Ramakrishnan

Dr. Ramakrishnan Vaidyanathan

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4 years, 361 days

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I have retired as Professor-Mechanical in Sri Venkateswara College of Engineering and Technology under Anna University affiliated colleges in Tamil Nadu, India. I have 19 years of Industrial and 20 years of teaching experience. I am learning Maple for the past four and half years hoping to make at least one appreciable maple presentation.

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These are replies submitted by Ramakrishnan


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sol1 := dsolve([diff(diff(y(x), x), x) = (diff(y(x), x))^2-2*x^2*y(x), y(-1) = 0, y(1) = 1], numeric, method = bvp);

 

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Download possiblesolnto_yr_problem.mwpossiblesolnto_yr_problem.mw

In my opinion, f(x) needs to be mentioned. possible soln by num method is attached.Hope this will be useful.

Ramakrishnan V

rukmini_ramki@hotmail.com

@Carl Love 

Thank you very much for the clarification that simplify command will do. Useful learning for me. Ramakrishnan V

@Thomas Dean

Sorry for the belated response. Thank you very much for the answer It is useful for me to know the basics in Maple. Ramakrishnan V 


 

 

With 25 terms it gives the expression with same terms but jumbled in order:

expr1 := t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16+t17+t18+t19+t20+t21+t22+t0-t0+t23;

0

(1)

 

With 24 terms it gives the expression with same terms  in the same order given by us:

expr3 := t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16+t17+t18+t19+t20+t21+t0-t0+t23;

0

(2)

 =

With 25 terms it gives the expression with same terms but jumbled in order (expr5):With 24 terms, it keeps the order given by us.(expr6)

expr5 := t2+t1+t3+t4+t23+t22+t20+t21+t19+t17+t18+t15+t16+t13+t14+t11+t12+t9+t10+t8+t6+t7+t5+t24-t25;

-t25

(3)

With 25 terms even if there is no arithmatics (maths) involved) it gives the expression with same terms given by us but jumbled in order:

expr7:=t1+t2+t3+t4+t5+t6+t7+t8+t9+t10+t11+t12+t13+t14+t15+t16+t17+t18+t19+t20+t21+t22+t24+t23+t25;

t2+t1+t3+t4+t23+t22+t20+t21+t19+t17+t18+t15+t16+t13+t14+t11+t12+t9+t10+t8+t6+t7+t5+t24+t25

(4)

 

   

     

NULL


Download jumbled_order_if_terms_are_more_than_24.mwjumbled_order_if_terms_are_more_than_24.mw

I am new to maple. However i see something in yr post and want to comment. I attach my attemps below. I find that the order is retained up to 24 terms with or without maths (subtraction t0-t0=0 in our case) involved.


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Eq := diff(y(x), x, x) = -(x^2+1)*y(x)+k;

diff(diff(y(x), x), x) = -(x^2+1)*y(x)+k

(1)

"(->)"

y(x) = WhittakerM(-(1/4)*I, 1/4, I*x^2)*_C2/x^(1/2)+WhittakerW(-(1/4)*I, 1/4, I*x^2)*_C1/x^(1/2)+(2*I)*k*((Int(x^(3/2)*WhittakerW(-(1/4)*I, 1/4, I*x^2)/((1+3*I)*WhittakerW(-(1/4)*I, 1/4, I*x^2)*WhittakerM(1-(1/4)*I, 1/4, I*x^2)+(4*I)*WhittakerW(1-(1/4)*I, 1/4, I*x^2)*WhittakerM(-(1/4)*I, 1/4, I*x^2)), x))*WhittakerM(-(1/4)*I, 1/4, I*x^2)-(Int(x^(3/2)*WhittakerM(-(1/4)*I, 1/4, I*x^2)/((1+3*I)*WhittakerW(-(1/4)*I, 1/4, I*x^2)*WhittakerM(1-(1/4)*I, 1/4, I*x^2)+(4*I)*WhittakerW(1-(1/4)*I, 1/4, I*x^2)*WhittakerM(-(1/4)*I, 1/4, I*x^2)), x))*WhittakerW(-(1/4)*I, 1/4, I*x^2))/x^(1/2)

(2)

"->"

 

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Download ODESolnTWOSent14-11-2015byRVtMaplePrimeoCommunity.mw

I forgot to attach my doc. Enclosed now the cube plot, an answer to the Linear ODE of second order problem for constant k

 

With constant k, the solution is cube. Answer given below.

 

@Kitonum 

Thanks for going through my step by step statements. Thanks for finding the mistake mede by me.

Ramakrishnan V

@Kitonum 

I am interested in knowing what the formula is for because it is not clear what logic leads to such a formula.

