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Stochastic Parabolic Equations in the Whole Space
| Content Provider | Scilit |
|---|---|
| Author | Chow, Pao-Liu |
| Copyright Year | 2014 |
| Description | This chapter is concerned with some parabolic Itoˆ equations in Rd. In Chapter 3 we considered such equations in a bounded domain D. The analysis was based mainly on the method of eigenfunction expansion for the associated elliptic boundary-valued problem. In Rd the spectrum of such an elliptic operator may be continuous and the method of eigenfunction expansion is no longer suitable. For an elliptic operator with constant coefficients, the method of Fourier transform is a natural substitution. Consider the initial-value or Cauchy problem for the heat equation on the real line: ∂u ∂t = ∂2u ∂x2 + f(x, t), −∞ < x <∞, t > 0, u(x, 0) = h(x), (4.1) where f and g are given smooth functions. Let uˆ(λ, t) denote a Fourier transform of u in the space variable defined by uˆ(λ, t) = √ 1 2π ∫ e−iλxu(x, t)dx with i = √−1, where λ ∈ R is a parameter and the integration is over the real line. By applying a Fourier transform to the heat equation, it yields a simple ordinary differential equation duˆ dt = −λ2uˆ+ fˆ(λ, t), uˆ(λ, 0) = hˆ(λ), (4.2) which can be easily solved to give uˆ(λ, t) = hˆ(λ)e−λ 2t + ∫ t e−λ 2(t−s)fˆ(λ, s)ds. Book Name: Stochastic Partial Differential Equations |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2014-0-38727-8&isbn=9780429101113&doi=10.1201/b17823-8&format=pdf |
| Ending Page | 144 |
| Page Count | 24 |
| Starting Page | 121 |
| DOI | 10.1201/b17823-8 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2014-12-10 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Stochastic Partial Differential Equations Mathematical Physics Differential Eigenfunction |
| Content Type | Text |
| Resource Type | Chapter |
