1% Title:  Examples of Laplace Transforms.
2
3% Author: L. Kazasov.
4
5% Date: 24 October 1988.
6
7order p;
8
9% Elementary functions with argument k*x, where x is object var.
10
11laplace(1,x,p);
12laplace(c,x,p);
13laplace(sin(k*x),x,p); laplace(sin(x/a),x,p);
14laplace(sin(17*x),x,p);
15laplace(sinh x,x,p);
16laplace(cosh(k*x),x,p);
17laplace(x,x,p); laplace(x**3,x,p);
18off mcd; laplace(e**(c*x) + a**x, x, s);
19laplace(e**x - e**(a*x) + x**2, x, p);
20laplace(one(k*t) + sin(a*t) - cos(b*t) - e**t, t, p);
21laplace(sqrt(x),x,p); laplace(x**(1/2),x,p); on mcd;
22laplace(x**(-1/2),x,p); laplace(x**(5/2),x,p);
23laplace(-1/4*x**2*c*sqrt(x), x, p);
24
25% Elementary functions with argument k*x - tau,
26%   where k>0, tau>=0, x is object var.
27
28laplace(cos(x-a),x,p);
29laplace(one(k*x-tau),x,p);
30laplace(sinh(k*x-tau),x,p); laplace(sinh(k*x),x,p);
31laplace((a*x-b)**c,x,p);
32% But ...
33off mcd; laplace((a*x-b)**2,x,p); on mcd;
34laplace(sin(2*x-3),x,p);
35on lmon; laplace(sin(2*x-3),x,p); off lmon;
36off mcd; laplace(cosh(t-a) - sin(3*t-5), t, p); on mcd;
37
38% More complicated examples - multiplication of functions.
39% We use here on lmon - a new switch that forces all
40% trigonometrical functions which depend on object var
41% to be represented as exponents.
42
43laplace(x*e**(a*x)*cos(k*x), x, p);
44laplace(x**(1/2)*e**(a*x), x, p);
45laplace(-1/4*e**(a*x)*(x-k)**(-1/2), x, p);
46laplace(x**(5/2)*e**(a*x), x, p);
47laplace((a*x-b)**c*e**(k*x)*const/2, x, p);
48off mcd; laplace(x*e**(a*x)*sin(7*x)/c*3, x, p); on mcd;
49laplace(x*e**(a*x)*sin(k*x-tau), x, p);
50% The next is unknown if lmon is off.
51laplace(sin(k*x)*cosh(k*x), x, p);
52laplace(x**(1/2)*sin(k*x), x, p);
53on lmon;  % But now is OK.
54laplace(x**(1/2)*sin(a*x)*cos(a*b), x, p);
55laplace(sin(x)*cosh(x), x, p);
56laplace(sin(k*x)*cosh(k*x), x, p);
57% Off exp leads to very messy output in this case.
58% off exp; laplace(sin(k*x-t)*cosh(k*x-t), x, p); on exp;
59laplace(sin(k*x-t)*cosh(k*x-t), x, p);
60laplace(cos(x)**2,x,p);laplace(c*cos(k*x)**2,x,p);
61laplace(c*cos(2/3*x)**2, x, p);
62laplace(5*sinh(x)*e**(a*x)*x**3, x, p);
63off exp; laplace(sin(2*x-3)*cosh(7*x-5), x, p); on exp;
64laplace(sin(a*x-b)*cosh(c*x-d), x, p);
65% To solve this problem we must tell the program which one-function
66% is rightmost shifted.  However, in REDUCE 3.4, this rule is still
67% not sufficient.
68for all x let one(x-b/a)*one(x-d/c) = one(x-b/a);
69laplace(sin(a*x-b)*cosh(c*x-d), x, p);
70for all x clear one(x-b/a)*one(x-d/c) ;
71off lmon;
72
73% Floating point arithmetic.
74% laplace(3.5/c*sin(2.3*x-4.11)*e**(1.5*x), x, p);
75on rounded;
76laplace(3.5/c*sin(2.3*x-4.11)*e**(1.5*x), x, p);
77laplace(x**2.156,x,p);
78laplace(x**(-0.5),x,p);
79off rounded; laplace(x**(-0.5),x,p); on rounded;
80laplace(x*e**(2.35*x)*cos(7.42*x), x, p);
81laplace(x*e**(2.35*x)*cos(7.42*x-74.2), x, p);
82% Higher precision works, but uses more memory.
83% precision 20; laplace(x**2.156,x,p);
84% laplace(x*e**(2.35*x)*cos(7.42*x-74.2), x, p);
85off rounded;
86
87% Integral from 0 to x, where x is object var.
