A fractional differential quadrature method for fractional differential equations and fractional eigenvalue problems

2020 ◽  
Author(s):  
Salwa A. Mohamed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Eigenvalue Problems ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Fractional Differential

Analysis of free vibration of an isotropic plate with surface or internal long crack using generalized differential quadrature method

2019 ◽  
Vol 55 (1-2) ◽  
pp. 42-52
Author(s):  
Milad Ranjbaran ◽  
Rahman Seifi
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Isotropic Plate ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential

This article proposes a new method for the analysis of free vibration of a cracked isotropic plate with various boundary conditions based on Kirchhoff’s theory. The isotropic plate is assumed to have a part-through surface or internal crack. The crack is considered parallel to one of the plate edges. Existence of the crack modified the governing differential equations which were formulated based on the line-spring model. Generalized differential quadrature method discretizes the obtained governing differential equations and converts them into an algebraic system of equations. Then, an eigenvalue analysis was used to determine the natural frequencies of the cracked plates. Some numerical results are given to demonstrate the accuracy and convergence of the obtained results. To demonstrate the efficiency of the method, the results were compared with finite element solutions and available literature. Also, effects of the crack depth, its location along the thickness, the length of the crack and different boundary conditions on the natural frequencies were investigated.

scholarly journals The Existence of Positive Solutions for Fractional Differential Equations with Sign Changing Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Weihua Jiang ◽  
Jiqing Qiu ◽  
Weiwei Guo
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Eigenvalue Problems ◽  
Existence Of Positive Solutions ◽  
Fractional Differential ◽  
Generalized Boundary Conditions

We investigate the existence of at least two positive solutions to eigenvalue problems of fractional differential equations with sign changing nonlinearities in more generalized boundary conditions. Our analysis relies on the Avery-Peterson fixed point theorem in a cone. Some examples are given for the illustration of main results.

Investigation of MHD Natural Convection Flow Exposed to Constant Magnetic Field via Generalized Differential Quadrature Method

2014 ◽  
Author(s):  
Elgiz Baskaya ◽  
Melih Fidanoglu ◽  
Guven Komurgoz ◽  
Ibrahim Ozkol
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Constant Magnetic Field ◽  
Volume Fraction ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Convection Flow ◽  
Quadrature Method ◽  
Generalized Differential Quadrature Method ◽  
Generalized Differential

In this work, nanofluid flow characteristics of an inclined channel flow exposed to constant magnetic field and pressure gradient is investigated. The nanofluid considered is water based Cu nanoparticles with a volume fraction of 0.06. The viscous dissipation is taken into account in the energy equation and the governing differential equations are nondimensionalized. The coupled one dimensional differential equations are solved via Generalized Differential Quadrature Method (GDQM) discretization followed by Newton Raphson method. Furthermore, the effect of magnetic field, inclination angle of the channel and volume fraction on nanoparticles in the nanofluid on velocity and temperature profiles are examined and represented by figures to give a thorough understanding of the system behavior. Designing systems utilizing nanofluids optimally, is highly dependent to achieving accurate model definitions figuring their inherent performance.

scholarly journals Inverse differential quadrature method: mathematical formulation and error analysis

2021 ◽  
Vol 477 (2248) ◽  
Author(s):  
Saheed O. Ojo ◽  
Luan C. Trinh ◽  
Hasan M. Khalid ◽  
Paul M. Weaver
Keyword(s):  
Differential Equations ◽  
Numerical Stability ◽  
Differential Quadrature Method ◽  
Mathematical Formulation ◽  
High Sensitivity ◽  
Numerical Differentiation ◽  
High Order ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Engineering Systems

Engineering systems are typically governed by systems of high-order differential equations which require efficient numerical methods to provide reliable solutions, subject to imposed constraints. The conventional approach by direct approximation of system variables can potentially incur considerable error due to high sensitivity of high-order numerical differentiation to noise, thus necessitating improved techniques which can better satisfy the requirements of numerical accuracy desirable in solution of high-order systems. To this end, a novel inverse differential quadrature method (iDQM) is proposed for approximation of engineering systems. A detailed formulation of iDQM based on integration and DQM inversion is developed separately for approximation of arbitrary low-order functions from higher derivatives. Error formulation is further developed to evaluate the performance of the proposed method, whereas the accuracy through convergence, robustness and numerical stability is presented through articulation of two unique concepts of the iDQM scheme, known as Mixed iDQM and Full iDQM. By benchmarking iDQM solutions of high-order differential equations of linear and nonlinear systems drawn from heat transfer and mechanics problems against exact and DQM solutions, it is demonstrated that iDQM approximation is robust to furnish accurate solutions without losing computational efficiency, and offer superior numerical stability over DQM solutions.

scholarly journals Dynamic Analysis of a Tapered Composite Thin-Walled Rotating Shaft Using the Generalized Differential Quadrature Method

2020 ◽  
Vol 2020 ◽  
pp. 1-14
Author(s):  
Weiyan Zhong ◽  
Feng Gao ◽  
Yongsheng Ren ◽  
Xiaoxiao Wu ◽  
Hongcan Ma
Keyword(s):  
Differential Equations ◽  
Dynamic Model ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Rotating Shaft ◽  
Generalized Differential Quadrature Method ◽  
Thin Walled ◽  
Generalized Differential

A dynamic model of a tapered composite thin-walled rotating shaft is presented. In this model, the transverse shear deformation, rotary inertia, and gyroscopic effects have been incorporated. The equations of motion are derived based on a refined variational asymptotic method (VAM) and Hamilton’s principle. The partial differential equations of motion are reduced to the ordinary differential equations of motion by using the generalized differential quadrature method (GDQM). The validity of the dynamic model is proved by comparing the numerical results with those obtained in the literature and by using ANSYS. The effects of taper ratio, boundary conditions, ply angle, length to mean radius ratios, and mean radius to thickness ratios on the natural frequencies and critical rotating speeds are investigated.

scholarly journals A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1336
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Dumitru Ţucu ◽  
Marioara Lăpădat ◽  
Mădălina Sofia Paşca
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Differential Quadrature ◽  
Test Problems ◽  
Quadrature Method ◽  
Partial Differential

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator

2020 ◽  
Vol 70 (1) ◽  
pp. 107-124
Author(s):  
Wengui Yang
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Eigenvalue Problems ◽  
Laplacian Operator ◽  
Hadamard Fractional Differential Equations ◽  
Existence And Nonexistence ◽  
Fractional Differential ◽  
Eigenvalue Intervals ◽  
The Green Function

AbstractThis paper is concerned with the existence and nonexistence of positive solutions for the eigenvalue problems of nonlinear Hadamard fractional differential equations with p-Laplacian operator. By applying the properties of the Green function and Guo-Krasnosel’skii fixed point theorem on cones, some existence and nonexistence results of positive solutions are obtained based on different eigenvalue intervals. Finally, some examples are presented to demonstrate the feasibility of our main results.

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