1,523 research outputs found
Parallel algorithm with spectral convergence for nonlinear integro-differential equations
We discuss a numerical algorithm for solving nonlinear integro-differential
equations, and illustrate our findings for the particular case of Volterra type
equations. The algorithm combines a perturbation approach meant to render a
linearized version of the problem and a spectral method where unknown functions
are expanded in terms of Chebyshev polynomials (El-gendi's method). This
approach is shown to be suitable for the calculation of two-point Green
functions required in next to leading order studies of time-dependent quantum
field theory.Comment: 15 pages, 9 figure
Semi-spectral Chebyshev method in Quantum Mechanics
Traditionally, finite differences and finite element methods have been by
many regarded as the basic tools for obtaining numerical solutions in a variety
of quantum mechanical problems emerging in atomic, nuclear and particle
physics, astrophysics, quantum chemistry, etc. In recent years, however, an
alternative technique based on the semi-spectral methods has focused
considerable attention. The purpose of this work is first to provide the
necessary tools and subsequently examine the efficiency of this method in
quantum mechanical applications. Restricting our interest to time independent
two-body problems, we obtained the continuous and discrete spectrum solutions
of the underlying Schroedinger or Lippmann-Schwinger equations in both, the
coordinate and momentum space. In all of the numerically studied examples we
had no difficulty in achieving the machine accuracy and the semi-spectral
method showed exponential convergence combined with excellent numerical
stability.Comment: RevTeX, 12 EPS figure
A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations
In this paper we introduce a numerical method for solving nonlinear Volterra
integro-differential equations. In the first step, we apply implicit trapezium
rule to discretize the integral in given equation. Further, the Daftardar-Gejji
and Jafari technique (DJM) is used to find the unknown term on the right side.
We derive existence-uniqueness theorem for such equations by using Lipschitz
condition. We further present the error, convergence, stability and bifurcation
analysis of the proposed method. We solve various types of equations using this
method and compare the error with other numerical methods. It is observed that
our method is more efficient than other numerical methods
Optimal control of systems with memory
The “Optimal Control of Systems with memory” is a PhD project that is borne
from the collaboration between the Department of Mechanical and Aerospace
Engineering of Sapienza University of Rome and CNR-INM the Institute for Marine
Engineering of the National Research Council of Italy (ex INSEAN). This project is
part of a larger EDA (European Defence Agency) project called ETLAT: Evaluation
of State of the Art Thin Line Array Technology. ETLAT is aimed at improving
the scientific and technical knowledge of potential performance of current Thin
Line Towed Array (TLA) technologies (element sensors and arrays) in view of
Underwater Surveillance applications.
A towed sonar array has been widely employed as an important tool for naval
defence, ocean exploitation and ocean research. Two main operative limitations
costrain the TLA design such as: a fixed immersion depth and the stabilization of
its horizontal trim. The system is composed by a towed vehicle and a towed line
sonar array (TLA). The two subsystems are towed by a towing cable attached to
the moving boat. The role of the vehicle is to guarantee a TLA’s constant depth of
navigation and the reduction of the entire system oscillations. The vehicle is also
called "depressor" and its motion generates memory effects that influence the proper
operation of the TLA. The dynamic of underwater towed system is affected by
memory effects induced by the fluid-structure interaction, namely: vortex shedding
and added damping due to the presence of a free surface in the fluid. In time
domain, memory effects are represented by convolution integral between special
kernel functions and the state of the system. The mathematical formulation of the
underwater system, implies the use of integral-differential equations in the time
domain, that requires a nonstandard optimal control strategy. The goal of this
PhD work is to developed a new optimal control strategy for mechanical systems
affected by memory effects and described by integral-differential equations. The
innovative control method presented in this thesis, is an extension of the Pontryagin
optimal solution which is normally applied to differential equations. The control is
based on the variational control theory implying a feedback formulation, via model
predictive control.
This work introduces a novel formulation for the control of the vehicle and cable
oscillations that can include in the optimal control integral terms besides the more
conventional differential ones. The innovative method produces very interesting
results, that show how even widely applied control methods (LQR) fail, while the
present formulation exhibits the advantage of the optimal control theory based on
integral-differential equations of motion
High Accuracy Combination Method For Solving the Systems of Nonlinear Volterra Integral and Integro-differential Equations with Weakly Singular Kernels of the Second Kind
This paper presents a high accuracy combination algorithm for solving the systems of nonlinear Volterra integral and integro-differential equations with weakly singular kernels of the second kind. Two quadrature algorithms for solving the systems are discussed, which possess high accuracy order and the asymptotic expansion of the errors. By means of combination algorithm, we may obtain a numerical solution with higher accuracy order than the original two quadrature algorithms. Moreover an a posteriori error estimation for the algorithm is derived. Both of the theory and the numerical examples show that the algorithm is effective and saves storage capacity and computational cost
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