88% Syntax is intl(<expr>,<var>,0,<obj.var>).
89
90laplace(c1/c2*intl(2*y**2,y,0,x), x,p);
91off mcd; laplace(intl(e**(2*y)*y**2+sqrt(y),y,0,x),x,p); on mcd;
92laplace(-2/3*intl(1/2*y*e**(a*y)*sin(k*y),y,0,x), x, p);
93
94% Use of delta function and derivatives.
95
96laplace(-1/2*delta(x), x, p); laplace(delta(x-tau), x, p);
97laplace(c*cos(k*x)*delta(x),x,p);
98laplace(e**(a*x)*delta(x), x, p);
99laplace(c*x**2*delta(x), x, p);
100laplace(-1/4*x**2*delta(x-pi), x, p);
101laplace(cos(2*x-3)*delta(x-pi),x,p);
102laplace(e**(-b*x)*delta(x-tau), x, p);
103on lmon;
104laplace(cos(2*x)*delta(x),x,p);
105laplace(c*x**2*delta(x), x, p);
106laplace(c*x**2*delta(x-pi), x, p);
107laplace(cos(a*x-b)*delta(x-pi),x,p);
108laplace(e**(-b*x)*delta(x-tau), x, p);
109off lmon;
110
111laplace(2/3*df(delta x,x),x,p);
112off exp; laplace(e**(a*x)*df(delta x,x,5), x, p); on exp;
113laplace(df(delta(x-a),x), x, p);
114laplace(e**(k*x)*df(delta(x),x), x, p);
115laplace(e**(k*x)*c*df(delta(x-tau),x,2), x, p);
116on lmon;laplace(e**(k*x)*sin(a*x)*df(delta(x-t),x,2),x,p);off lmon;
117
118% But if tau is positive, Laplace transform is not defined.
119
120laplace(e**(a*x)*delta(x+tau), x, p);
121laplace(2*c*df(delta(x+tau),x), x, p);
122laplace(e**(k*x)*df(delta(x+tau),x,3), x, p);
123
124% Adding new let rules for Laplace operator. Note the syntax.
125
126for all x let laplace(log(x),x) = -log(gam*il!&)/il!&;
127laplace(-log(x)*a/4, x, p); laplace(-log(x),x,p);
128laplace(a*log(x)*e**(k*x), x, p);
129for all x clear laplace(log(x),x);
130
131operator f; for all x let
132    laplace(df(f(x),x),x) = il!&*laplace(f(x),x) - sub(x=0,f(x));
133for all x,n such that numberp n and fixp n let
134    laplace(df(f(x),x,n),x) = il!&**n*laplace(f(x),x) -
135      for i:=n-1 step -1 until 0 sum
136        sub(x=0, df(f(x),x,n-1-i)) * il!&**i ;
137for all x let laplace(f(x),x) = f(il!&);
138
139laplace(1/2*a*df(-2/3*f(x)*c,x), x,p);
140laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);
141laplace(1/2*a*e**(k*x)*df(-2/3*f(x)*c,x,2), x,p);
142clear f;
143
144% Or if the boundary conditions are known and assume that
145% f(i,0)=sub(x=0,df(f(x),x,i)) the above may be overwritten as:
146operator f; for all x let
147    laplace(df(f(x),x),x) = il!&*laplace(f(x),x) - f(0,0);
148for all x,n such that numberp n and fixp n let
149    laplace(df(f(x),x,n),x) = il!&**n*laplace(f(x),x) -
150      for i:=n-1 step -1 until 0 sum il!&**i * f(n-1-i,0);
151for all x let laplace(f(x),x) = f(il!&);
152let f(0,0)=0, f(1,0)=1, f(2,0)=2, f(3,0)=3;
153laplace(1/2*a*df(-2/3*f(x)*c,x), x,p);
154laplace(1/2*a*df(-2/3*f(x)*c,x,4), x,p);
155clear f(0,0), f(1,0), f(2,0), f(3,0); clear f;
156
157% Very complicated examples.
158
159on lmon;
160laplace(sin(a*x-b)**2, x, p);
161off mcd; laplace(x**3*(sin x)**4*e**(5*k*x)*c/2, x,p);
162a:=(sin x)**4*e**(5*k*x)*c/2; laplace(x**3*a,x,p); clear a; on mcd;
163% And so on, but is very time consuming.
