A Method for Exact Series Solutions in Structural Mechanics

1999 ◽  
Vol 66 (2) ◽  
pp. 380-387 ◽  
Author(s):  
J. T.-S. Wang ◽  
C.-C. Lin
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Integral Form ◽  
Structural Mechanics ◽  
Series Solutions ◽  
Analysis Method ◽  
Weighting Functions ◽  
Systematic Analysis ◽  
Unified Method

A systematic analysis method for solving boundary value problems in structural mechanics is presented. Euler-Lagrange differential equations are transformed into integral form with respect to sinusoidal weighting functions. General solutions are represented by complete sets of functions without being concerned with boundary conditions in advance while all boundary conditions are satisfied in the process. The convergence of results is assured, and the procedure leads to pointwise exact solutions. A number of simple structural mechanics problems of stress, buckling, and vibration analyses are presented for illustrative purposes. All results have verified the exactness of solutions, and indicate that this unified method is simple to use and effective.

Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model

2018 ◽  
Vol 24 (3) ◽  
pp. 559-572 ◽  
Author(s):  
Yuanbin Wang ◽  
Kai Huang ◽  
Xiaowu Zhu ◽  
Zhimei Lou
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Nonlocal Model ◽  
Bernoulli Beam ◽  
Two Phase ◽  
Differential Model ◽  
Euler Bernoulli Beam

Eringen’s nonlocal differential model has been widely used in the literature to predict the size effect in nanostructures. However, this model often gives rise to paradoxes, such as the cantilever beam under end-point loading. Recent studies of the nonlocal integral models based on Euler–Bernoulli beam theory overcome the aforementioned inconsistency. In this paper, we carry out an analytical study of the bending problem based on Eringen’s two-phase nonlocal model and Timoshenko beam theory, which accounts for a better representation of the bending behavior of short, stubby nanobeams where the nonlocal effect and transverse shear deformation are significant. The governing equations are established by the principal of virtual work, which turns out to be a system of integro-differential equations. With the help of a reduction method, the complicated system is reduced to a system of differential equations with mixed boundary conditions. After some detailed calculations, exact analytical solutions are obtained explicitly for four types of boundary conditions. Asymptotic analysis of the exact solutions reveals clearly that the nonlocal parameter has the effect of increasing the deflections. In addition, as compared with nonlocal Euler–Bernoulli beam, the shear effect is evident, and an additional scale effect is captured, indicating the importance of applying higher-order beam theories in the analysis of nanostructures.

scholarly journals Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Z. Pashazadeh Atabakan ◽  
A. Kazemi Nasab ◽  
A. Kılıçman ◽  
Zainidin K. Eshkuvatov
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Solution ◽  
Homotopy Analysis Method ◽  
Existence And Uniqueness ◽  
Lagrange Interpolation ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution ◽  
Spectral Homotopy Analysis Method

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

Rotating Flow of a Micropolar Fluid Induced by a Stretching Surface

2010 ◽  
Vol 65 (10) ◽  
pp. 829-843 ◽  
Author(s):  
Tariq Javed ◽  
Iftikhar Ahmad ◽  
Zaheer Abbas ◽  
Tasawar Hayat
Keyword(s):  
Differential Equations ◽  
Micropolar Fluid ◽  
Nonlinear Partial Differential Equations ◽  
Rotating Flow ◽  
Nonlinear Ordinary Differential Equations ◽  
Series Solutions ◽  
Analysis Method ◽  
Stretching Surface ◽  
Homotopy Analysis ◽  
Layer Flow

This investigation deals with the boundary layer flow of a micropolar fluid over a stretching surface. The flow is considered in a rotating frame of reference. The governing nonlinear partial differential equations are reduced to coupled nonlinear ordinary differential equations. The set of similarity equations has been solved analytically employing the homotopy analysis method (HAM). The series solutions are given for velocity and microrotation, and the convergence of these solutions are explicitly discussed. Attention has been focused to the variations of the emerging parameters on the velocity and microrotation are discussed through graphs.

scholarly journals Method of Singular Integral Equations for Analysis of Strip Structures and Experimental Confirmation

