solution to differential equations system looks strange. What is wrong with my code?

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susman 2021-1-29

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https://ww2.mathworks.cn/matlabcentral/answers/730008-solution-to-differential-equations-system-looks-strange-what-is-wrong-with-my-code

评论： Walter Roberson 2021-1-30

I am writing a code to solve the differential equations. I have written the following code but my solution looks wierd. First, I get long digits numbers, which I do not get if I solve the system by hand. secondly, I get constants like C1 and C2 etc... Though I have defined the values of all parameters of functions. Please tell me what is wrong in my code:

%defined parameters 
hf12 = exp(-21.6079+0.0099).*exp(-0.0134.*t) 
hf13 = 0
hf14 = 0
hf15 = exp(-9.65573+0.01844+0.0821).*exp(0.02246*t)
hf11 = -(hf12+hf15)
hf21 = 0
hf23 = exp(-1.666-0.1116).*exp(-0.0025.*t)
hf24 = exp(-8.96236+0.07691).*exp(0.00978.*t)
hf25 = exp(-9.65573+0.08218).*exp(0.02246.*t)
hf22 = -(hf23+hf24+hf25)
   hf31 = 0
   hf32 = exp(-21.6079+0.0676+0.0099).*exp(-0.0134.*t) 
   hf34 = 0
   hf35 = exp(-9.65573-(0.11853)+0.08218).*exp(0.02246.*t)
   hf33 = -(hf32+hf35)
  hf41 = 0
hf42 = exp(-21.609-0.4176+0.0099).*exp(-0.0134.*t)
hf43 = 0
hf45 = exp(-9.65573-0.00415+0.08218).*exp(0.02246.*t)
hf44 = -(hf42+hf45)
%% solution 
syms p11(t) p22(t) p33(t) p44(t) p55(t) p12(t) p13(t) p14(t) p15(t) p21(t) p23(t) p24(t) p25(t) ...
     p31(t) p32(t) p34(t) p35(t) p41(t) p42(t) p43(t) p45(t) p51(t) p52(t) p53(t) p54(t) p55(t)
ode = [diff(p11,t) == exp(hf12+hf15),...
        diff(p22,t) == exp(hf23+hf24+hf25),...
        diff(p33,t) == exp(+hf32+hf35),...
        diff(p44,t) == exp(+hf42+hf45),...
        diff(p12,t) == p11.*hf12+p12*hf22,...
        diff(p13,t) == 0,...
        diff(p14,t) == 0,...
        diff(p15,t) == p11.*hf15+p12*hf25,...
        diff(p21,t) == 0,...
        diff(p23,t) == p22.*hf23+p23*hf33,...
        diff(p24,t) == p22.*hf24+p24*hf44,...
        diff(p25,t) == p22.*hf25+p23*hf35+p24.*hf45,...
        diff(p31,t) == 0,...
        diff(p32,t) == p32.*hf22+p33*hf32,...
        diff(p34,t) == 0,...
        diff(p35,t) == p32.*hf25+p33*hf35,...
        diff(p41,t) == 0,...
        diff(p42,t) == p42.*hf22+p44*hf42,...
        diff(p43,t) == 0,...
        diff(p45,t) == p42.*hf25+p44*hf45,...
        diff(p51,t) == 0,...
        diff(p52,t) == 0,...
        diff(p53,t) == 0,...
        diff(p54,t) == 0,...
        diff(p55,t) == 1]
    cond1 = p11(t)+p12(t)+p13(t)+p14(t)+p15(t) == 1;
    cond2 = p21(t)+p22(t)+p23(t)+p24(t)+p25(t) == 1;
    cond3 = p31(t)+p32(t)+p33(t)+p34(t)+p35(t) == 1;
    cond4 = p41(t)+p42(t)+p43(t)+p44(t)+p45(t) == 1;
    cond5 = p51(t)+p52(t)+p53(t)+p54(t)+p55(t) == 1;
S = dsolve(ode)

These equations are solutions to probabilities therefore, I am expecting the answers between 0 and 1.

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susman 2021-1-29

where, t can be any time period like t = 20

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Answer 1

Walter Roberson 2021-1-29

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/730008-solution-to-differential-equations-system-looks-strange-what-is-wrong-with-my-code#answer_608973

编辑：Walter Roberson 2021-1-29

在 MATLAB Online 中打开

Q = @(v) sym(v)

Q = function_handle with value:

