The Continuous Galerkin Method for an Integro-differential Equation Modeling Dynamic Fractional Order Viscoelasticity
نویسندگان
چکیده
We consider a fractional order integro-differential equation with a weakly singular convolution kernel. The equation with homogeneuos Dirichlet boundary conditions is reformulated as an abstract Cauchy problem, and well-posedness is verified in the context of linear semigroup theory. Then we formulate a continuous Galerkin method for the problem, and we prove stability estimates. These are then used to prove a priori error estimates. The theory is illustrated by a numerical example.
منابع مشابه
Discontinuous Galerkin Method for an Integro-differential Equation Modeling Dynamic Fractional Order Viscoelasticity
An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem. Stability estimates of the discrete problem are proved, that are used to prove optimal order a priori error estimate...
متن کاملDiscretization of Integro-Differential Equations Modeling Dynamic Fractional Order Viscoelasticity
We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integrodifferential equation with a weakly singular convolution kernel. We discretize in the spatial variable by a standard Galerkin finite element method. We prove stability and regularity estimates which show how the convolution term introduces dissipation into the equa...
متن کاملDynamic Simulation and Control of a Continuous Bioreactor Based on Cell Population Balance Model
Saccharomyces cerevisiae (baker’s yeast) can exhibit sustained oscillations during the operation in a continuous bioreactor that adversely affects its stability and productivity. Because of heterogeneous nature of cell populations, the cell population balance equation (PBE) can be used to capture the dynamic behavior of such cultures. In this work, an unstructured-segregated model is used f...
متن کاملThe Petrov-Galerkin Method and Chebyshev Multiwavelet Basis for Solving Integro-Differential Equations
Abstract: There are some methods for solving integro-differential equations. In this work, we solve the general-order Feredholm integro-differential equations. The Petrov-Galerkin method by considering Chebyshev multiwavelet basis is used. By using the orthonormality property of basis elements in discretizing the equation, we can reduce an equation to a linear system with small dimension. For ...
متن کاملModified Laplace decomposition method for fractional Volterra-Fredholm integro-differential equations
This paper successfully applies the Adomian decomposition and the modified Laplace Adomian decomposition methods to find the approximate solution of a nonlinear fractional Volterra-Fredholm integro-differential equation. The reliability of the methods and reduction in the size of the computational work give these methods a wider applicability. Also, the behavior of the solution can be formall...
متن کامل