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‎docs/RK1_euler_doc.html‎

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</style></head><body><div class="content"><h1><tt>RK1_euler</tt></h1><!--introduction--><p>Propagates the state vector forward one time step using the Euler (first-order) method.</p><p><a href="index.html">Back to ODE Solver Toolbox Contents</a>.</p><!--/introduction--><h2>Contents</h2><div><ul><li><a href="#1">Syntax</a></li><li><a href="#2">Description</a></li><li><a href="#3">Input/Output Parameters</a></li><li><a href="#4">Example</a></li><li><a href="#8">See also</a></li></ul></div><h2 id="1">Syntax</h2><pre class="language-matlab">y_next = RK1_euler(f,t,y,h)
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</pre><h2 id="2">Description</h2><p><tt>y_next = RK1_euler(f,t,y,h)</tt> returns the state vector at the next sample time, <tt>y_next</tt>, given the current state vector <tt>y</tt> at time <tt>t</tt>, the function <tt>f(t,y)</tt> defining the ODE <img src="RK1_euler_doc_eq13964145860186194730.png" alt="$\dot{\mathbf{y}}=\mathbf{f}(t,\mathbf{y})$" style="width:50px;height:11px;">, and the step size <tt>h</tt>.</p><h2 id="3">Input/Output Parameters</h2><p>
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<table border=1>
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<tr>
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<td></td>
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<td style="text-align:center"><b>Variable</b></td>
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<td style="text-align:center"><b>Symbol</b></td>
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<td style="text-align:center"><b>Description</b></td>
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<td style="text-align:center"><b>Format</b></td>
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</tr>
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<tr>
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<td rowspan="4" style="text-align:center"><b>Input</b></td>
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<td style="text-align:center"><TT>f</TT></td>
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<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}(t,\mathbf{y})" title="" /></td>
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<td>multivariate, vector-valued function (<img
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src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}:\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{p}"
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title="" />) defining the ordinary differential equation <img src="https://latex.codecogs.com/svg.latex?\inline&space;\frac{d\mathbf{y}}{dt}=\mathbf{f}(t,\mathbf{y})" title="" />
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<BR> - inputs to <TT>f</TT> are the current time (<TT>t</TT>, 1×1 double) and the current state vector (<TT>y</TT>, p×1 double)
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<BR> - output of <TT>f</TT> is the state vector derivative (<TT>dydt</TT>, p×1 double) at the current time/state</td>
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<td style="text-align:center">1×1<BR>function_handle</td>
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</tr>
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<tr>
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<td style="text-align:center"><TT>t</TT></td>
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<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;t_{n}" title="" /></td>
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<td>current sample time</td>
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<td style="text-align:center">1×1<BR>double</td>
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</tr>
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<tr>
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<td style="text-align:center"><TT>y</TT></td>
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<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{y}_{n}=\mathbf{y}(t_{n})" title="" /></td>
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<td>state vector (i.e. solution) at the current sample time</td>
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<td style="text-align:center">p×1<BR>double</td>
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</tr>
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<tr>
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<td style="text-align:center"><TT>h</TT></td>
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<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;h" title="h" /></td>
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<td>step size</td>
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<td style="text-align:center">1×1<BR>double</td>
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</tr>
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<tr>
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<td rowspan="1" style="text-align:center"><b>Output</b></td>
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<td style="text-align:center"><TT>y_next</TT></td>
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<td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\mathbf{y}_{n+1}=\mathbf{y}(t_{n+1})" title="" /></td>
