Search results for "Mathematical analysis"

Mellin transform approach for the solution of coupled systems of fractional differential equations

2015

In this paper, the solution of a multi-order, multi-degree-of-freedom fractional differential equation is addressed by using the Mellin integral transform. By taking advantage of a technique that relates the transformed function, in points of the complex plane differing in the value of their real part, the solution is found in the Mellin domain by solving a linear set of algebraic equations. The approximate solution of the differential (or integral) equation is restored, in the time domain, by using the inverse Mellin transform in its discretized form.

Weighted ENO interpolation and applications

2004

Abstract Data-dependent interpolatory techniques such as essentially non-oscillatory (ENO) technique [J. Comput. Phys. 71 (1987) 231] have long been used as a reconstruction process in multiresolution schemes. In this work we analyze the weighted ENO (WENO) technique introduced by Liu et al. in the context of conservation laws [J. Comput. Phys. 115 (1994) 200] and improved by Jiang and Shu [J. Comput. Phys. 126 (1996) 202], and apply it to the compression of images, using multiresolution techniques.

Active macro-zone approach for incremental elastoplastic-contact analysis

2013

SUMMARY The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact-detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem. A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem-elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) betw…

Corrigendum to “Fractional differential equations solved by using Mellin transform” [Commun Nonlinear Sci Numer Simul 19(7) (2014) 2220–2227]

2015

Order optimal preconditioners for fully implicit Runge-Kutta schemes applied to the bidomain equations

2010

The partial differential equation part of the bidomain equations is discretized in time with fully implicit Runge–Kutta methods, and the resulting block systems are preconditioned with a block diagonal preconditioner. By studying the time-stepping operator in the proper Sobolev spaces, we show that the preconditioned systems have bounded condition numbers given that the Runge–Kutta scheme is A-stable and irreducible with an invertible coefficient matrix. A new proof of order optimality of the preconditioners for the one-leg discretization in time of the bidomain equations is also presented. The theoretical results are verified by numerical experiments. Additionally, the concept of weakly po…

Controllability method for the Helmholtz equation with higher-order discretizations

2007

We consider a controllability technique for the numerical solution of the Helmholtz equation. The original time-harmonic equation is represented as an exact controllability problem for the time-dependent wave equation. This problem is then formulated as a least-squares optimization problem, which is solved by the conjugate gradient method. Such an approach was first suggested and developed in the 1990s by French researchers and we introduce some improvements to its practical realization. We use higher-order spectral elements for spatial discretization, which leads to high accuracy and lumped mass matrices. Higher-order approximation reduces the pollution effect associated with finite elemen…

Quasi-isometries associated to A-contractions

2014

Abstract Given two operators A and T ( A ≥ 0 , ‖ A ‖ = 1 ) on a Hilbert space H satisfying T ⁎ A T ≤ A , we study the maximum subspace of H which reduces M = A 1 / 2 T to a quasi-isometry, that is on which the equality M ⁎ M = M ⁎ 2 M 2 holds. In some cases, this subspace coincides with the maximum subspace which reduces M to a normal partial isometry, for example when A = T T ⁎ , and in particular if T ⁎ is a cohyponormal contraction. In this case the corresponding subspace can be completely described in terms of asymptotic limit of the contraction T. When M is quasinormal and M ⁎ M = A then the former above quoted subspace reduces to the kernel of A − A 2 . The case of an arbitrary contra…

Monotonic solution of heterogeneous anisotropic diffusion problems

2013

Anisotropic problems arise in various areas of science and engineering, for example groundwater transport and petroleum reservoir simulations. The pure diffusive anisotropic time-dependent transport problem is solved on a finite number of nodes, that are selected inside and on the boundary of the given domain, along with possible internal boundaries connecting some of the nodes. An unstructured triangular mesh, that attains the Generalized Anisotropic Delaunay condition for all the triangle sides, is automatically generated by properly connecting all the nodes, starting from an arbitrary initial one. The control volume of each node is the closed polygon given by the union of the midpoint of…

Time-harmonic elasticity with controllability and higher-order discretization methods

2008

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions.The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situa…

Solution of time-independent Schrödinger equation by the imaginary time propagation method

2007

Numerical solution of eigenvalues and eigenvectors of large matrices originating from discretization of linear and non-linear Schrodinger equations using the imaginary time propagation (ITP) method is described. Convergence properties and accuracy of 2nd and 4th order operator-splitting methods for the ITP method are studied using numerical examples. The natural convergence of the method is further accelerated with a new dynamic time step adjustment method. The results show that the ITP method has better scaling with respect to matrix size as compared to the implicitly restarted Lanczos method. An efficient parallel implementation of the ITP method for shared memory computers is also demons…