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Intuitive Understanding of Solutions of Partially Differential Equations

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Title: Intuitive Understanding of Solutions of Partially Differential Equations
Language: English
Authors: Kobayashi, Y.
Source: International Journal of Mathematical Education in Science and Technology. Apr 2008 39(3):365-371.
Availability: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals/default.html
Peer Reviewed: Y
Physical Description: PDF
Page Count: 7
Publication Date: 2008
Document Type: Journal Articles
Reports - Descriptive
Descriptors: Mathematics Education, Calculus, Problem Solving, Thermodynamics, Scientific Concepts, Visual Perception, Visual Learning, Demonstrations (Educational)
DOI: 10.1080/00207390600913343
ISSN: 0020-739X
Abstract: This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based on the transition rule are useful for understanding the process. (Contains 8 figures.)
Abstractor: Author
Number of References: 3
Entry Date: 2008
Accession Number: EJ789594
Database: ERIC
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  Value: <anid>AN0031428519;imt01apr.08;2020Feb28.13:00;v2.2.500</anid> <title id="AN0031428519-1">Intuitive understanding of solutions of partially differential equations. </title> <p>This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based on the transition rule are useful for understanding the process.</p> <p>Keywords: partially differential equation; wave equation; heat conduction equation; visual thinking</p> <hd id="AN0031428519-2">1. Introduction</hd> <p>The main purpose of mathematics for the engineering student is to develop the ability to analyse any physical phenomenon in a logical method and to apply to its solution the basic principles [<reflink idref="bib1" id="ref1">1</reflink>]. Most useful mathematical models of physical phenomena are described in terms of differential equations. Standard text books of analytics, mechanics, electromagnetics and so on explain the key to solution in a logical and rigorous sequence. The learning process, however, is inductive and thus, it is difficult for the engineering student to see intuitively why the solutions are true. This article introduces an intuitive approach to the solutions of the one-dimensional wave equation and heat conduction equation. It is useful to introduce simple diagrams before solving the equations as well as to analyse them mechanically. This is emphasized through out this article. Visual thinking is essential to applying mathematical models to engineering design [<reflink idref="bib2" id="ref2">2</reflink>].</p> <hd id="AN0031428519-3">2. Wave equation</hd> <p>The wave equation, , is most generally satisfied by a wave function with the form</p> <p>Graph</p> <p>where <emph>c</emph> is a positive constant [<reflink idref="bib3" id="ref3">3</reflink>]. The functions <emph>G</emph> and <emph>H</emph> are twice differentiable and indicate two waves travelling in opposite directions along the <emph>x</emph>-axis with the same velocity. How are we convinced that the solution of the wave equation are represented by the form (<reflink idref="bib1" id="ref4">1</reflink>)? Let us remember the principle of numerical analysis. Divide the interval 0 ≤ <emph>x</emph> ≤ <emph>X</emph> into <emph>M</emph> equal parts and consider the respective point <emph>x</emph><subs>0</subs>, <emph>x</emph><subs>1</subs>, ... , <emph>x</emph><subs><emph>M</emph></subs>. Here <emph>x</emph><subs>0</subs> = 0 and <emph>x</emph><subs><emph>M</emph></subs> = <emph>X</emph>. Similarly divide the time interval 0 ≤ <emph>t</emph> ≤ <emph>T</emph> into <emph>N</emph> equal parts. Thus the difference <emph>h</emph> is <emph>h</emph> = <emph>X</emph>/<emph>M</emph> and the time interval <emph>k</emph> is <emph>k</emph> = <emph>T</emph>/<emph>N</emph>. The differential quotients and are approximated with difference equations</p> <p>Graph</p> <p>where <emph>u</emph><subs><emph>j</emph>,<emph>n</emph></subs> is the value of <emph>f</emph>(<emph>x</emph>, <emph>t</emph>) and <emph>v</emph><subs><emph>j</emph>,<emph>n</emph></subs> is the value of at the point (<emph>x</emph><subs><emph>j</emph></subs>, <emph>t</emph><subs><emph>n</emph></subs>) = (<emph>jh</emph>, <emph>nk</emph>). After arranging the expression, we obtain</p> <p>Graph</p> <p>For simplicity, let us consider <emph>c</emph> = 1, <emph>h</emph> = 1, <emph>k</emph> = 1. We represent a pulselike wave as a line of '1's and '0's. Choosing an initial condition and a boundary condition and rearranging the 1s and 0s in accordance with the transition rule (<reflink idref="bib3" id="ref5">3</reflink>), we can see pictorially the propagation of the wave. It is instructive to compare the rearrangement of 1s and 0s with the numerical solution of the wave equation.