Scientific Computing with MATLAB, Second Edition

https://www.crcpress.com/Scientific-Computing-with-MATLAB-Second-Edition/Xue-Chen/9781498757775

https://bbs.pinggu.org/thread-5317824-1-1.html

1. Example 5.1 For a given time domain function f(t) = t2e
   
2. −2t sin(t + π), compute its
   
3. Laplace transform function F(s).
   
4. Solution From the original problem, it can be seen that the time domain variable t should
   
5. be declared first. With the MATLAB statements, the function f(t) can be specified. Then,
   
6. the laplace() function can be used to derive the Laplace transform of the original function
   
7. >> syms t; f=t^2*exp(-2*t)*sin(t+pi); F=laplace(f) % the 3-step procedure

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1. Example 5.2 Assume that the original function is given by f(x) = x2e
   
2. −2x sin(x+π),
   
3. compute the Laplace transform, and then, take inverse Laplace transform and see whether
   
4. the original function can be recovered.
   
5. Solution Similarly, the laplace() function can still be used
   
6. >> syms x w; f=x^2*exp(-2*x)*sin(x+pi); F=laplace(f,x,w)

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1. Example 5.4 For the function f(t) given in Example 5.1, explore the relationship between
   
2. L[d5f(t)/ dt5] and s5L[f(t)].
   
3. Solution To solve the problem, the fifth-order derivative to the given function f(t) can be
   
4. obtained by function diff(). Then, the Laplace transform can be obtained
   
5. >> syms t s; f=t^2*exp(-2*t)*sin(t+pi); % declare variable and function
   
6. F=simplify(laplace(diff(f,t,5))) % evaluate Laplace transform

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1. Example 5.7 Please solve the inverse Laplace transform problem numerically to the
   
2. function given in Example 5.3.
   
3. Solution It can be seen from the earlier example that, although the analytical solution does
   
4. not exist, a high-precision numerical solution can be found with Symbolic Math Toolbox. For
   
5. the same function, the variable x can be substituted by s and can be converted to a string
   
6. with char() function. Numerical inverse Laplace transform can be obtained. Compared with
   
7. exact method, the maximum relative error is 0.005826%.
   
8. >> syms x t; % declare symbolic variables and the function
   
9. G=(-17*x^5-7*x^4+2*x^3+x^2-x+1)...
   
10. /(x^6+11*x^5+48*x^4+106*x^3+125*x^2+75*x+17);
    
11. f=ilaplace(G,x,t); fun=char(subs(G,x,’s’)); % convert to string of s
    
12. [t1,y1]=INVLAP(fun,0.01,5,100); tic, y0=subs(f,t,t1); toc
    
13. y0=double(y0); err=norm((y1-y0)./y0) % evaluate the error

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