Abstract: In this work, we investigate the exponential stability of a linear system described by x˙ (t) = Ax(t) − ρBx(t). Here, A generates a semigroup S(t) on a Hilbert space, the operator B is supposed to be of Desch-Schappacher type, which makes the investigation more interesting in many applications. The case of Miyadera-Voigt perturbations is also considered. Sufficient conditions are formulated in terms of admissibility and observability inequalities and the approach is based on some energy estimates. Finally, the obtained results are applied to prove the uniform exponential stabilization of bilinear partial differential equations.
Abstract: In this work, we present an efficient approach for
solving variable-order time-fractional partial differential equations,
which are based on Legendre and Laguerre polynomials. First, we
introduced the pseudo-operational matrices of integer and variable
fractional order of integration by use of some properties of
Riemann-Liouville fractional integral. Then, applied together with
collocation method and Legendre-Laguerre functions for solving
variable-order time-fractional partial differential equations. Also, an
estimation of the error is presented. At last, we investigate numerical
examples which arise in physics to demonstrate the accuracy of the
present method. In comparison results obtained by the present method
with the exact solution and the other methods reveals that the method
is very effective.
Abstract: Minimizing the weight in flexible structures means
reducing material and costs as well. However, these structures could
become prone to vibrations. Attenuating these vibrations has become
a pivotal engineering problem that shifted the focus of many research
endeavors. One technique to do that is to design and implement
an active control system. This system is mainly composed of a
vibrating structure, a sensor to perceive the vibrations, an actuator
to counteract the influence of disturbances, and finally a controller to
generate the appropriate control signals. In this work, two different
techniques are explored to create two different mathematical models
of an active control system. The first model is a finite element model
with a reduced number of nodes and it is called a super-element.
The second model is in the form of state-space representation, i.e.
a set of partial differential equations. The damping coefficients are
calculated and incorporated into both models. The effectiveness of
these models is demonstrated when the system is excited by its first
natural frequency and an active control strategy is developed and
implemented to attenuate the resulting vibrations. Results from both
modeling techniques are presented and compared.
Abstract: The convective and radiative heat transfer performance and entropy generation on forced convection through a direct absorption solar collector (DASC) is investigated numerically. Four different fluids, including Cu-water nanofluid, Al2O3-waternanofluid, TiO2-waternanofluid, and pure water are used as the working fluid. Entropy production has been taken into account in addition to the collector efficiency and heat transfer enhancement. Penalty finite element method with Galerkin’s weighted residual technique is used to solve the governing non-linear partial differential equations. Numerical simulations are performed for the variation of mass flow rate. The outcomes are presented in the form of isotherms, average output temperature, the average Nusselt number, collector efficiency, average entropy generation, and Bejan number. The results present that the rate of heat transfer and collector efficiency enhance significantly for raising the values of m up to a certain range.
Abstract: This study proposes the transformation of nonlinear
Magnetic Levitation System into linear one, via state and feedback
transformations using explicit algorithm. This algorithm allows
computing explicitly the linearizing state coordinates and feedback
for any nonlinear control system, which is feedback linearizable,
without solving the Partial Differential Equations. The algorithm is
performed using a maximum of N-1 steps where N being the
dimension of the system.
Abstract: We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.
Abstract: In this study, a new reliable technique use to handle the foam drainage equation. This new method is resulted from VIM by a simple modification that is Reconstruction of Variational Iteration Method (RVIM). The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. Foaming occurs in many distillation and absorption processes. Results are compared with those of Adomian’s decomposition method (ADM).The comparisons show that the Reconstruction of Variational Iteration Method is very effective and overcome the difficulty of traditional methods and quite accurate to systems of non-linear partial differential equations.
Abstract: The motion of a sphere moving along the axis of a
rotating viscous fluid is studied at high Reynolds numbers and
moderate values of Taylor number. The Higher Order Compact
Scheme is used to solve the governing Navier-Stokes equations. The
equations are written in the form of Stream function, Vorticity
function and angular velocity which are highly non-linear, coupled
and elliptic partial differential equations. The flow is governed by
two parameters Reynolds number (Re) and Taylor number (T). For
very low values of Re and T, the results agree with the available
experimental and theoretical results in the literature. The results are
obtained at higher values of Re and moderate values of T and
compared with the experimental results. The results are fourth order
accurate.
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 work, we obtain some analytic solutions for the (3+1)-dimensional breaking soliton after obtaining its Hirota-s bilinear form. Our calculations show that, three-wave method is very easy and straightforward to solve nonlinear partial differential equations.
Abstract: A novel PDE solver using the multidimensional wave
digital filtering (MDWDF) technique to achieve the solution of a 2D
seismic wave system is presented. In essence, the continuous physical
system served by a linear Kirchhoff circuit is transformed to an
equivalent discrete dynamic system implemented by a MD wave
digital filtering (MDWDF) circuit. This amounts to numerically
approximating the differential equations used to describe elements of a
MD passive electronic circuit by a grid-based difference equations
implemented by the so-called state quantities within the passive
MDWDF circuit. So the digital model can track the wave field on a
dense 3D grid of points. Details about how to transform the continuous
system into a desired discrete passive system are addressed. In
addition, initial and boundary conditions are properly embedded into
the MDWDF circuit in terms of state quantities. Graphic results have
clearly demonstrated some physical effects of seismic wave (P-wave
and S–wave) propagation including radiation, reflection, and
refraction from and across the hard boundaries. Comparison between
the MDWDF technique and the finite difference time domain (FDTD)
approach is also made in terms of the computational efficiency.
Abstract: Piezoelectric transformers are electronic devices made
from piezoelectric materials. The piezoelectric transformers as the
name implied are used for changing voltage signals from one level to another. Electrical energy carried with signals is transferred by means of mechanical vibration. Characterizing in both electrical and
mechanical properties leads to extensively use and efficiency enhancement of piezoelectric transformers in various applications. In
this paper, study and analysis of electrical and mechanical properties of multi-layer piezoelectric transformers in forms of potential and
displacement distribution throughout the volume, respectively. This
paper proposes a set of quasi-static mathematical model of electromechanical
coupling for piezoelectric transformer by using a set of
partial differential equations. Computer-based simulation utilizing the three-dimensional finite element method (3-D FEM) is exploited
as a tool for visualizing potentials and displacements distribution
within the multi-layer piezoelectric transformer. This simulation was
conducted by varying a number of layers. In this paper 3, 5 and 7 of
the circular ring type were used. The computer simulation based on
the use of the FEM has been developed in MATLAB programming environment.
Abstract: In this work, we derive two numerical schemes for
solving a class of nonlinear partial differential equations. The first
method is of second order accuracy in space and time directions, the
scheme is unconditionally stable using Von Neumann stability
analysis, the scheme produced a nonlinear block system where
Newton-s method is used to solve it. The second method is of fourth
order accuracy in space and second order in time. The method is
unconditionally stable and Newton's method is used to solve the
nonlinear block system obtained. The exact single soliton solution
and the conserved quantities are used to assess the accuracy and to
show the robustness of the schemes. The interaction of two solitary
waves for different parameters are also discussed.