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 140
Author(s):  
Liudmila Nickelson ◽  
Raimondas Pomarnacki ◽  
Tomyslav Sledevič ◽  
Darius Plonis
Keyword(s):  
Differential Equations ◽  
Integral Representation ◽  
Boundary Conditions ◽  
Integral Equations ◽  
Singular Integral ◽  
Singular Integral Equations ◽  
Integral Form ◽  
Hankel Function ◽  
Cauchy Type ◽  
Constitutive Parameters

This paper presents a rigorous solution of the Helmholtz equation for regular waveguide structures with the finite sizes of all cross-section elements that may have an arbitrary shape. The solution is based on the theory of Singular Integral Equations (SIE). The SIE method proposed here is used to find a solution to differential equations with a point source. This fundamental solution of the equations is then applied in an integral representation of the general solution for our boundary problem. The integral representation always satisfies the differential equations derived from the Maxwell’s ones and has unknown functions μe and μh that are determined by the implementation of appropriate boundary conditions. The waveguide structures under consideration may contain homogeneous isotropic materials such as dielectrics, semiconductors, metals, and so forth. The proposed algorithm based on the SIE method also allows us to compute waveguide structures containing materials with high losses. The proposed solution allows us to satisfy all boundary conditions on the contour separating materials with different constitutive parameters and the condition at infinity for open structures as well as the wave equation. In our solution, the longitudinal components of the electric and magnetic fields are expressed in the integral form with the kernel consisting of an unknown function μe or μh and the Hankel function of the second kind. It is important to note that the above-mentioned integral representation is transformed into the Cauchy type integrals with the density function μe or μh at certain singular points of the contour of integration. The properties and values of these integrals are known under certain conditions. Contours that limit different materials of waveguide elements are divided into small segments. The number of segments can determine the accuracy of the solution of a problem. We assume for simplicity that the unknown functions μe and μh, which we are looking for, are located in the middle of each segment. After writing down the boundary conditions for the central point of every segment of all contours, we receive a well-conditioned algebraic system of linear equations, by solving which we will define functions μe and μh that correspond to these central points. Knowing the densities μe, μh, it is easy to calculate the dispersion characteristics of the structure as well as the electromagnetic (EM) field distributions inside and outside the structure. The comparison of our calculations by the SIE method with experimental data is also presented in this paper.

scholarly journals An exact solution method for Fredholm integro-differential equations

2019 ◽  
pp. 2-8 ◽  
Author(s):  
Kyriaki Tsilika
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Solution Method ◽  
Initial Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Abstract Operator ◽  
Nonlocal Integral Boundary Conditions

Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom consideredwith integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocalintegral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocalintegral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existencecriterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditionsand obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for thespecial case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for theexistence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct methodanalytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations.If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existenceand uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by twoexamples. The article uses computer algebra system Mathematica to demonstrate the results.

The Stefan Problem of a Polymorphous Material

1979 ◽  
Vol 46 (4) ◽  
pp. 789-794 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Similarity Solution ◽  
Existence And Uniqueness ◽  
Series Solutions ◽  
Special Cases ◽  
Interfacial Boundary ◽  
Infinite Region ◽  
Uniformly Convergent

The problem of freezing or melting of a polymorphous material in a semi-infinite region with arbitrarily prescribed initial and boundary conditions is studied. Exact solutions of the problem are established. The solutions of temperature of all phases are expressed in polynomials and functions in the error integral family and time t and the position of the interfacial boundaries in power series of t1/2. Existence and uniqueness of the series solutions are considered and proved. It is also shown that these series are absolutely and uniformly convergent. The paper concludes with some remarks on density changes at the interfacial boundary and various special cases, one of which is the similarity solution.

scholarly journals Exact Solutions for the Axial Couette Flow of a Fractional Maxwell Fluid in an Annulus

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
M. Imran ◽  
A. U. Awan ◽  
Mehwish Rana ◽  
M. Athar ◽  
M. Kamran
Keyword(s):  
Shear Stress ◽  
Velocity Field ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Couette Flow ◽  
Maxwell Fluid ◽  
Circular Cylinders ◽  
Rotational Flow ◽  
Material Parameters

The velocity field and the adequate shear stress corresponding to the rotational flow of a fractional Maxwell fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. The solutions that have been obtained are presented in terms of generalized Ga,b,c(·,t) and Ra,b(·,t) functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases of our general solutions. Finally, the influence of the material parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations.

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