Abstract: In this article, an adaptive least-squares mixed finite element method is studied for pseudo-parabolic integro-differential equations. The solutions of least-squares mixed weak formulation and mixed finite element are proved. A posteriori error estimator is constructed based on the least-squares functional and the posteriori errors are obtained.
Abstract: In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.
Abstract: In this article, a simulation method called the Homotopy Perturbation Method (HPM) is employed in the steady flow of a Walter's B' fluid in a vertical channel with porous wall. We employed Homotopy Perturbation Method to derive solution of a nonlinear form of equation obtained from exerting similarity transforming to the ordinary differential equation gained from continuity and momentum equations of this kind of flow. The results obtained from the Homotopy Perturbation Method are then compared with those from the Runge–Kutta method in order to verify the accuracy of the proposed method. The results show that the Homotopy Perturbation Method can achieve good results in predicting the solution of such problems. Ultimately we use this solution to obtain the other terms of velocities and physical discussion about it.
Abstract: In this paper, by constructing a special set and utilizing fixed point index theory, we study the existence of solution for the boundary value problem of second-order singular differential equations in Banach space, which improved and generalize the result of related paper.
Abstract: The main objective of the present paper is to derive an easy numerical technique for the analysis of the free vibration through the stepped regions of plates. Based on the utilities of the step by step integration initial values IV and Finite differences FD methods, the present improved Initial Value Finite Differences (IVFD) technique is achieved. The first initial conditions are formulated in convenient forms for the step by step integrations while the upper and lower edge conditions are expressed in finite difference modes. Also compatibility conditions are created due to the sudden variation of plate thickness. The present method (IVFD) is applied to solve the fourth order partial differential equation of motion for stepped plate across two different panels under the sudden step compatibility in addition to different types of end conditions. The obtained results are examined and the validity of the present method is proved showing excellent efficiency and rapid convergence.
Abstract: In this paper, an analytical approach is used to study the coupled lateral-torsional vibrations of laminated composite beam. It is known that in such structures due to the fibers orientation in various layers, any lateral displacement will produce a twisting moment. This phenomenon is modeled by the bending-twisting material coupling rigidity and its main feature is the coupling of lateral and torsional vibrations. In addition to the material coupling, the effects of shear deformation and rotary inertia are taken into account in the definition of the potential and kinetic energies. Then, the governing differential equations are derived using the Hamilton-s principle and the mathematical model matches the Timoshenko beam model when neglecting the effect of bending-twisting rigidity. The equations of motion which form a system of three coupled PDEs are solved analytically to study the free vibrations of the beam in lateral and rotational modes due to the bending, as well as the torsional mode caused by twisting. The analytic solution is carried out in three steps: 1) assuming synchronous motion for the kinematic variables which are the lateral, rotational and torsional displacements, 2) solving the ensuing eigenvalue problem which contains three coupled second order ODEs and 3) imposing different boundary conditions related to combinations of simply, clamped and free end conditions. The resulting natural frequencies and mode shapes are compared with similar results in the literature and good agreement is achieved.
Abstract: In this paper, we study the application of Extreme
Learning Machine (ELM) algorithm for single layered feedforward
neural networks to non-linear chaotic time series problems. In this
algorithm the input weights and the hidden layer bias are randomly
chosen. The ELM formulation leads to solving a system of linear
equations in terms of the unknown weights connecting the hidden
layer to the output layer. The solution of this general system of
linear equations will be obtained using Moore-Penrose generalized
pseudo inverse. For the study of the application of the method we
consider the time series generated by the Mackey Glass delay
differential equation with different time delays, Santa Fe A and
UCR heart beat rate ECG time series. For the choice of sigmoid,
sin and hardlim activation functions the optimal values for the
memory order and the number of hidden neurons which give the
best prediction performance in terms of root mean square error are
determined. It is observed that the results obtained are in close
agreement with the exact solution of the problems considered
which clearly shows that ELM is a very promising alternative
method for time series prediction.
Abstract: An accurate procedure to determine free vibrations of
beams and plates is presented.