164% laplace(e**(k*x)*x**2*sin(a*x-b)**2, x, p);
165% for all x let one(a*x-b)*one(c*x-d) = one(c*x-d);
166% laplace(x*e**(-2*x)*cos(a*x-b)*sinh(c*x-d), x, p);
167% for all x clear one(a*x-b)*one(c*x-d) ;
168% laplace(x*e**(c*x)*sin(k*x)**3*cosh(x)**2*cos(a*x), x, p);
169off lmon;
170
171% Error messages.
172
173laplace(sin(-x),x,p);
174on lmon; laplace(sin(-a*x), x, p); off lmon;
175laplace(e**(k*x**2), x, p);
176laplace(sin(-a*x+b)*cos(c*x+d), x, p);
177laplace(x**(-5/2),x,p);
178% With int arg, can't be shifted.
179laplace(intl(y*e**(a*y)*sin(k*y-tau),y,0,x), x, p);
180laplace(cosh(x**2), x, p);
181laplace(3*x/(x**2-5*x+6),x,p);
182laplace(1/sin(x),x,p);   % But ...
183laplace(x/sin(-3*a**2),x,p);
184% Severe errors.
185% laplace(sin x,x,cos y);
186% laplace(sin x,x,y+1);
187% laplace(sin(x+1),x+1,p);
188
189
190Comment  Examples of Inverse Laplace transformations;
191
192symbolic(ordl!* := nil);   % To nullify previous order declarations.
193
194order t;
195
196% Elementary ratio of polynomials.
197
198invlap(1/p, p, t);
199invlap(1/p**3, p, t);
200invlap(1/(p-a), p, t); invlap(1/(2*p-a),p,t); invlap(1/(p/2-a),p,t);
201invlap(e**(-k*p)/(p-a), p, t); invlap(b**(-k*p)/(p-a), p, t);
202invlap(1/(p-a)**3, p, t);
203invlap(1/(c*p-a)**3, p, t); invlap(1/(p/c-a)**3, p, t);
204invlap((c*p-a)**(-1)/(c*p-a)**2, p, t);
205invlap(c/((p/c-a)**2*(p-a*c)), p, t);
206invlap(1/(p*(p-a)), p, t);
207invlap(c/((p-a)*(p-b)), p, t);
208invlap(p/((p-a)*(p-b)), p, t);
209off mcd; invlap((p+d)/(p*(p-a)), p, t);
210invlap((p+d)/((p-a)*(p-b)), p, t);
211invlap(1/(e**(k*p)*p*(p+1)), p, t); on mcd;
212off exp; invlap(c/(p*(p+a)**2), p, t); on exp;
213invlap(1, p, t); invlap(c1*p/c2, p, t);
214invlap(p/(p-a), p, t); invlap(c*p**2, p, t);
215invlap(p**2*e**(-a*p)*c, p, t);
216off mcd;invlap(e**(-a*p)*(1/p**2-p/(p-1))+c/p, p, t);on mcd;
217invlap(a*p**2-2*p+1, p, x);
218
219% P to non-integer power in denominator - i.e. gamma-function case.
220
221invlap(1/sqrt(p), p, t); invlap(1/sqrt(p-a), p, t);
222invlap(c/(p*sqrt(p)), p, t); invlap(c*sqrt(p)/p**2, p, t);
223invlap((p-a)**(-3/2), p, t);
224invlap(sqrt(p-a)*c/(p-a)**2, p, t);
225invlap(1/((p-a)*b*sqrt(p-a)), p, t);
226invlap((p/(c1-3)-a)**(-3/2), p, t);
227invlap(1/((p/(c1-3)-a)*b*sqrt(p/(c1-3)-a)), p, t);
228invlap((p*2-a)**(-3/2), p, t);
229invlap(sqrt(2*p-a)*c/(p*2-a)**2, p, t);
230invlap(c/p**(7/2), p, t); invlap(p**(-7/3), p, t);
231invlap(gamma(b)/p**b,p,t); invlap(c*gamma(b)*(p-a)**(-b),p,t);
232invlap(e**(-k*p)/sqrt(p-a), p, t);
233
234% Images that give elementary object functions.
235% Use of new switches lmon, lhyp.
236
237invlap(k/(p**2+k**2), p, t);
238% This is made more readable by :
239on ltrig; invlap(k/(p**2+k**2), p, t);
240invlap(p/(p**2+1), p, t);
241invlap((p**2-a**2)/(p**2+a**2)**2, p, t);
242invlap(p/(p**2+a**2)**2, p, t);
243invlap((p-a)/((p-a)**2+b**2), p, t); off ltrig;
244on lhyp; invlap(s/(s**2-k**2), s, t);
245invlap(e**(-tau/k*p)*p/(p**2-k**2), p, t); off lhyp;
246% But it is not always possible to convert expt. functions, e.g.:
247on lhyp; invlap(k/((p-a)**2-k**2), p, t); off lhyp;
248on ltrig; invlap(e**(-tau/k*p)*k/(p**2+k**2), p, t); off ltrig;
249% In such situations use the default switches:
250invlap(k/((p-a)**2-k**2), p, t); % i.e. e**(a*t)*cosh(k*t).
