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Question: a system of differential equations

Question:a system of differential equations

akbarhp 20 Maple
solve rootfinding differential-equations
December 17 2023
2
> 

restart:
> 

with(LinearAlgebra):
> 

with(plots):
> 

with(Maplets[Examples]):
> 

with(Student[Calculus1]):
> 

Digits:=100:
> 

 

NULL

``

``lambda[C] := .5

.5

 (1)

lambda[L] := .5

.5

 (2)

beta := 0.1e-1

0.1e-1

 (3)

alpha := .5

.5

 (4)

NULL

X[0]:=0;X[1]:=0.5;X[2]:=1;

1

 (5)

 #A: Governing Equations:

 EQ[1] := lambda[C]^2*(diff(v[1](x), x, x, x, x, x, x))-(diff(v[1](x), x, x, x, x))

.25*(diff(diff(diff(diff(diff(diff(v[1](x), x), x), x), x), x), x))-(diff(diff(diff(diff(v[1](x), x), x), x), x))

 (6)

EQ[2] := lambda[C]^2*(diff(v[2](x), x, x, x, x, x, x))-(diff(v[2](x), x, x, x, x))

.25*(diff(diff(diff(diff(diff(diff(v[2](x), x), x), x), x), x), x))-(diff(diff(diff(diff(v[2](x), x), x), x), x))

 (7)

EQ[3] := s[1](x)+diff(v[1](x), x)-lambda[C]^2*(diff(s[1](x), x, x)+diff(v[1](x), x, x, x))-beta*(diff(v[1](x), x, x, x)-lambda[C]^2*(diff(v[1](x), x, x, x, x, x)))

s[1](x)+diff(v[1](x), x)-.25*(diff(diff(s[1](x), x), x))-.26*(diff(diff(diff(v[1](x), x), x), x))+0.25e-2*(diff(diff(diff(diff(diff(v[1](x), x), x), x), x), x))

 (8)

``

EQ[4] := s[2](x)+diff(v[2](x), x)-lambda[C]^2*(diff(s[2](x), x, x)+diff(v[2](x), x, x, x))-beta*(diff(v[2](x), x, x, x)-lambda[C]^2*(diff(v[2](x), x, x, x, x, x)))

s[2](x)+diff(v[2](x), x)-.25*(diff(diff(s[2](x), x), x))-.26*(diff(diff(diff(v[2](x), x), x), x))+0.25e-2*(diff(diff(diff(diff(diff(v[2](x), x), x), x), x), x))

 (9)

assign(dsolve({EQ[1], EQ[2], EQ[3], EQ[4]}, {s[1](x), s[2](x), v[1](x), v[2](x)}))

NULL

V[1] := subs(_C1 = A[1], _C2 = A[2], _C3 = A[3], _C4 = A[4], _C5 = A[5], _C6 = A[6], _C7 = A[7], _C8 = A[8], _C9 = A[9], _C10 = A[10], _C11 = A[11], _C12 = A[12], _C13 = A[13], _C14 = A[14], _C15 = A[15], _C16 = A[16], v[1](x))

NULL

V[2] := subs(_C1 = A[1], _C2 = A[2], _C3 = A[3], _C4 = A[4], _C5 = A[5], _C6 = A[6], _C7 = A[7], _C8 = A[8], _C9 = A[9], _C10 = A[10], _C11 = A[11], _C12 = A[12], _C13 = A[13], _C14 = A[14], _C15 = A[15], _C16 = A[16], v[2](x))

NULL

S[1] := subs(_C1 = A[1], _C2 = A[2], _C3 = A[3], _C4 = A[4], _C5 = A[5], _C6 = A[6], _C7 = A[7], _C8 = A[8], _C9 = A[9], _C10 = A[10], _C11 = A[11], _C12 = A[12], _C13 = A[13], _C14 = A[14], _C15 = A[15], _C16 = A[16], s[1](x))

NULL``

S[2] := subs(_C1 = A[1], _C2 = A[2], _C3 = A[3], _C4 = A[4], _C5 = A[5], _C6 = A[6], _C7 = A[7], _C8 = A[8], _C9 = A[9], _C10 = A[10], _C11 = A[11], _C12 = A[12], _C13 = A[13], _C14 = A[14], _C15 = A[15], _C16 = A[16], s[2](x))

``

# B: Costitutive Boundary Conditions:

eq[1] := evalf(eval((alpha*lambda[C]^4+lambda[C]^2*lambda[L]^2)*(diff(V[1], x, x, x, x, x))-alpha*lambda[C]^3*(diff(V[1], x, x, x, x))+(-alpha*lambda[C]^2+lambda[C]^2-lambda[L]^2)*(diff(V[1], x, x, x))+(alpha*lambda[C]-lambda[C])*(diff(V[1], x, x)), x = X[0]))