</p> <hd id="AN0031428519-4">Example 1</hd> <p></p> <ulist> <item> Initial condition: , ∂<emph>f</emph>(<emph>x</emph>, <emph>t</emph>)/∂<emph>t</emph> = 0</item> <p></p> <item> Boundary condition: ,</item> <p></p> </ulist> <p>• </p> <p>In Figure 1, waves at <emph>t</emph> = 0, 0.001, 2, 4, 6, 8, 10, 12, 14 are shown and in Figure 2, the rearrangements of 1s and 0s are shown.</p> <p>Graph: Figure 1. Waves at t = 0, 0.001, 2, 4, 6, 8, 10, 12, 14.</p> <p>Graph: Figure 2. Rearrangement of 1s and 0s. Note: From the top, n = 1, 2, ... , 25.</p> <hd id="AN0031428519-5">Example 2</hd> <p></p> <ulist> <item> Initial condition: , ∂<emph>f</emph>(<emph>x</emph>, <emph>t</emph>)/∂<emph>t</emph> = 0</item> <p></p> <item> Boundary condition: ,</item> </ulist> <p>In Figure 3, waves at <emph>t</emph> = 0, 0.001, 2, 4, 6, 8, 10, 12, 14 are shown and in Figure 4, the rearrangements of 1s and 0s are shown.</p> <p>Graph: Figure 3. Waves at t = 0, 0.001, 2, 4, 6, 8, 10, 12, 14.</p> <p>Graph: Figure 4. Rearrangement of 1s and 0s. Note: From the top, n = 1, 2, ... , 27.</p> <hd id="AN0031428519-6">3. Heat conduction equation</hd> <p>The heat conduction equation, , can be solved given an initial condition and a boundary condition. The solutions indicate the process of diffusion as seen from Figures 5 and 6. By observing the calculation process through the following transition rule (<reflink idref="bib4" id="ref6">4</reflink>), we can also be convinced that these particular solutions are true. The differential equations are approximated with difference equation</p> <p>Graph</p> <p>where <emph>u</emph><subs><emph>j</emph>,<emph>n</emph></subs> is the value of <emph>f</emph>(<emph>x</emph>, <emph>t</emph>) at the point (<emph>x</emph><subs><emph>j</emph></subs>, <emph>t</emph><subs><emph>n</emph></subs>) = (<emph>jh</emph>, <emph>nk</emph>). For simplicity, let us consider κ = 1, <emph>h</emph> = 1, <emph>k</emph> = 1. We represent the region of high temperature as a line of '1' and the other region as a line of '0'. For the present purpose, it is sufficient to consider <emph>u</emph><subs><emph>j</emph>,<emph>n</emph> + 1</subs> = (1 − 2κ<emph>k</emph>/<emph>h</emph><sups>2</sups>) (<emph>u</emph><subs><emph>j</emph>+1,<emph>n</emph></subs> − 2<emph>u</emph><subs><emph>j</emph>,<emph>n</emph></subs> + <emph>u</emph><subs><emph>j</emph>−1,<emph>n</emph></subs>) (mod 2).</p> <hd id="AN0031428519-7">Example 3</hd> <p></p> <ulist> <item> Initial condition:</item> <p></p> <item> Boundary condition: ,</item> </ulist> <p>In Figure 5, waves at <emph>t</emph> = 0, 0.001, 2, 4, 6, 8 are shown and in Figure 6, the rearrangements of 1s and 0s are shown.</p> <p>Graph: Figure 5. Waves at t = 0, 0.001, 2, 4, 6, 8.</p> <p>Graph: Figure 6. Rearrangement of 1s and 0s. Note: From the top, n = 1, 2, ... , 15.</p> <hd id="AN0031428519-8">Example 4</hd> <p></p> <ulist> <item> Initial condition: , where θ(<emph>x</emph>) is Heaviside step function.</item> <p></p> <item> Boundary condition: <emph>f</emph>(−8, <emph>t</emph>) = 0,</item> </ulist> <p>In Figure 7, waves at <emph>t</emph> = 0, 0.001, 2, 4, 6, 8, 10, 12, 14 are shown and in Figure 8, the rearrangements of 1s and 0s are shown.</p> <p>Graph: Figure 7. Waves at t = 0, 0.001, 2, 4, 6, 8, 10, 12, 14.</p> <p>Graph: Figure 8. Rearrangement of 1s and 0s. Note: From the top, n = 1, 2, ... , 21.</p> <hd id="AN0031428519-9">4. Concluding remarks</hd> <p>Although the analytical approach is significant, we cannot observe the mechanisms of the key to solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based on the transition rule can help understand the process to the solutions of the wave equation and heat conduction equation.</p> <ref id="AN0031428519-10"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Beer, FP and Johnston, ERJr. 1987. Mechanics for Engineers Statics, New York: McGraw-Hill.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref2" type="bt">2</bibl> <bibtext> Nelsen, RB. 1993. Proofs without Words, Washington, DC: The Mathematical Association of America.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref3" type="bt">3</bibl> <bibtext> Crawford Jr, FS. 1968. Waves, New York: McGraw-Hill.</bibtext> </blist> </ref> <aug> <p>By Y. Kobayashi</p> <p>Reported by Author</p> </aug> <nolink nlid="nl1" bibid="bib4" firstref="ref6"></nolink>
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  Data: This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based on the transition rule are useful for understanding the process. (Contains 8 figures.)
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