Search results for "Partial"
showing 10 items of 1477 documents
Explicit solutions for second-order operator differential equations with two boundary-value conditions. II
1992
AbstractBoundary-value problems for second-order operator differential equations with two boundary-value conditions are studied for the case where the companion operator is similar to a block-diagonal operator. This case is strictly more general than the one treated in an earlier paper, and it provides explicit closed-form solutions of boundary-value problem in terms of data without increasing the dimension of the problem.
CQ *-algebras of measurable operators
2022
Abstract We study, from a quite general point of view, a CQ*-algebra (X, 𝖀0) possessing a sufficient family of bounded positive tracial sesquilinear forms. Non-commutative L 2-spaces are shown to constitute examples of a class of CQ*-algebras and any abstract CQ*-algebra (X, 𝖀0) possessing a sufficient family of bounded positive tracial sesquilinear forms can be represented as a direct sum of non-commutative L 2-spaces.
The MAST FV/FE scheme for the simulation of two-dimensional thermohaline processes in variable-density saturated porous media
2009
A novel methodology for the simulation of 2D thermohaline double diffusive processes, driven by heterogeneous temperature and concentration fields in variable-density saturated porous media, is presented. The stream function is used to describe the flow field and it is defined in terms of mass flux. The partial differential equations governing system is given by the mass conservation equation of the fluid phase written in terms of the mass-based stream function, as well as by the advection-diffusion transport equations of the contaminant concentration and of the heat. The unknown variables are the stream function, the contaminant concentration and the temperature. The governing equations sy…
Efficient numerical methods for pricing American options under stochastic volatility
2007
Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved u…
Numerical Study of Two Sparse AMG-methods
2003
A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.
An iterative method for pricing American options under jump-diffusion models
2011
We propose an iterative method for pricing American options under jump-diffusion models. A finite difference discretization is performed on the partial integro-differential equation, and the American option pricing problem is formulated as a linear complementarity problem (LCP). Jump-diffusion models include an integral term, which causes the resulting system to be dense. We propose an iteration to solve the LCPs efficiently and prove its convergence. Numerical examples with Kou@?s and Merton@?s jump-diffusion models show that the resulting iteration converges rapidly.
Implicit analytic solutions for a nonlinear fractional partial differential beam equation
2020
Abstract Analytic solutions in implicit form are derived for a nonlinear partial differential equation (PDE) with fractional derivative elements, which can model the dynamics of a deterministically excited Euler-Bernoulli beam resting on a viscoelastic foundation. Specifically, the initial-boundary value problem for the corresponding PDE is reduced to an initial value problem for a nonlinear ordinary differential equation in a Hilbert space. Next, by employing the cosine and sine families of operators, a variation of parameters representation of the solution map is introduced. Due to the presence of a nonlinear term, a local fixed point theorem is employed to prove the local existence and u…
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…