Partial integro-differential equations of parabolic type arise naturally in the modeling of many phenomena in various fields of physics, engineering, and economics. The main aim of this thesis is to study finite element methods with numerical quadrature for this class of equations. Both one- and two-dimensional problems are considered. We investigate the stability and convergence properties of the schemes and obtain superconvergence error estimates. It is important to note that these superconvergence results hold also for the equivalent finite difference methods and, in this context, they stand without restrictions on the spatial mesh. In the derivation of these results, we introduce an approach to error analysis that deviates from the traditional one. The significant advantage of this modified strategy is that less regularity for the solution of the continuous problem is needed. The discretization in time using an implicit-explicit method is also addressed, and stability and convergence estimates are derived.
The mathematical modeling and numerical simulation of non-reactive solute transport in porous media is also in the scope of this thesis. Among many other applications, this fluid dynamic problem plays a major role in hydrology...
Static and free vibration analysis of carbon nano wires with rectangular cross section based on Timoshenko beam theory is studied in this research. Differential quadrature method (DQM) is employed to solve the governing equations. From the knowledge of author, it is the first time that free vibration of nano wires is investigated. It is also the first time that differential quadrature method is used for bending analysis of nano wires.
As a first endeavor, bending analysis of tapered nano wires with circular cross section is investigated. In this research, nonlocal elasticity theory based on Euler-Bernoulli beam theory is used to formulate the equations. Differential quadrature method (DQM) is employed to solve the governing equations. Different parameters such as nonlocal parameter, length and radius of tapered nano wires are also considered. The results of present work can be used as bench marks for future works.
Bending analysis of functionally graded single walled carbon nano tubes is presented in this paper. Carbon nano tubes are modeled as Euler-Bernoulli beam theory in this study. Harmonic differential quadrature (HDQ) method is used to discretize the governing equations. In order to show the accuracy of present work, the results are compared with those of other existing results. Then the effects of different parameters such as power law index, inner and outer radius of nano tubes and length nano tubes of are studied, too.
In this study, static analysis of the two-dimensional rectangular nanoplates are investigated by the Differential Quadrature Method (DQM). Numerical solution procedures are proposed for deflection of an embedded nanoplate under distributed nanoparticles based on the DQM within the framework of Kirchhoff and Mindlin plate theories. The governing equations and the related boundary conditions are derived by using nonlocal elasticity theory. The difference between the two models is discussed and bending properties of the nanoplate are illustrated. Consequently, the DQM has been successfully applied to analyze nanoplates with discontinuous loading and various boundary conditions for solving Kirchhoff and Mindlin plates with small-scale effect, which are not solvable directly. The results show that the above mentioned effects play an important role on the static behavior of the nanoplates.
The method of differential quadrature is employed to analyze the free vibration of a cracked cantilever beam resting on elastic foundation. The beam is made of a functionally graded material and rests on a Winkler–Pasternak foundation. The crack action is simulated by a line spring model. Also, the differential quadrature method with a geometric mapping are applied to study the free vibration of irregular plates. The obtained results agreed with the previous studies in the literature. Further, a parametric study is introduced to investigate the effects of geometric and elastic characteristics of the problem on the natural frequencies.
This paper presents effects of boundary conditions and axial loading on frequency characteristics of rotating laminated conical shells with meridional and circumferential stiffeners, i.e., stringers and rings, using Generalized Differential Quadrature Method (GDQM). Hamilton's principle is applied when the stiffeners are treated as discrete elements. The conical shells are stiffened at uniform intervals and it is assumed that the stiffeners have similar material and geometric properties. Equations of motion as well as equations of the boundary condition are transformed into a set of algebraic equations by applying the GDQM. Obtained results discuss the effects of parameters such as rotating velocities, depth to width ratios of the stiffeners, number of stiffeners, cone angles, and boundary conditions on natural frequency of the shell. The results will then be compared with those of other published works particularly with a non-stiffened conical shell and a special case where angle of the stiffened conical shell approaches zero, i.e. a stiffened cylindrical shell. In addition, another comparison is made with present FE method for a non-rotating stiffened conical shell. These comparisons confirm reliability of the present work as a measure to approximate solutions to the problem of rotating stiffened conical shells.