@(v)sym(v)

syms t real

%defined parameters

hf12 = exp(Q(-21.6079)+Q(0.0099)).*exp(Q(-0.0134).*t)

hf12 =

hf13 = 0

hf14 = 0

hf15 = exp(Q(-9.65573)+Q(0.01844)+Q(0.0821)).*exp(Q(0.02246)*t)

hf15 =

hf11 = -(hf12+hf15)

hf11 =

hf21 = 0

hf23 = exp(Q(-1.666)-Q(0.1116)).*exp(Q(-0.0025).*t)

hf23 =

hf24 = exp(Q(-8.96236)+Q(0.07691)).*exp(Q(0.00978).*t)

hf24 =

hf25 = exp(Q(-9.65573)+Q(0.08218)).*exp(Q(0.02246).*t)

hf25 =

hf22 = -(hf23+hf24+hf25)

hf22 =

hf31 = 0

hf32 = exp(Q(-21.6079)+Q(0.0676)+Q(0.0099)).*exp(Q(-0.0134).*t)

hf32 =

hf34 = 0

hf35 = exp(Q(-9.65573)-Q(0.11853)+Q(0.08218)).*exp(Q(0.02246).*t)

hf35 =

hf33 = -(hf32+hf35)

hf33 =

hf41 = 0

hf42 = exp(Q(-21.609)-Q(0.4176)+Q(0.0099)).*exp(Q(-0.0134).*t)

hf42 =

hf43 = 0

hf45 = exp(Q(-9.65573)-Q(0.00415)+Q(0.08218)).*exp(Q(0.02246).*t)

hf45 =

hf44 = -(hf42+hf45)

hf44 =

%% solution

syms p11(t) p22(t) p33(t) p44(t) p55(t) p12(t) p13(t) p14(t) p15(t) p21(t) p23(t) p24(t) p25(t) ...

p31(t) p32(t) p34(t) p35(t) p41(t) p42(t) p43(t) p45(t) p51(t) p52(t) p53(t) p54(t) p55(t)

ode = [diff(p11,t) == exp(hf12+hf15),...

diff(p22,t) == exp(hf23+hf24+hf25),...

diff(p33,t) == exp(+hf32+hf35),...

diff(p44,t) == exp(+hf42+hf45),...

diff(p12,t) == p11.*hf12+p12*hf22,...

diff(p13,t) == 0,...

diff(p14,t) == 0,...

diff(p15,t) == p11.*hf15+p12*hf25,...

diff(p21,t) == 0,...

diff(p23,t) == p22.*hf23+p23*hf33,...

diff(p24,t) == p22.*hf24+p24*hf44,...

diff(p25,t) == p22.*hf25+p23*hf35+p24.*hf45,...

diff(p31,t) == 0,...

diff(p32,t) == p32.*hf22+p33*hf32,...

diff(p34,t) == 0,...

diff(p35,t) == p32.*hf25+p33*hf35,...

diff(p41,t) == 0,...

diff(p42,t) == p42.*hf22+p44*hf42,...

diff(p43,t) == 0,...

diff(p45,t) == p42.*hf25+p44*hf45,...

diff(p51,t) == 0,...

diff(p52,t) == 0,...

diff(p53,t) == 0,...

diff(p54,t) == 0,...

diff(p55,t) == 1].';

ode = ode(t);

cond1 = p11(t)+p12(t)+p13(t)+p14(t)+p15(t) == 1;

cond2 = p21(t)+p22(t)+p23(t)+p24(t)+p25(t) == 1;

cond3 = p31(t)+p32(t)+p33(t)+p34(t)+p35(t) == 1;

cond4 = p41(t)+p42(t)+p43(t)+p44(t)+p45(t) == 1;

cond5 = p51(t)+p52(t)+p53(t)+p54(t)+p55(t) == 1;

S = dsolve(ode)

S = struct with fields:

p11: [1×1 sym] p12: [1×1 sym] p13: [1×1 sym] p14: [1×1 sym] p15: [1×1 sym] p21: [1×1 sym] p22: [1×1 sym] p23: [1×1 sym] p24: [1×1 sym] p25: [1×1 sym] p31: [1×1 sym] p32: [1×1 sym] p33: [1×1 sym] p34: [1×1 sym] p35: [1×1 sym] p41: [1×1 sym] p42: [1×1 sym] p43: [1×1 sym] p44: [1×1 sym] p45: [1×1 sym] p51: [1×1 sym] p52: [1×1 sym] p53: [1×1 sym] p54: [1×1 sym] p55: [1×1 sym]

S.p11

ans =

vpa(S.p11, 5)

ans =

Sc = struct2cell(S);

Sv = simplify(vertcat(Sc{:}), 'steps', 10)

Sv =

symvar(Sv)

ans =

conds = [cond1; cond2; cond3; cond4; cond5]

conds =

temp = subs(conds, S);

%do not simplify() the equations directly, as it will test to see

%whether the left side equals the right side and will say "symfalse"

conds_subs = simplify(lhs(temp)-rhs(temp), 'steps', 10)

conds_subs =

symvar(conds_subs)

ans =

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Walter Roberson 2021-1-30

Your first four p5* have derivative 0 so they must be constant. Your p55 derivative is 1, so the function must be 1*t plus a constant. The five together total 1. But the five added are a series of constants plus t. That cannot be constant unless t is restricted to 0. Therefore your equations are wrong.

Walter Roberson 2021-1-30

Notice that you have exp() of terms that are already exp(). See sigma 19 in the above "where" list to see it come in as a factor. You can be fairly sure that you will not be able to find a closed form solution for integrals or ode that include such terms.

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solution to differential equations system looks strange. What is wrong with my code?

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