# 微积分网课代修|偏微分方程代写Partial Differential Equation代考|AMATH569 Poisson’s Equation with Complex 2-D Geometry

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Poisson’s Equation with Complex 2-D Geometry

This example shows how to solve the Poisson’s equation, $-\Delta u=f$ using the PDE app. This problem requires configuring a 2-D geometry with Dirichlet and Neumann boundary conditions.

To start the PDE app, type the command pdetool at the MATLAB prompt. The PDE app looks similar to the following figure, with exception of the grid. Turn on the grid by selecting Grid from the Options menu. Also, enable the “snap-to-grid” feature by selecting Snap from the Options menu. The “snap-to-grid” feature simplifies aligning the solid objects.

The first step is to draw the geometry on which you want to solve the PDE. The PDE app provides four basic types of solid objects: polygons, rectangles, circles, and ellipses. The objects are used to create a Constructive Solid Geometry model (CSG model). Each solid object is assigned a unique label, and by the use of set algebra, the resulting geometry can be made up of a combination of unions, intersections, and set differences. By default, the resulting CSG model is the union of all solid objects.
To select a solid object, either click the button with an icon depicting the solid object that you want to use, or select the object by using the Draw pull-down menu. In this case, rectangle/square objects are selected. To draw a rectangle or a square starting at a corner, click the rectangle button without a $+$ sign in the middle. The button with the $+$ sign is used when you want to draw starting at the center. Then, put the cursor at the desired corner, and click-and-drag using the left mouse button to create a rectangle with the desired side lengths. (Use the right mouse button to create a square.) Click and drag from $(-1, .2)$ to $(1,-.2)$. Notice how the “snap-to-grid” feature forces the rectangle to line up with the grid. When you release the mouse, the CSG model is updated and redrawn. At this stage, all you have is a rectangle. It is assigned the label R1. If you want to move or resize the rectangle, you can easily do so. Click-and-drag an object to move it, and doubleclick an object to open a dialog box, where you can enter exact location coordinates. From the dialog box, you can also alter the label. If you are not satisfied and want to restart, you can delete the rectangle by clicking the Delete key or by selecting Clear from the Edit menu.

Next, draw a circle by clicking the button with the ellipse icon with the $+$ sign, and then click-and-drag in a similar way, starting near the point $(-.5,0)$ with radius $.4$, using the right mouse button, starting at the circle center.

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|PDE App Shortcuts

The toolbar consists of three different parts: the five leftmost buttons for draw mode functions, the next six buttons for different boundary, mesh, solution, and plot functions, and the rightmost button for activating the zoom feature.

Five buttons on the left let you draw the geometry. Double-click a button makes it “stick,” and you can then continue to draw solid objects of the selected type until you single-click the button to “release” it.
In draw mode, you can create the 2 -D geometry using the constructive solid geometry (CSG) model paradigm. A set of solid objects (rectangle, circle, ellipse, and polygon) is provided. These objects can be combined using set formulas in a flexible way.

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|PDE App Shortcuts

PDE 应用程序工具栏提供对菜单中也可用的关键操作的快速访问。