Trotter products and reaction-diffusion equations. (English) Zbl 1179.65113
Summary: We study a class of generalized diffusion-reaction equations of the form
\[ \frac{\partial u}{\partial t}(x,t)=(Au(\cdot ,t))(x)+f(x,u(x,t)), \]
where \(A\) is a pseudodifferential operator which generates a Feller semigroup. Using the Trotter product formula we give a corresponding discrete time integro-difference equation for numerical solutions.
\[ \frac{\partial u}{\partial t}(x,t)=(Au(\cdot ,t))(x)+f(x,u(x,t)), \]
where \(A\) is a pseudodifferential operator which generates a Feller semigroup. Using the Trotter product formula we give a corresponding discrete time integro-difference equation for numerical solutions.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35K57 | Reaction-diffusion equations |
| 35R11 | Fractional partial differential equations |
| 47G30 | Pseudodifferential operators |
| 47D07 | Markov semigroups and applications to diffusion processes |
Keywords:
reaction-diffusion equation; pseudodifferential operator; Feller semigroup; fractional diffusion equation; Trotter product formula; discrete time integro-difference equationReferences:
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