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<td>state vector (i.e. solution) at the next sample time, <img src="https://latex.codecogs.com/svg.latex?t_{n+1}=t_{n}+h" title="" /></td>
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<td style="text-align:center">p×1<BR>double</td>
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</tr>
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</table>
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</p><h2 id="4">Example</h2><p><i>Consider the initial value problem</i></p><p><img src="RK1_euler_doc_eq12868018520907325398.png" alt="$$\frac{dy}{dt}=y,\quad y(2)=3$$" style="width:89px;height:23px;"></p><p><i>Find the solution <img src="RK1_euler_doc_eq17784377804790973832.png" alt="$y(t)$" style="width:18px;height:11px;"> until <img src="RK1_euler_doc_eq08325516970591449802.png" alt="$t=10$" style="width:29px;height:8px;"> using <tt>RK1_euler</tt>. Then, compare your result to the solution found by <tt>oderk</tt> using the Euler method.</i></p><p>First, let's define our ODE (<img src="RK1_euler_doc_eq15749537601348477239.png" alt="$\frac{dy}{dt}=f(t,y)$" style="width:53px;height:15px;">) and initial condition in MATLAB.</p><pre class="codeinput">f = @(t,y) y;
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y2 = 3;
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</pre><p>Let's define a time vector between <img src="RK1_euler_doc_eq11906475312356421895.png" alt="$t=2$" style="width:24px;height:8px;"> and <img src="RK1_euler_doc_eq08325516970591449802.png" alt="$t=10$" style="width:29px;height:8px;"> with a spacing of <img src="RK1_euler_doc_eq10072498013177321932.png" alt="$h=0.01$" style="width:40px;height:8px;">.</p><pre class="codeinput">h = 0.01;
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t = (2:h:10)';
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</pre><p>Solving for <img src="RK1_euler_doc_eq17784377804790973832.png" alt="$y(t)$" style="width:18px;height:11px;"> using <tt>RK1_euler</tt> and comparing the result to the result obtained using <tt>oderk</tt> with the Euler method,</p><pre class="codeinput"><span class="comment">% preallocate vector to store solution</span>
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y = zeros(size(t));
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<span class="comment">% store initial condition</span>
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y(1) = y2;
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<span class="comment">% solving using "RK1_euler"</span>
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<span class="keyword">for</span> i = 1:(length(t)-1)
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y(i+1) = RK1_euler(f,t(i),y(i),h);
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<span class="keyword">end</span>
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<span class="comment">% solving using "oderk"</span>
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[t_oderk,y_oderk] = oderk(f,[2,10],y2,h,<span class="string">'RK1_euler'</span>);
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<span class="comment">% maximum absolute error between the two results</span>
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max(abs(y_oderk-y))
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</pre><pre class="codeoutput">
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ans =
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0
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</pre><p>As expected, the two methods obtain identical results.</p><h2 id="8">See also</h2><p><a href="RK2_doc.html"><tt>RK2</tt></a> | <a href="RK2_heun_doc.html"><tt>RK2_heun</tt></a> | <a href="RK2_ralston_doc.html"><tt>RK2_ralston</tt></a> | <a href="RK3_doc.html"><tt>RK3</tt></a> | <a href="RK3_heun_doc.html"><tt>RK3_heun</tt></a> | <a href="RK3_ralston_doc.html"><tt>RK3_ralston</tt></a> | <a href="SSPRK3_doc.html"><tt>SSPRK3</tt></a> | <a href="RK4_doc.html"><tt>RK4</tt></a> | <a href="RK4_ralston_doc.html"><tt>RK4_ralston</tt></a> | <a href="RK4_38_doc.html"><tt>RK4_38</tt></a></p><p class="footer"><br><a href="https://www.mathworks.com/products/matlab/">Published with MATLAB&reg; R2021b</a><br></p></div><!--
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##### SOURCE BEGIN #####
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%% |RK1_euler|
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% Propagates the state vector forward one time step using the Euler
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% (first-order) method.
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%
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% <index.html Back to ODE Solver Toolbox Contents>.
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%% Syntax
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% y_next = RK1_euler(f,t,y,h)
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%% Description
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% |y_next = RK1_euler(f,t,y,h)| returns the state vector at the next
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% sample time, |y_next|, given the current state vector |y| at time |t|,
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% the function |f(t,y)| defining the ODE
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% $\dot{\mathbf{y}}=\mathbf{f}(t,\mathbf{y})$, and the step size |h|.
