2019학년도 2학기

                   반도체 공학과 공학수학2


   주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition

   부교재: 이상구 외 4인, 최신공학수학 Ⅱ   1st Edition  

   실습실: http://www.hanbit.co.kr/EM/sage/

          http://matrix.skku.ac.kr/2018-album/R-Sage-Stat-Lab-1.html  

          http://matrix.skku.ac.kr/2018-album/R-Sage-Stat-Lab-2.html 

   강의시간:  공학수학2, (화16:30-17:45)   (목15:00-16:15)

   담당교수:  김응기 박사



Week 6


6주차

12.6: Heat Equation: Steady Two-Dimensional Heat Problems. Dirichlet Problem

  12.4 확산방정식


12.7: Heat Equation: Solution by Fourier Integrals and Transforms

  http://www.hanbit.co.kr/EM/sage/2_chap12.html 

  http://matrix.skku.ac.kr/2018-EM/EM-2-W4-lab.html



12.6 Heat Equation:

Steady Two-Dimensional Heat Problems.

Dirichlet Problem



The two-dimensional heat equation

    

for steady problems.

Then and the heat equation reduces to Laplace’s Equation

(14)   

A heat problem then consists of this PDE to be considered in some region of the plane and a given boundary condition on the boundary curve of


This is a boundary value problem (BVP).


First BVP or Dirichlet Problem if is prescribed on (Dirichlet boundary condition)

Second BVP or Neumann Problem if the normal derivative is prescribed on (Neumann boundary condition)

Third BVP, Mixed BVP or Robin Problem if is prescribed on a portion of (Robin boundary condition).



Dirichlet Problem in a Rectangle R.

We consider a Dirichlet Problem for Laplace’s Equation (14) in a rectangle , assuming that the temperature equals a given function on the upper side and on the other three sides of the rectangle.

Substituting into (14) written as

         

         

            dividing by ,

and equating both sides to a negative constant, we obtain

          .

    .

         ,

and the left and the right boundary condition imply

       and   .


        


Case 1  For negative .

          

A general solution of the second-order ODE is

    .

 imply

 imply

          (No interest).


Case 2  For .

    

A general solution of (5) is

    .

    

 imply

          (No interest).


Case 3  For positive .

          

A general solution of the second-order ODE is

    .

    

 imply

                (No interest).

 since otherwise . Hence . Thus

    ,   hence      .

Setting , a solution of the second-order ODE is

(15)    ,     .

             ,

The ODE for with then becomes

       .

Solution are

       .

Now the boundary condition on the lower side of implies that ; that is,

         .

This gives

       .

We obtain as the eigenfunctions of our problem

(16)   

               .

These solution satisfy the boundary condition on the left, right, and lower sides.

To get a solution also satisfy the boundary condition on the upper side, we consider the infinite series

        .

From this and (16) with we obtain

        .

We can write this in the form

        .

The Fourier coefficients of

        .

From this and (16) we see that the solution of our problem is

        

where

        .

The series for , , and have the right sums. and are continuous and is piecewise continuous on the interval .




12.7 Heat Equation:

Solution by Fourier Integrals and Transforms



Model of bars of infinite length

Heat equation

(1)     


Initial Conditions

(2)          

where is the given initial temperature of the bar.


Substituting into (1).

The two ODEs

(3)     

and

(4)     

The solutions of is

        

where and are any constants.

The solutions of is

        .

Hence a solution of (1) is

        .



Use of Fourier Integrals


Heat equation is linear and homogeneous

(6)     

is then a solution of (1).



Determination of and from the Initial Condition

(7)     

where    and  

      

             

             

             .


     

Using the formula

We choose as a new variable of integration and set .

Then and , so that (10) becomes

     .

     

            

            

              where

If is bounded for all values of and integrable in every finite interval.



Example  1  Temperature in an Infinite Bar

Find the temperature in the infinite bar if the initial temperature is

       

    

Solution

       

                

            ,          

              


Example  2  Temperature in the Infinite Bar in Example 1

Solve Example 1 using the Fourier transform.

Solution

 : Fourier transform of , regarded as a function of

Heat equation : 

Interchange the order of differentiation and integration

        

General solution :

Initial condition      

    

    

    

           

           

           

           


Sage Coding

https://sagecell.sagemath.org/ 



Example  3  Solution in Example by the Method of Convolution

Solve the heat problem in Example by the method of convolution.

Solution

          

          

          

(18)      *

    

with a suitable . With or , using (17) we obtain

    

Hence has the inverse

    .

Replacing with and substituting this into (18) we finally have

    *

           

             

           


http://matrix.skku.ac.kr/sglee/ 


[한빛 아카데미] Engineering Mathematics with Sage:

[저자] 이상 구, 김영 록, 박준 현, 김응 기, 이재 화


Contents

 A. 공학수학 1 – 선형대수, 상미분방정식+ Lab

Chapter 01 벡터와 선형대수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-1.html


Chapter 02 미분방정식의 이해 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-2.html

Chapter 03 1계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-3.html    

Chapter 04 2계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-4.html

Chapter 05 고계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-5.html

Chapter 06 연립미분방정식, 비선형미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-6.html

Chapter 07 상미분방정식의 급수해법, 특수함수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-7.html  

Chapter 08 라플라스 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-8.html

 

B. 공학수학 2 - 벡터미적분, 복소해석 + Lab

Chapter 09 벡터미분, 기울기, 발산, 회전 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-9.html

Chapter 10 벡터적분, 적분정리 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-10.html

Chapter 11 푸리에 급수, 적분 및 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-11.html

Chapter 12 편미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-12.html

Chapter 13 복소수와 복소함수, 복소미분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-13.html

Chapter 14 복소적분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-14.html

Chapter 15 급수, 유수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-15.html

Chapter 16 등각사상 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-16.html



Made by Prof. Sang-Gu LEE  sglee at skku.edu

http://matrix.skku.ac.kr/sglee/   with Dr. Jae Hwa LEE


Copyright @ 2019 SKKU Matrix Lab. All rights reserved.
Made by Manager: Prof. Sang-Gu Lee and Dr. Jae Hwa Lee
*This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03035865).