In this paper, the free vibration behavior of post-buckled functionally graded (FG) Mindlin rectangular microplates are described based on the modified couple stress theory (MCST). This theory enables the consideration of the size-effect through introducing material length scale parameters. The FG microplates made of a mixture of metal and ceramic are considered whose volume fraction of components is expressed by a power law function. By means of Hamilton's principle, the nonlinear governing equations and associated boundary conditions are derived for FG micro-plates in the postbuckling domain. The governing equations and boundary conditions are then discretized by using the generalized differential quadrature (GDQ) method before solving numerically by the pseudo-arclength continuation technique. In the solution procedure, the postbuckling problem of microplates is investigated first. Afterwards, the free vibration of microplates around the buckled configuration is discussed. The effects of dimensionless length scale parameter, material gradient index and aspect ratio on the on the postbuckling path and frequency of FG microplates subject to arbitrary edge supports are thoroughly discussed.
To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead o
In this paper, a numerical solution of the two dimensional nonlinear coupled
viscous Burgers equation is discussed with the appropriate initial and boundary
conditions using the modified cubic B spline differential quadrature method. In
this method, the weighting coefficients are computed using the modified cubic B
spline as a basis function in the differential quadrature method. Thus, the
coupled Burgers equations are reduced into a system of ordinary differential
equations (ODEs). An optimal five stage and fourth order strong stability
preserving Runge Kutta scheme is applied to solve the resulting system of ODEs.
The accuracy of the scheme is illustrated via two numerical examples. Computed
results are compared with the exact solutions and other results available in
the literature. Numerical results show that the MCB DQM is efficient and
reliable scheme for solving the two dimensional coupled Burgers equation.; Comment: 15 pages, 3 figures
In this article, a numerical simulation of two dimensional nonlinear
sine-Gordon equation with Neumann boundary condition is obtained by using a
composite scheme referred to as a modified cubic B spline differential
quadrature method. The modified cubic B-spline serves as a basis function in
the differential quadrature method to compute the weighting coefficients. Thus,
the sine-Gordon equation is converted into a system of second order ordinary
differential equations (ODEs). We solve the resulting system of ODEs by an
optimal five stage and fourth-order strong stability preserving Runge Kutta
scheme. Both damped and undamped cases are considered for the numerical
simulation with Josephson current density function with value minus one. The
computed results are found to be in good agreement with the exact solutions and
other numerical results available in literature.
In this article, we study the numerical solution of the one dimensional
nonlinear sine-Gordon by using the modified cubic B-spline differential
quadrature method. The scheme is a combination of a modified cubic B spline
basis function and the differential quadrature method. The modified cubic B
spline is used as a basis function in the differential quadrature method to
compute the weighting coefficients. Thus, the sine Gordon equation is converted
into a system of ordinary differential equations (ODEs). The resulting system
of ODEs is solved by an optimal five stage and fourth order strong stability
preserving Runge Kutta scheme. The accuracy and efficiency of the scheme are
successfully described by considering the three numerical examples of the
nonlinear sine Gordon equation having the exact solutions.; Comment: 15 pages, 6 figures. arXiv admin note: substantial text overlap with
Multilevel quadrature methods for parametric operator equations such as the
multilevel (quasi-) Monte Carlo method are closely related to the sparse tensor
product approximation between the spatial variable and the stochastic variable.
In this article, we employ this fact and reverse the multilevel quadrature
method via the sparse grid construction by applying differences of quadrature
rules to finite element discretizations of different resolution. Besides being
more efficient if the underlying quadrature rules are nested, this way of
performing the sparse tensor product approximation enables the use of
non-nested and even adaptively refined finite element meshes. Especially, the
multilevel quadrature is non-intrusive and allows the use of standard finite
element solvers. Numerical results are provided to illustrate the approach.
The Hadamard and SJT product of matrices are two types of special matrix
product. The latter was first defined by Chen. In this study, they are applied
to the differential quadrature (DQ) solution of geometrically nonlinear bending
of isotropic and orthotropic rectangular plates. By using the Hadamard product,
the nonlinear formulations are greatly simplified, while the SJT product
approach minimizes the effort to evaluate the Jacobian derivative matrix in the
Newton-Raphson method for solving the resultant nonlinear formulations. In
addition, the coupled nonlinear formulations for the present problems can
easily be decoupled by means of the Hadamard and SJT product. Therefore, the
size of the simultaneous nonlinear algebraic equations is reduced by two-thirds
and the computing effort and storage requirements are alleviated greatly. Two
recent approaches applying the multiple boundary conditions are employed in the
present DQ nonlinear computations. The solution accuracies are improved
obviously in comparison to the previously given by Bert et al. The numerical
results and detailed solution procedures are provided to demonstrate the superb
efficiency, accuracy and simplicity of the new approaches in applying DQ method
for nonlinear computations.; Comment: Welcome any comments to email@example.com or
This paper shows that the weighting coefficient matrices of the differential
quadrature method (DQM) are centrosymmetric or skew-centrosymmetric if the grid
spacings are symmetric irrespective of whether they are equal or unequal. A new
skew centrosymmetric matrix is also discussed. The application of the
properties of centrosymmetric and skew centrosymmetric matrix can reduce the
computational effort of the DQM for calculations of the inverse, determinant,
eigenvectors and eigenvalues by 75%. This computational advantage are also
demonstrated via several numerical examples.