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%% Input/Output Parameters
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% <html>
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% <table border=1>
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% <tr>
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% <td></td>
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% <td style="text-align:center"><b>Variable</b></td>
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% <td style="text-align:center"><b>Symbol</b></td>
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% <td style="text-align:center"><b>Description</b></td>
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% <td style="text-align:center"><b>Format</b></td>
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% </tr>
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% <tr>
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% <td rowspan="4" style="text-align:center"><b>Input</b></td>
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% <td style="text-align:center"><TT>f</TT></td>
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% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}(t,\mathbf{y})" title="" /></td>
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% <td>multivariate, vector-valued function (<img
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% src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{f}:\mathbb{R}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{p}"
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% title="" />) defining the ordinary differential equation <img src="https://latex.codecogs.com/svg.latex?\inline&space;\frac{d\mathbf{y}}{dt}=\mathbf{f}(t,\mathbf{y})" title="" />
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% <BR> - inputs to <TT>f</TT> are the current time (<TT>t</TT>, 1×1 double) and the current state vector (<TT>y</TT>, p×1 double)
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% <BR> - output of <TT>f</TT> is the state vector derivative (<TT>dydt</TT>, p×1 double) at the current time/state</td>
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% <td style="text-align:center">1×1<BR>function_handle</td>
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% </tr>
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% <tr>
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% <td style="text-align:center"><TT>t</TT></td>
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% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;t_{n}" title="" /></td>
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% <td>current sample time</td>
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% <td style="text-align:center">1×1<BR>double</td>
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% </tr>
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% <tr>
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% <td style="text-align:center"><TT>y</TT></td>
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% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;\mathbf{y}_{n}=\mathbf{y}(t_{n})" title="" /></td>
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% <td>state vector (i.e. solution) at the current sample time</td>
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% <td style="text-align:center">p×1<BR>double</td>
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% </tr>
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% <tr>
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% <td style="text-align:center"><TT>h</TT></td>
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% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\inline&space;h" title="h" /></td>
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% <td>step size</td>
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% <td style="text-align:center">1×1<BR>double</td>
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% </tr>
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% <tr>
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% <td rowspan="1" style="text-align:center"><b>Output</b></td>
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% <td style="text-align:center"><TT>y_next</TT></td>
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% <td style="text-align:center"><img src="https://latex.codecogs.com/svg.latex?\mathbf{y}_{n+1}=\mathbf{y}(t_{n+1})" title="" /></td>
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% <td>state vector (i.e. solution) at the next sample time, <img src="https://latex.codecogs.com/svg.latex?t_{n+1}=t_{n}+h" title="" /></td>
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% <td style="text-align:center">p×1<BR>double</td>
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% </tr>
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% </table>
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% </html>
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%% Example
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% _Consider the initial value problem_
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%
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% $$\frac{dy}{dt}=y,\quad y(2)=3$$
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%
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% _Find the solution $y(t)$ until $t=10$ using |RK1_euler|. Then, compare
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% your result to the solution found by |oderk| using the Euler method._
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%
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% First, let's define our ODE ($\frac{dy}{dt}=f(t,y)$) and initial
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% condition in MATLAB.
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f = @(t,y) y;
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y2 = 3;
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%%
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% Let's define a time vector between $t=2$ and $t=10$ with a spacing of
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% $h=0.01$.
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h = 0.01;
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t = (2:h:10)';
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%%
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% Solving for $y(t)$ using |RK1_euler| and comparing the result to the
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% result obtained using |oderk| with the Euler method,
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% preallocate vector to store solution
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y = zeros(size(t));
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% store initial condition
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y(1) = y2;
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% solving using "RK1_euler"
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for i = 1:(length(t)-1)
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y(i+1) = RK1_euler(f,t(i),y(i),h);
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end
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% solving using "oderk"
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[t_oderk,y_oderk] = oderk(f,[2,10],y2,h,'RK1_euler');
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% maximum absolute error between the two results
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max(abs(y_oderk-y))
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%%
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% As expected, the two methods obtain identical results.
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%% See also
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% <RK2_doc.html |RK2|> |
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% <RK2_heun_doc.html |RK2_heun|> |
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% <RK2_ralston_doc.html |RK2_ralston|> |
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% <RK3_doc.html |RK3|> |
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% <RK3_heun_doc.html |RK3_heun|> |
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% <RK3_ralston_doc.html |RK3_ralston|> |
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% <SSPRK3_doc.html |SSPRK3|> |
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% <RK4_doc.html |RK4|> |
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% <RK4_ralston_doc.html |RK4_ralston|> |
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% <RK4_38_doc.html |RK4_38|>
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##### SOURCE END #####
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--></body></html>

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