The natural frequencies are exact solutions of governing vibration
equations witch load to a nonlinear homogeny system.
The bilinear and linear structures considered simulate a bridge.
The dynamic behavior of this one is analyzed by using the theory of
the orthotropic plate simply supported on two sides and free on the
two others. The plate can be excited by a convoy of constant or
harmonic loads. The determination of the dynamic response of the
structures considered requires knowledge of the free frequencies and
the shape modes of vibrations. Our work is in this context. Indeed,
we are interested to develop a self-consistent calculation of the Eigen
frequencies.
The formulation is based on the determination of the solution of
the differential equations of vibrations. The boundary conditions
corresponding to the shape modes permit to lead to a homogeneous
system. Determination of the noncommonplace solutions of this
system led to a nonlinear problem in Eigen frequencies.
We thus, develop a computer code for the determination of the
eigenvalues. It is based on a method of bisection with interpolation
whose precision reaches 10 -12. Moreover, to determine the
corresponding modes, the calculation algorithm that we develop uses
the method of Gauss with a partial optimization of the "pivots"
combined with an inverse power procedure. The Eigen frequencies
of a plate simply supported along two opposite sides while
considering the two other free sides are thus analyzed. The results
could be generalized with the case of a beam by regarding it as a
plate with low width.
We give, in this paper, some examples of treated cases. The
comparison with results presented in the literature is completely
satisfactory.
Abstract: Our aim in this piece of work is to demonstrate the
power of the Laplace Adomian decomposition method (LADM) in
approximating the solutions of nonlinear differential equations
governing the two-dimensional viscous flow induced by a shrinking
sheet.
Abstract: Complex assemblies of interacting proteins carry out
most of the interesting jobs in a cell, such as metabolism, DNA
synthesis, mitosis and cell division. These physiological properties
play out as a subtle molecular dance, choreographed by underlying
regulatory networks that control the activities of cyclin-dependent
kinases (CDK). The network can be modeled by a set of nonlinear
differential equations and its behavior predicted by numerical
simulation. In this paper, an innovative approach has been proposed
that uses genetic algorithms to mine a set of behavior data output by
a biological system in order to determine the kinetic parameters of
the system. In our approach, the machine learning method is
integrated with the framework of existent biological information in a
wiring diagram so that its findings are expressed in a form of system
dynamic behavior. By numerical simulations it has been illustrated
that the model is consistent with experiments and successfully shown
that such application of genetic algorithms will highly improve the
performance of mathematical model of the cell division cycle to
simulate such a complicated bio-system.
Abstract: Shear walls are used in most of the tall buildings for
carrying the lateral load. When openings for doors or windows are
necessary to be existed in the shear walls, a special type of the shear
walls is used called "coupled shear walls" which in some cases is
stiffened by specific beams and so, called "stiffened coupled shear
walls".
In this paper, a mathematical method for geometrically nonlinear
analysis of the stiffened coupled shear walls has been presented.
Then, a suitable formulation for determining the critical load of the
stiffened coupled shear walls under gravity force has been proposed.
The governing differential equations for equilibrium and deformation
of the stiffened coupled shear walls have been obtained by setting up
the equilibrium equations and the moment-curvature relationships for
each wall. Because of the complexity of the differential equation, the
energy method has been adopted for approximate solution of the
equations.
