22 research outputs found
Mathematical analysis of the optimizing acquisition and retention over time problem
No full textWhile making informed decisions regarding
investments in customer retention and acquisition becomes a
pressing managerial issue, formal models and analysis, which may
provide insight into this topic, are still scarce. In this study
we examine two dynamic models for optimal acquisition and
retention models of a monopoly, the total cost and the cost per
customer models.
These models are analytically analyzed using classical, direct,
methods and asymptotic expansions (for the total cost model).
In order to numerically simulated the models, an innovative
numerical method was developed for solving ODE systems with
initial/final value problems.
Multi-Dimensional Asymptotically Stable 4th Order Accurate Schemes for the Diffusion Equation
No full textAn algorithm is presented which solves the multi-dimensional diffusion equation on co mplex shapes to 4th-order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions fail
Multi-Dimensional Asymptotically Stable Finite Difference Schemes for the Advection-Diffusion Equation
No full textAn algorithm is presented which solves the multi-dimensional advection-diffusion equation on complex shapes to 2 -order accuracy and is asymptotically stable in time. This bounded-error result is achieved by constructing, on a rectangular grid, a differentiation matrix whose symmetric part is negative definite. The differentiation matrix accounts for the Dirichlet boundary condition by imposing penalty like terms. Numerical examples in 2-D show that the method is effective even where standard schemes, stable by traditional definitions, fail. It gives accurate, non oscillatory results even when boundary layers are not resolved
Artificial Boundary Conditions for the Simulation of the Heat Equation in an Infinite Domain
No full textBounded Error Schemes for the Wave Equation on Complex Domains
No full textThis paper considers the application of the method of boundary penalty terms ("SAT") to the numerical solution of the wave equation on complex shapes with Dirichlet boundary conditions. A theory is developed, in a semi-discrete setting, that allows the use of a Cartesian grid on complex geometries, yet maintains the order of accuracy with only a linear temporal error-bound. A numerical example, involving the solution of Maxwell's equations inside a 2-D circular wave-guide demonstrates the efficacy of this method in comparison to others (e.g. the staggered Yee scheme) - we achieve a decrease of two orders of magnitude in the level of the L2-error
On Error Bounds of Finite Difference Approximations to Partial Differential EquationsTemporal Behavior and Rate of Convergence
No full textThis paper considers a family of spatially semi-discrete approximations, includ-ing boundary treatments, to hyperbolic and parabolic equations. We derive the dependence of the error-bounds on time as well as on mesh size. KEY WORDS: Finite difference; error bounds. 1
DOI: 10.1007/s10915-006-9112-x Efficient Solution of Ax (k) = b (k) Using A −1
No full textwhere b (k+1) = f(x (k)). We show that when A is a full n × n matrix and K � cn, where c ≪ 1 depends on the specific software and hardware setup, it is faster to solve Ax (k) = b (k) for k = 1,...,K by explicitly evaluating the inverse matrix A −1 rather than through the LU decomposition of A. We also show that the forward error is comparable in both methods, regardless of the condition number of A. KEY WORDS: Matrix inversion; linear systems
On error bounds of finite difference approximations to partial differential equations : Temporal behavior and rate of convergence
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