Civan and Sliepcevich [1, 2] suggested that special matrix solver should be
developed to further reduce the computing effort in applying the differential
quadrature (DQ) method for the Poisson and convection-diffusion equations.
Therefore, the purpose of the present communication is to introduce and apply
the Lyapunov formulation which can be solved much more efficiently than the
Gaussian elimination method. Civan and Sliepcevich  first presented DQ
approximate formulas in polynomial form for partial derivatives in
tow-dimensional variable domain. For simplifying formulation effort, Chen et
al.  proposed the compact matrix form of these DQ approximate formulas. In
this study, by using these matrix approximate formulas, the DQ formulations for
the Poisson and convection-diffusion equations can be expressed as the Lyapunov
algebraic matrix equation. The formulation effort is simplified, and a simple
and explicit matrix formulation is obtained. A variety of fast algorithms in
the solution of the Lyapunov equation [4-6] can be successfully applied in the
DQ analysis of these two-dimensional problems, and, thus, the computing effort
can be greatly reduced. Finally, we also point out that the present reduction
technique can be easily extended to the three-dimensional cases.
The Lane-Emden type equations are employed in the modelling of several
phenomena in the areas of mathematical physics and astrophysics . In this paper
a new numerical method is applied to investigate some well-known classes of
Lane-Emden type equations which are nonlinear ordinary differential equations
on the semi-infinite domain. We will apply a mesh-less method based on radial
basis function differential quadrature method. In RBFs-DQ the derivative value
of function with respect to a point is directly approximated by a linear
combination of all functional values in the global domain . The main aim of
this method is the determination of weight coefficients. Here we concentrate on
Gaussian(GS) as a radial function for approximating the solution of the
mentioned equation. The comparison of the results with the other numerical
methods shows the efficiency and accuracy of this method.; Comment: 9 figures, 29 pages. arXiv admin note: substantial text overlap with
Current research made contribution to the numerical analysis of highly oscillatory ordinary differential equations. Highly oscillatory functions appear to be at the forefront of the research in numerical analysis. In this work we developed efficient numerical algorithms for solving highly oscillatory differential equations. The main important achievements are: to the contrary of classical methods, our numerical methods share the feature that asymptotically the approximation to the exact solution improves as the frequency of oscillation grows; also our methods are computationally feasible and as such do not require fine partition of the integration interval. In this work we show that our methods introduce better accuracy of approximation as compared with the state of the art solvers in Matlab and Maple.; This thesis presents methods for efficient numerical approximation of linear and non-linear systems of highly oscillatory ordinary differential equations.
Phenomena of high oscillation is considered a major computational problem occurring in Fourier analysis, computational harmonic analysis, quantum mechanics, electrodynamics and fluid dynamics. Classical methods based on Gaussian quadrature fail to approximate oscillatory integrals. In this work we introduce numerical methods which share the remarkable feature that the accuracy of approximation improves as the frequency of oscillation increases. Asymptotically...
This paper proposes a second-order scheme of precision integration for dynamic analysis with respect to long-term integration. Rather than transforming into first-order equations, a recursive scheme is presented in detail for direct solution of the homogeneous part of second-order algebraic and differential equations. The sine and cosine matrices involved in the scheme are calculated using the so-called 2N algorithm. Numerical tests show that both the efficiency and the accuracy of homogeneous equations can be improved considerably with the second-order scheme. The corresponding particular solution is also presented, incorporated with the second-order scheme where the excitation vector is approximated by the truncated Taylor series.
Based on the interpolation technique with the aid of boundary integral equations, a new differential quadrature method has been developed (boundary integral equation supported differential quadrature method, BIE-DQM) to solve boundary value problems over