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.

Natural vibration and critical velocity of translating Timoshenko beam with non-homogeneous boundaries

In most practical engineering applications,the translating belt wraps around two fixed wheels.The boundary conditions of the dynamic model are typically specified as simply supported or fixed boundaries.In this paper,non-homogeneous boundaries are introduced by the support wheels.Utilizing the translating belt as the mechanical prototype,the vibration characteristics of translating Timoshenko beam models with nonhomogeneous boundaries are investigated for the first time.The governing equations of Timoshenko beam are deduced by employing the generalized Hamilton's principle.The effects of parameters such as the radius of wheel and the length of belt on vibration characteristics including the equilibrium deformations,critical velocities,natural frequencies,and modes,are numerically calculated and analyzed.The numerical results indicate that the beam experiences deformation characterized by varying curvatures near the wheels.The radii of the wheels play a pivotal role in determining the change in trend of the relative difference between two beam models.Comparing the results unearths that the relative difference in equilibrium deformations between the two beam models is more pronounced with smaller-sized wheels.When the two wheels are of equal size,the critical velocities of both beam models reach their respective minima.In addition,the relative difference in natural frequencies between the two beam models exhibits nonlinear variation and can easily exceed 50%.Furthermore,as the axial velocities increase,the impact of non-homogeneous boundaries on modal shape of translating beam becomes more significant.Although dealing with non-homogeneous boundaries is challenging,beam models with non-homogeneous boundaries are more sensitive to parameters,and the differences between the two types of beams undergo some interesting variations under the influence of non-homogeneous 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.

CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

Non-Homogeneous Riemann Boundary Value Problem with Radicals

The solution of the non-homogeneous Riemann boundary value problem with radicals (1. 2) together with its condition of solvability is obtained for arbitrary positive integersp andq, which generalizes the results for the casep=q=2.

UNIQUENESS FOR SOLUTIONS OF NONHOMOGENEOUS A -HARMONIC EQUATIONS WITH VERY WEAK BOUNDARY VALUES

A uniqueness result for nonhomogeneous quasilinear elliptic partial differential equations with very weak boundary values was proved.

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.

Global Solutions for a Strongly-coupled Parabolic System with Initial Boundary Values

In this paper, we consider a strongly-coupled parabolic system with initial boundary values. Under the appropriate conditions, using Gagliard-Nirenberg inequality, Poincare inequality, Gronwall inequality and imbedding theorem, we obtain the global existence of solutions.

Matrix Boundary Value Problem on Hyperbola

We study a special class of lower trigonometric matrix value boundary value problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial on hyperbola is given in bilinear form and it is shown that it is the only one. Secondly, a special boundary value problem of lower triangular matrix is presented and transformed into four related boundary value problems. Finally, Liouville theorem and Painlevé theorem and pseudo-orthogonal polynomials are used to give solutions.

A Sparse Kernel Approximate Method for Fractional Boundary Value Problems

In this paper,the weak pre-orthogonal adaptive Fourier decomposition(W-POAFD)method is applied to solve fractional boundary value problems(FBVPs)in the reproducing kernel Hilbert spaces(RKHSs)W_(0)^(4)[0,1] and W^(1)[0,1].The process of the W-POAFD is as follows:(i)choose a dictionary and implement the pre-orthogonalization to all the dictionary elements;(ii)select points in[0,1]by the weak maximal selection principle to determine the corresponding orthonormalized dictionary elements iteratively;(iii)express the analytical solution as a linear combination of these determined dictionary elements.Convergence properties of numerical solutions are also discussed.The numerical experiments are carried out to illustrate the accuracy and efficiency of W-POAFD for solving FBVPs.

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

The present paper proposes a mathematical method to numerically treat a class of third-order linear Boundary Value Problems (BVPs). This method is based on the combination of the Adomian Decomposition Method (ADM) and, the modified shooting method. A complete derivation of the proposed method has been provided, in addition to its numerical implementation and, validation via the utilization of the Runge-Kutta method and, other existing methods. The method has been applied to diverse test problems and turned out to perform remarkably. Lastly, the simulated numerical results have been graphically illustrated and, also supported by some absolute error comparison tables.

Local Regularity for a 1D Compressible Viscous Micropolar Fluid Model with Non-homogeneous Temperature Boundary

In this paper,we discuss the local existence of H^i(i=2,4)solutions for a 1D compressible viscous micropolar fluid model with non-homogeneous temperature boundary.The proof is based on the local existence of solutions in[1].

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

The isogeometric boundary element technique(IGABEM)is presented in this study for steady-state inhomogeneous heat conduction analysis.The physical unknowns in the boundary integral formulations of the governing equations are discretized using non-uniform rational B-spline(NURBS)basis functions,which are utilized to build the geometry of the structures.To speed up the assessment of NURBS basis functions,the Bezier extraction´approach is used.To solve the extra domain integrals,we use a radial integration approach.The numerical examples show the potential of IGABEM for dimension reduction and smooth integration of CAD and numerical analysis.

Classification of Selfish and Regular Nodes Based on Reputation Values in MANET Using Adaptive Decision Boundary

A MANET is a cooperative network in which each node has dual responsibilities of forwarding and routing thus node strength is a major factor because a lesser number of nodes reduces network performance. The existing reputation based methods have limitation due to their stricter punishment strategy because they isolate nodes from network participation having lesser reputation value and thus reduce the total strength of nodes in a network. In this paper we have proposed a mathematical model for the classification of nodes in MANETs using adaptive decision boundary. This model classifies nodes in two classes: selfish and regular node as well as it assigns the grade to individual nodes. The grade is computed by counting how many passes are required to classify a node and it is used to define the punishment strategy as well as enhances the reputation definition of traditional reputation based mechanisms. Our work provides the extent of noncooperation that a network can allow depending on the current strength of nodes for the given scenario and thus includes selfish nodes in network participation with warning messages. We have taken a leader node for reputation calculation and classification which saves energy of other nodes as energy is a major challenge of MANET. The leader node finally sends the warning message to low grade nodes and broadcasts the classification list in the MANET that is considered in the routing activity.

Gradient Estimate of Solutions to a Class of Mean Curvature Equations with Prescribed Contact Angle Boundary Problem

This paper studies the prescribed contact angle boundary value problem of a certain type of mean curvature equation.Applying the maximum principle and the moving frame method and based on the location of the maximum point,the boundary gradient estimation of the solutions to the equation is obtained.

Solving Neumann Boundary Problem with Kernel-Regularized Learning Approach

We provide a kernel-regularized method to give theory solutions for Neumann boundary value problem on the unit ball. We define the reproducing kernel Hilbert space with the spherical harmonics associated with an inner product defined on both the unit ball and the unit sphere, construct the kernel-regularized learning algorithm from the view of semi-supervised learning and bound the upper bounds for the learning rates. The theory analysis shows that the learning algorithm has better uniform convergence according to the number of samples. The research can be regarded as an application of kernel-regularized semi-supervised learning.

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.