.

ON THE EXISTENCE AND NODAL CHARACTER OF THE SOLUTIONS OF SINGULAR NONLINEAR BOUNDARY VALUE PROBLEMS

1. Introduction We consider the singular nonlinear boundary value problem where l=v+3/v-1,l+1 is the critical exponent of the embedding of weighted Sobolev space Wt21,2（O, +∞） into Lt2q（O, ∞）, v>2. When v=N-1 we can get the radial solutions of problem where 2*=2N/N-2 is the critical exponent of the Sobolev embedding H1（Rn）→LQ（RN）. Kurtz has discussed the existence of κ-node solution of （1.1）, （1.2） for each κ∈N U{0} when the growth rate of |u|l-1u+f（u） is lower then |u|v+3/v-1 i.e.

BOUNDARY VALUE PROBLEM FOR A SINGULARLY PERTURBED NONLINEAR SYSTEM

In this paper, by the technique and the method of diagonalization, the boundary value problem for second order singularly perturbed nonlinear system as follows is dealt with: epsilon y '=f(t, y, y', epsilon), y(0, epsilon)=a(epsilon), y(1,epsilon)=b(epsilon) The existance of the solution and its asymptotic properties are discussed when the eigenvalues of Jacobi matrix f(y') has K negative real parts and N-K positve real parts.

EXISTENCE OF POSITIVE SOLUTIONS FOR A CLASS OF SINGULAR TWO POINT BOUNDARY VALUE PROBLEMS OF SECOND ORDER NONLINEAR EQUATION

In this paper, we establish the existence of positive solutions of (|y'| p-2g' )'+f(t,y)= 0 (P>1 ). y (0)=y (1) = 0. The function f is allowed to be singular when y= 0.

Singular perturbation of a second-order three-point boundary value problem for nonlinear systems

This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.

SINGULAR PERTURBATIONS FOR A CLASS OF BOUNDARY VALUE PROBLEMS OF HIGHER ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, it has been studied that the singular perturbations for the higherorder nonlinear boundary value problem of the formε2y(n)=f(t, ε, y. '', y(n-2))pj(ε)y(1)(0, ε)-qj(ε)y(j+1)(0. ε)=Aj(ε) (0≤j≤n-3)a1(ε)u(n-2)(0.ε)-a2(ε)y(n-1)(0, ε)=B(ε)b1(ε)y(n-2)(1, ε)+b2(ε)y(n-1),(1. ε)=C(ε)by the method of higher order differential inequalities and boundary layer corrections.Under some mild conditions, the existence of the perturbed solution is proved and itsuniformly efficient asymptotic expansions up to its n-th order derivative function aregiven out. Hence, the existing results are extended and improved.

SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

In this paper, we study the following perturbed nonlinear boundary value problemof the form:εx' =f(t, x,y, ε)εy' =g(t, x,y, ε)x(0)= A(ξ1,ξ2 ,x(1) -x(0), y(1 )- y(0), ε)y(0)=B(ξ1, ξ2,x(1)-￣x(0 ),y(1 )-g(0 ),ε)where ξ1, ξ2 are functions of ε. 0<ε<<1. Under some suitable conditions, we give the asymptotic expansion of solution of any order, and obtain the estimation of remaindet term by using the comparison theorem.

SINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEM FOR A KIND OF VOLTERRA TYPE FUNCTIONAL DIFFERENTIAL EQUATION

By employing the theory of differential inequality and some analysis methods, a nonlinear boundary value problem subject to a general kind of second_order Volterra functional differential equation was considered first. Then, by constructing the right_side layer function and the outer solution, a nonlinear boundary value problem subject to a kind of second_order Volterra functional differential equation with a small parameter was studied further. By using the differential mean value theorem and the technique of upper and lower solution, a new result on the existence of the solutions to the boundary value problem is obtained, and a uniformly valid asymptotic expansions of the solution is given as well.

SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR SEMI-LINEAR RETARDED DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

A boundary value problems for functional differenatial equations, with nonlinear boundary condition, is studied by the theorem of differential inequality. Using new method to construct the upper solution and lower solution, sufficient conditions for the existence of the problems' solution are established. A uniformly valid asymptotic expansions of the solution is also given.

Symmetric Positive Solutions of Nonlinear Singular Second-order Three-point Boundary Value Problem

In this paper, the second-order three-point boundary value problem u(t) + λa(t)f(t, u(t)) = 0, 0 < t < 1,u(t) = u(1- t), u(0)- u(1) = u(12)is studied, where λ is a positive parameter, under various assumption on a and f, we establish intervals of the parameter λ, which yield the existence of positive solution, our proof based on Krasnosel'skii fixed-point theorem in cone.{u"(t)+λa(t)f(t,u(t))=0,0<t<1,u(t)=u(1-t),u′(0)-u′(1)=u(1/2)is studied,where A is a positive parameter,under various assumption on a and f,we establish intervals of the parameter A,which yield the existence of positive solution,our proof based on Krasnosel'skii fixed-point theorem in cone.

SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEMS FOR SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS ON INFINITE INTERVAL(Ⅱ)

In this paper the existence of solutions of the singularly perturbed boundary value problems on infinite interval for the second order nonlinear equation containing a small parameterε>0,εy'=f(x,y,y'),y'(0)=a,y(∞)=βis examined,where are constants,and i=0,1.Moreover,asymptotic estimates of the solutions for the above problems are given.

The Singularly Perturbed Boundary Value Problems for Elliptic Equation with Turning Point

The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.

The Singularly Perturbed Nonlinear Nonlocal Initial Boundary Value Problems for Reaction Diffusion Equations

The singularly perturbed nonlinear noniocal initial boundary value problem for reaction diffusion equations is discussed. Under suitable conditions, the outer solution of the original problem is obtained. By using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. By using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems are studied, and by educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are considered.