251invlap(e**(-tau/k*p)*k/(p**2+k**2), p, t); % i.e. sin(k*t-tau).
252
253% More complicated examples.
254
255off exp,mcd; invlap((p+d)/(p**2*(p-a)), p, t);
256invlap(e**(-tau/k*p)*c/(p*(p-a)**2), p, t);
257invlap(1/((p-a)*(p-b)*(p-c)), p, t);
258invlap((p**2+g*p+d)/(p*(p-a)**2), p, t); on exp,mcd;
259invlap(k*c**(-b*p)/((p-a)**2+k**2), p, t);
260on ltrig; invlap(c/(p**2*(p**2+a**2)), p, t);
261invlap(1/(p**2-p+1), p, t); invlap(1/(p**2-p+1)**2, p, t);
262invlap(2*a**2/(p*(p**2+4*a**2)), p, t);
263% This is (sin(a*t))**2 and you can get this by using the let rules :
264for all x let sin(2*x)=2*sin x*cos x, cos(2*x)=(cos x)**2-(sin x)**2,
265(cos x)**2 =1-(sin x)**2;
266invlap(2*a**2/(p*(p**2+4*a**2)), p, t);
267for all x clear sin(2*x),cos(2*x),cos(x)**2;  off ltrig;
268on lhyp;invlap((p**2-2*a**2)/(p*(p**2-4*a**2)),p,t);
269off lhyp; % Analogously, the above is (cosh(a*t))**2.
270
271% Floating arithmetic.
272
273invlap(2.55/((0.5*p-2.0)*(p-3.3333)), p, t);
274on rounded;
275invlap(2.55/((0.5*p-2.0)*(p-3.3333)), p, t);
276invlap(1.5/sqrt(p-0.5), p, t);
277invlap(2.75*p**2-0.5*p+e**(-0.9*p)/p, p, t);
278invlap(1/(2.0*p-3.0)**3, p, t); invlap(1/(2.0*p-3.0)**(3/2), p, t);
279invlap(1/(p**2-5.0*p+6), p, t);
280off rounded;
281
282% Adding new let rules for the invlap operator. note the syntax:
283
284for all x let invlap(log(gam*x)/x,x) = -log(lp!&);
285invlap(-1/2*log(gam*p)/p, p, t);
286invlap(-e**(-a*p)*log(gam*p)/(c*p), p, t);
287for all x clear invlap(1/x*log(gam*x),x);
288
289% Very complicated examples and use of factorizer.
290
291off exp,mcd; invlap(c**(-k*p)*(p**2+g*p+d)/(p**2*(p-a)**3), p, t);
292on exp,mcd;
293invlap(1/(2*p**3-5*p**2+4*p-1), p, t);
294on ltrig,lhyp; invlap(1/(p**4-a**4), p, t);
295invlap(1/((b-3)*p**4-a**4*(2+b-5)), p, t); off ltrig,lhyp;
296% The next three examples are the same:
297invlap(c/(p**3/8-9*p**2/4+27/2*p-27)**2,p,t);invlap(c/(p/2-3)**6,p,t);
298off exp; a:=(p/2-3)**6; on exp; invlap(c/a, p, t); clear a;
299% The following two examples are the same :
300invlap(c/(p**4+2*p**2+1)**2, p, t); invlap(c/((p-i)**4*(p+i)**4),p,t);
301% The following three examples are the same :
302invlap(e**(-k*p)/(2*p-3)**6, p, t);
303invlap(e**(-k*p)/(4*p**2-12*p+9)**3, p, t);
304invlap(e**(-k*p)/(8*p**3-36*p**2+54*p-27)**2, p, t);
305
306% Error messages.
307
308invlap(e**(a*p)/p, p, t);
309invlap(c*p*sqrt(p), p, t);
310invlap(sin(p), p, t);
311invlap(1/(a*p**3+b*p**2+c*p+d),p,t);
312invlap(1/(p**2-p*sin(p)+a**2),p,t);
313on rounded; invlap(1/(p**3-1), p, t); off rounded;
314% Severe errors:
315%invlap(1/(p**2+1), p+1, sin(t) );
316%invlap(p/(p+1)**2, sin(p), t);
317
318end;
319