166 lines
2.9 KiB
Plaintext
166 lines
2.9 KiB
Plaintext
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\lyxformat 474
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\begin_document
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\begin_header
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\textclass article
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\use_default_options true
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\maintain_unincluded_children false
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\language english
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\font_tt_scale 100
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\graphics default
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\bibtex_command default
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\use_package amsmath 1
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\use_package amssymb 1
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\use_package cancel 1
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\use_package esint 1
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\use_package mathdots 1
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\use_package mathtools 1
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\use_package mhchem 1
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\use_package stackrel 1
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\use_package stmaryrd 1
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\cite_engine basic
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\index Index
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\shortcut idx
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\color #008000
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\end_index
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\end_header
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\begin_body
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\begin_layout Title
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Boundary layer flow
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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\frac{\partial u}{\partial t}-\frac{1}{s^{2}}\frac{\partial^{2}u}{\partial y^{2}}=K(t)
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula $y=0:u=0$
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\end_inset
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,
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\begin_inset Formula $y=1,\frac{\partial u}{\partial y}=0$
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\end_inset
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\end_layout
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\begin_layout Standard
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Discretization, FTCD:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}-\frac{1}{s^{2}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{\Delta y^{2}}=K^{n}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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Rewriting:
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\begin_inset Note Note
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status open
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\begin_layout Plain Layout
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\begin_inset Formula $u_{i}^{n+1}-u_{i}^{n}-\frac{\Delta t}{\Delta y^{2}s^{2}}\frac{u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}}{}=K^{n}\Delta t$
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Formula $u_{i}^{n+1}=\Delta tK^{n}+u_{i}^{n}+\frac{\Delta t}{\Delta y^{2}s^{2}}\left(u_{i+1}^{n}-2u_{i}^{n}-u_{i-1}^{n}\right)$
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Formula $u_{0}=0$
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\end_inset
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\end_layout
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\begin_layout Plain Layout
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and
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\end_layout
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\begin_layout Plain Layout
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\begin_inset Formula $u_{N}-u_{n-1}=0$
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\end_inset
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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\]
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\end_inset
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\end_layout
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\end_body
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\end_document
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