Closed form solutions for free vibrations of rectangular Mindlin plates

A new two-eigenfunctions theory, using the amplitude deflection and the generalized curvature as two fundamental eigenfunctions, is proposed for the free vibration solutions of a rectangular Mindlin plate. The three classical eigenvalue differential equations of a Mindlin plate are reformulated to arrive at two new eigenvalue differential equations for the proposed theory. The closed form eigensolutions, which are solved from the two differential equations by means of the method of separation of variables are identical with those via Kirchhoff plate theory for thin plate, and can be employed to predict frequencies for any combinations of simply supported and clamped edge conditions. The free edges can also be dealt with if the other pair of opposite edges are simply supported. Some of the solutions were not available before. The frequency parameters agree closely with the available ones through pb-2 Rayleigh-Ritz method for different aspect ratios and relative thickness of plate.

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material (FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching-bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-of-variables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.

Closed form solution of stress intensity factors for cracks emanating from surface semi-spherical cavity in finite body with energy release rate method

In this paper, a new semi-analytical and semi-engineering method of the closed form solution of stress intensity factors （SIFs） of cracks emanating from a surface semi-spherical cavity in a finite body is derived using the energy release rate theory. A mode of crack opening displacements of a normal slice is established, and the normal slice relevant functions are introduced. The proposed method is both effective and accurate for the problem of three-dimensional cracks emanating from a surface cavity. A series of useful results of SIFs are obtained.

Novel Closed-Form Solution for Analyzing Mutual Coupling Between Cylindrical Comformal Rectangular Microstrip Patch Antennas

Based on the integral equation formulations and the moment method, a novel closed form solution for analyzing the mutual coupling effect between the cylindrical comformal rectangular microstrip patch antennas is presented. By using this algorithm, the elements of the impedance matrix and exciting vector are cast into closed forms, thus the computational efficiency is improved dramatically. Numerical results are presented to verify the validity and reliability of the algorithm.

Closed form solution and numerical analysis for Eshelby's elliptic inclusion in plane elasticity

This paper presents a closed form solution and numerical analysis for Es- helby＇s elliptic inclusion in an infinite plate. The complex variable method and the confor- real mapping technique are used. The continuity conditions for the traction and displace- ment along the interface in the physical plane are reduced to the similar conditions along the unit circle of the mapping plane. The properties of the complex potentials defined in the finite elliptic region are analyzed. From the continuity conditions, one can separate and obtain the relevant complex potentials defined in the inclusion and the matrix. From the obtained complex potentials, the dependence of the real strains and stresses in the inclusion from the assumed eigenstrains is evaluated. In addition, the stress distribution on the interface along the matrix side is evaluated. The results are obtained in the paper for the first time.

A CLOSED FORM SOLUTION OF STRESS INTENSITY FACTORS FOR THREE DIMENSIONAL FINITE BODIES WITH ECCENTRIC CRACKS

In this paper, a new analytical-engineering method of closed form solution about stress intensity factors for three dimensional finite bodies with eccentric cracks is derived by means of energy release rate method. The results of stress intensity factors can be obtained. The results provided ir this method are in nice agreement with those of the famous alternating method by which only special cases can be solved.

On Study of Solutions of Kac-van Moerbeke Lattice and Self-dual Network Equations

The closed form of solutions of Kac-van Moerbeke lattice and self-dual network equations are considered by proposing transformations based on Riccati equation, using symbolic computation. In contrast to the numerical computation of travelling wave solutions for differential difference equations, our method obtains exact solutions which have physical relevance.

Comparison of the closed analytical solution on the parameters of the alignment and surveying of the elements of accelerator under random errors

Purpose Compared with the iterative approximation solution,the analytically rigorous solution of the parameters of the accelerator alignment and surveying is the closed theoretical solution.To make the calculated error meet the negligible principle,in this paper,such examples of the zero,small angle and small translation,only large translation and no angle,only large angle without translation and large translation and large angle parameters under random errors are constructed.Methods Furthermore,Singular Value Decomposition,Eigen Value Decomposition,Polar Decomposition,Orthogonal Procrustes Analysis,and Successive Equivalent Differences Transformation are compared.Results In all the above examples,the mutual calculation deviation of Singular Value Decomposition,Eigen Value Decomposition,Polar Decomposition,and Orthogonal Procrustes analysis methods is less than 10 um.Its calculation accuracy is the highest,and Successive Equivalent Differences Transformation-3 has the largest calculation error,and Successive Equivalent Differences Transformation-N is in the middle.The results show that the coordinate residuals,parameters offsets and point position errors of the above solutions with random errors are all nearly zero.Conclusion It shows that the closed analytical solution is a rigorous solution.Moreover,it also shows that the constructed method of the coordinate point pairs is correct.

New Formulation for Arbitrary Cracks Problem and Its Stress Intensity Factor of Plane Elasticity

Aim The general arbitrary cracked problem in an elastic plane was discussed. Methods For the purpose of acquiring the solution of the problem, a new formulation on the problem was proposed. Compared with the classical plane elastic crack model, only the known conditions were revised in the new formulation, which are greatly convenient to solve the problem, and no other new condition was given. Results and Conclusion The general exact analytic solution is given here based on the formulation though the problem is very complicated. Furthermore, the stress intensity factors K Ⅰ, K Ⅱ of the problem are also given.