Abstract: Solving Ordinary Differential Equations (ODEs) by
using Partitioning Block Intervalwise (PBI) technique is our aim in
this paper. The PBI technique is based on Block Adams Method and
Backward Differentiation Formula (BDF). Block Adams Method
only use the simple iteration for solving while BDF requires Newtonlike
iteration involving Jacobian matrix of ODEs which consumes a
considerable amount of computational effort. Therefore, PBI is
developed in order to reduce the cost of iteration within acceptable
maximum error
Abstract: A method based on the power series solution is proposed to solve the natural frequency of flapping vibration for the rotating inclined Euler beam with constant angular velocity. The vibration of the rotating beam is measured from the position of the corresponding steady state axial deformation. In this paper the governing equations for linear vibration of a rotating Euler beam are derived by the d'Alembert principle, the virtual work principle and the consistent linearization of the fully geometrically nonlinear beam theory in a rotating coordinate system. The governing equation for flapping vibration of the rotating inclined Euler beam is linear ordinary differential equation with variable coefficients and is solved by a power series with four independent coefficients. Substituting the power series solution into the corresponding boundary conditions at two end nodes of the rotating beam, a set of homogeneous equations can be obtained. The natural frequencies may be determined by solving the homogeneous equations using the bisection method. Numerical examples are studied to investigate the effect of inclination angle on the natural frequency of flapping vibration for rotating inclined Euler beams with different angular velocity and slenderness ratio.
Abstract: An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.
Abstract: Both the minimum energy consumption and
smoothness, which is quantified as a function of jerk, are generally
needed in many dynamic systems such as the automobile and the
pick-and-place robot manipulator that handles fragile equipments.
Nevertheless, many researchers come up with either solely
concerning on the minimum energy consumption or minimum jerk
trajectory. This research paper considers the indirect minimum Jerk
method for higher order differential equation in dynamics
optimization proposes a simple yet very interesting indirect jerks
approaches in designing the time-dependent system yielding an
alternative optimal solution. Extremal solutions for the cost functions
of indirect jerks are found using the dynamic optimization methods
together with the numerical approximation. This case considers the
linear equation of a simple system, for instance, mass, spring and
damping. The simple system uses two mass connected together by
springs. The boundary initial is defined the fix end time and end
point. The higher differential order is solved by Galerkin-s methods
weight residual. As the result, the 6th higher differential order shows
the faster solving time.
Abstract: The RK5GL3 method is a numerical method for solving
initial value problems in ordinary differential equations, and is based
on a combination of a fifth-order Runge-Kutta method and 3-point
Gauss-Legendre quadrature. In this paper we describe the propagation
of local errors in this method, and show that the global order of
RK5GL3 is expected to be six, one better than the underlying Runge-
Kutta method.
Abstract: The main aim of this study is to describe and introduce a method of numerical analysis in obtaining approximate solutions for the SIR-SI differential equations (susceptible-infectiverecovered for human populations; susceptible-infective for vector populations) that represent a model for dengue disease transmission. Firstly, we describe the ordinary differential equations for the SIR-SI disease transmission models. Then, we introduce the numerical analysis of solutions of this continuous time, discrete space SIR-SI model by simplifying the continuous time scale to a densely populated, discrete time scale. This is followed by the application of this numerical analysis of solutions of the SIR-SI differential equations to the estimation of relative risk using continuous time, discrete space dengue data of Kuala Lumpur, Malaysia. Finally, we present the results of the analysis, comparing and displaying the results in graphs, table and maps. Results of the numerical analysis of solutions that we implemented offers a useful and potentially superior model for estimating relative risks based on continuous time, discrete space data for vector borne infectious diseases specifically for dengue disease.
Abstract: The effect of variable chemical reaction on heat and mass transfer characteristics over unsteady stretching surface embedded in a porus medium is studied. The governing time dependent boundary layer equations are transformed into ordinary differential equations containing chemical reaction parameter, unsteadiness parameter, Prandtl number and Schmidt number. These equations have been transformed into a system of first order differential equations. MATHEMATICA has been used to solve this system after obtaining the missed initial conditions. The velocity gradient, temperature, and concentration profiles are computed and discussed in details for various values of the different parameters.
Abstract: In this paper, we study the existence of solution of
the four-point boundary value problem for second-order differential
equations with impulses by using Leray-Schauder theory:
Abstract: Beginning from the creator of integro-differential
equations Volterra, many scientists have investigated these
equations. Classic method for solving integro-differential
equations is the quadratures method that is successfully applied up
today. Unlike these methods, Makroglou applied hybrid methods
that are modified and generalized in this paper and applied to the
numerical solution of Volterra integro-differential equations. The
way for defining the coefficients of the suggested method is also
given.