Can you give the detail calculation with numeric numbers term by erm for M = 5 and K =4

M:=5:  K:=4:  K1:=K-2*K2:

 

Sum(M!/(M-K1-K2)! * K!/(K1! * K2)! * 1/M^K * 1/2^K2, K2=0..K/2);

value(%);

Step 1

k2=0

5!/(5-4-0)! * 4!/1 * 1/5^4 *1/2 ^0

value(%);

Step 2

k2=1

5!/(5-2-1)! * 4!/(2! * 1)! * 1/5^4 *1/2 ^1

 

Step 2

k2=2

5!/(5-1-4)! * 4!/(1 * 4)! * 1/5^4 *1/2 ^2

 

Is this ok?

What is the significance?

Ramakrishnan V

@J4James Thanks. I tried yr problem. Answers is x(t) =e-t

and y(t) = e-t - e-1.

I shall proceed on this line. Thanks.

@Rouben Rostamian  

Thank you so much. Your points are very clear, simple and understandable. I shall do the home work and come back with more details to get help from mapleprimes.

Just for information only i give below two statements. It may or may not help you to help me.

1. The problem is to find temperature variation inside a cylinder (IC engine) varying with cam angle (piston position varies with theta(t))

2. Combustion takes place during one stroke (third stroke i.e. first half of second revolution) only out of four strokes each 180 degrees.

I shall anyway come back with mathematical parts only soon to get possible help from all of ou in maleprimes.

 

Ramakrishnan V

@J4James I am amazed at the support provided by J4James and Rouben. Thank you both of you. I will be coming with formulation and boundary conditions for my model. I am getting confidence that i will be able to solve. Please bear with me for some more time to come back to you. Thanks a lot once again.

i give below an example

polynomial_example.mw 

 

@Preben Alsholm 

Thank you for the answer. It was interesting to read and understand more ideas from your comments and suggestions. I shall tryDEtools and learn more in a few days time and hope to come back with clear ideas. Thanks once again. Cheers. Ramakrishnan V

@Preben Alsholm 

y(t) = a*exp(-t)*sin(.57736*t)+b*exp(-t)*cos(.57736*t);
`assign%`;
print(`output redirected...`); # input placeholder
assign%
a, b := 2, 1;
plot(y(t), t = 0 .. 5);
Error, (in Explore) No parameters to explore

Animation command for the above answer wld help me greatly!

Solution i.e. graph obtained after plot command. To animate i went to explore, but error indicated.

Hope the previous and this efforts on my maple worksheet wld give you atleast some light upon the answer.

Cheers and thanks for yr active help.

Ramakrishnan V

@Preben Alsholm 

dl := 3*(diff(y(t), t, t))+6*(diff(y(t), t))+4*y(t) = 0;
print(`output redirected...`); # input placeholder
/ d / d \\ / d \
3 |--- |--- y(t)|| + 6 |--- y(t)| + 4 y(t) = 0
\ dt \ dt // \ dt /
"(->)"
/1 (1/2) \ /1 (1/2) \
y(t) = _C1 exp(-t) sin|- 3 t| + _C2 exp(-t) cos|- 3 t|
\3 / \3 /

Above copy paste i do not know how distorted! Sorry.

Line by line i hv given below.

 

 dl := 3*(diff(y(t), t, t))+6*(diff(y(t), t))+4*y(t) = 0

 y(t) = _C1*exp(-t)*sin((1/3)*sqrt(3)*t)+_C2*exp(-t)*cos((1/3)*sqrt(3)*t)

 

DEplot(dl, y(t), t = 0 .. 10, [seq([y(0) = y0, (D(y))(0) = 0], y0 = -3 .. 3)]);
print(`output redirected...`); # input placeholder
DEplot(dl, y(t), t = 0 .. 10, [[y(0) = -3, D(y)(0) = 0],

[y(0) = -2, D(y)(0) = 0], [y(0) = -1, D(y)(0) = 0],

[y(0) = 0, D(y)(0) = 0], [y(0) = 1, D(y)(0) = 0],

[y(0) = 2, D(y)(0) = 0], [y(0) = 3, D(y)(0) = 0]])

Plot does not appear

The following worked and a curve from t=0 to 4 sec appeared based on comments received in maple prime.

Explore(plot(a*exp(-1.*t)*sin(.57736*t)+b*exp(-1.*t)*cos(.57736*t), t = 0 .. 4))

 

The following command also worked as per yr suggestion.

 plots:-animate(DEplot, [ode, y(t), t = 0 .. 5, [seq([y(0) = y0, (D(y))(0) = y1], y0 = -3 .. 3)]], y1 = -2 .. 2)

y(t) = _C1 exp(-1. t) sin(0.57736 t)

Flg is answer not understandable! but graph has come.

+ _C2 exp(-1. t) cos(0.57736 t)
-1. + 0.57736 I
-1. - 0.57736 I
2
lambda + 2. lambda + 1.333344570
/ d \ / d / d \\
1.333344570 y(t) + 2. |--- y(t)| + |--- |--- y(t)|| = 0
\ dt / \ dt \ dt //

I want the previous curve animated from 0 to 2 seconds. Like wise all curves in yr solution from t=0 to 4 sec.

can i attach mw files here?

Ramakrishnan V

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