Abundant closed form wave solutions to some nonlinear evolution equations in mathematical physics

The propagation of waves in dispersive media,liquid flow containing gas bubbles,fluid flow in elastic tubes,oceans and gravity waves in a smaller domain,spatio-temporal rescaling of the nonlinear wave motion are delineated by the compound Korteweg-de Vries(KdV)-Burgers equation,the(2+1)-dimensional Maccari system and the generalized shallow water wave equation.In this work,we effectively derive abundant closed form wave solutions of these equations by using the double(G′/G,1/G)-expansion method.The obtained solutions include singular kink shaped soliton solutions,periodic solution,singular periodic solution,single soliton and other solutions as well.We show that the double(G′/G,1/G)-expansion method is an efficient and powerful method to examine nonlinear evolution equations(NLEEs)in mathematical physics and scientific application.

The closed form solutions of simplified MCH equation and third extended fifth order nonlinear equation

The investigation of closed form solutions for nonlinear evolution equations(NLEEs)is being an attractive subject in the different branches of mathematical and physical sciences.In this article,the enhanced(G'=G)-expansion method has been applied to find the closed form solutions for NLEEs,such as the simplified MCH equation and third extended fifth order nonlinear equations which are very important in mathematical physics.Plentiful closed form solutions with arbitrary parameters are successfully obtained by this method which are expressed in terms of hyperbolic and trigonometric functions.It is shown that the obtained solutions are more general and fresh and can be helpful to analyze the NLEES in mathematical physics and engineering problems.

REMARKS OF DIFFERENTIAL EQUATIONS ON CLOSED SUBSETS IN A BANACH SPACE

We give some extensions of Monch-Harton inequalities with respect to measures of noncompactness. As an example of the application, we obtain two existencetheorems of solutions for Cauchy problems of differential equations on closed setsunder weaker compactness conditions.

Abundant analytical soliton solutions and different wave profiles to the Kudryashov-Sinelshchikov equation in mathematical physics

The generalized exponential rational function(GERF)method is used in this work to obtain analytic wave solutions to the Kudryashov-Sinelshchikov(K S)equation.The K S equation depicts the occurrence of pressure waves in mixtures of liquid-gas bubbles while accounting for thermal expansion and viscosity.By applying the GERF method to the KS equation,we obtain analytic solutions in terms of trigonometric,hyperbolic,and exponential functions,among others.These solutions include solitary wave solutions,dark-bright soliton solutions,singular soliton solutions,singular bell-shaped solutions,traveling wave solutions,rational form solutions,and periodic wave solutions.We discuss the two-dimensional and three-dimensional graphics of some obtained solutions under the accurate range space by selecting appropriate values for the involved arbitrary parameters to make this research more praiseworthy.The obtained analytic wave solutions specify the GERF method’s dependability,capability,trustworthiness,and efficiency.

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods:the Lie symmetry analysis,the generalized Riccati equation mapping approach,and the modified Kudryashov approach,we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation(NCDR)with power-law nonlinearity and density-dependent diffusion.Lie symmetry analysis is one of the pow-erful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables.By the Lie group technique,we obtain one-parameter in-variant transformations,determining equations and corresponding vectors for the considered convection-diffusion-reaction equation.By treating the parameters of the governing equation as constants,the ap-plied methods yield a variety of new closed-form solutions,including inverse function solutions,periodic solutions,exponential function solutions,dark solitons,singular solitons,combo bright-singular solitons,and the combine of bright-dark solitons and dark-bright solitons.Moreover,using the Bäcklund transfor-mation of the generalized Riccati equation and modified Kudryashov method,we can construct multiple solitons and other solutions of the considered equation.The obtained new solutions of this work demon-strate that the used approaches are powerful and effective in dealing with nonlinear equations,and that these solutions are required to explain many biological and physical phenomena.Comparing our obtained solutions of this paper with the ones obtained in the literature,we see that our solutions are new and not reported elsewhere.These newly formed soliton solutions will be more beneficial in the various dis-ciplines of ocean engineering,plasma physics,and nonlinear sciences.

Fast Linearized Augmented Lagrangian Method for Euler’s Elastica Model

Recently,many variational models involving high order derivatives have been widely used in image processing,because they can reduce staircase effects during noise elimination.However,it is very challenging to construct efficient algo-rithms to obtain the minimizers of original high order functionals.In this paper,we propose a new linearized augmented Lagrangian method for Euler’s elastica image denoising model.We detail the procedures of finding the saddle-points of the aug-mented Lagrangian functional.Instead of solving associated linear systems by FFTor linear iterative methods(e.g.,the Gauss-Seidel method),we adopt a linearized strat-egy to get an iteration sequence so as to reduce computational cost.In addition,we give some simple complexity analysis for the proposed method.Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method,and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.

Elastic Deformation Analysis on MHD Viscous Dissipative Flow of Viscoelastic Fluid:An Exact Approach

This communication is devoted to analyze elastic deformation on electrically conducted viscoelastic fluid in the presence of viscous dissipation effects. Non-linear analysis is computed through exact solutions for velocity,temperature and concentration profiles. Special emphasis is provided for elastic deformation in the presence of magnetohydrodynamics effects. Concentration profile is discussed significantly in the presence constructive and destructive chemical reaction. Results are displayed through graphs and discussed for physical parameters that are used in present analysis. Notable findings include that temperature and thermal boundary layer thickness is an increasing function of Prandtl number and a decreasing function of elastic deformation. In addition, heat transfer rate is enhanced by increasing the conjugate parameter(γ) which measures the strength of surface heating.