-4.00000*A[16]-.750*A[14]-.50*A[13]

 (10)

NULL

eq[2] := evalf(eval((alpha*lambda[C]^4+lambda[C]^2*lambda[L]^2)*(diff(V[2], x, x, x, x, x))+alpha*lambda[C]^3*(diff(V[2], x, x, x, x))+(-alpha*lambda[C]^2+lambda[C]^2-lambda[L]^2)*(diff(V[2], x, x, x))+(-alpha*lambda[C]+lambda[C])*(diff(V[2], x, x)), x = X[2]))

10.87312731383618094144114988541064999102898837479983829986787051089630652141419037828552871410066571*A[9]+.50*A[7]

 (11)

````

eq[3] := evalf(eval(lambda[C]^2*(diff(S[1], x)+diff(V[1], x, x))-lambda[C]*(S[1]+diff(V[1], x))+alpha*beta*lambda[C]*(diff(V[1], x, x, x)-lambda[C]^2*(diff(V[1], x, x, x, x, x))), x = X[0]))

-1.00*A[3]+0.8000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-1*A[16]-0.1500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-1*A[14]

 (12)

 

eq[4] := evalf(eval(lambda[C]^2*(diff(S[2], x)+diff(V[2], x, x))+lambda[C]*(S[2]+diff(V[2], x))-alpha*beta*lambda[C]*(diff(V[2], x, x, x)-lambda[C]^2*(diff(V[2], x, x, x, x, x))), x = X[2]))

7.389056098930650227230427460575007813180315570551847324087127822522573796079057763384312485079121795*A[2]+.5911244879144520181784341968460006250544252456441477859269702258018059036863246210707449988063297436*A[9]+0.1500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-1*A[8]

 (13)

NULLNULL

eq[5] := evalf(eval(alpha*lambda[C]^4*(diff(V[1], x, x, x, x, x))-alpha*lambda[C]^3*(diff(V[1], x, x, x, x))+(-alpha*lambda[C]^2+lambda[C]^2)*(diff(V[1], x, x, x))+(alpha*lambda[C]-lambda[C])*(diff(V[1], x, x))+(1-alpha)*(int(exp((z-x)/lambda[C])*(diff(V[1], x, x)-lambda[C]^2*(diff(V[1], x, x, x, x))), z = X[0] .. X[1]))-lambda[L]^2*(diff(int(exp((z-x)/lambda[C])*(diff(V[1], x, x, x)-lambda[C]^2*(diff(V[1], x, x, x, x, x))), z = X[0] .. x), x)), x = X[1]))

-1.471517764685769286382095080645843469783244524127071338031347206789845982979599213428589097383678575*A[16]-0.777287426357452235899284828632869517530750448214776276426328038192883654260245575535813672783191983e-1*A[14]-.1839397205857211607977618850807304337229055655158839172539184008487307478724499016785736371729598218*A[13]

 (14)

NULL

NULL``

NULL

eq[6] := evalf(eval(alpha*lambda[C]^4*(diff(V[2], x, x, x, x, x))+alpha*lambda[C]^3*(diff(V[2], x, x, x, x))+(-alpha*lambda[C]^2+lambda[C]^2)*(diff(V[2], x, x, x))+(-alpha*lambda[C]+lambda[C])*(diff(V[2], x, x))-(1-alpha)*(int(exp((x-z)/lambda[C])*(diff(V[2], x, x)-lambda[C]^2*(diff(V[2], x, x, x, x))), z = X[1] .. X[2]))+lambda[L]^2*(diff(int(exp((x-z)/lambda[C])*(diff(V[2], x, x, x)-lambda[C]^2*(diff(V[2], x, x, x, x, x))), z = x .. X[2]), x)), x = X[1]))

10.87312731383618094144114988541064999102898837479983829986787051089630652141419037828552871410066571*A[9]+.4740904191214182588033571723789043494156416517261741241191223987269038781913251474821395442405602671*A[8]+.1839397205857211607977618850807304337229055655158839172539184008487307478724499016785736371729598218*A[7]

 (15)

NULL

eq[7] := evalf(eval(lambda[C]^2*(diff(S[1], x)+diff(V[1], x, x))-lambda[C]*(S[1]+diff(V[1], x))+alpha*beta*lambda[C]*(diff(V[1], x, x, x)-lambda[C]^2*(diff(V[1], x, x, x, x, x)))+(1-alpha)*beta*(int(exp((z-x)/lambda[C])*(diff(V[1], x, x, x)-lambda[C]^2*(diff(V[1], x, x, x, x, x))), z = X[0] .. X[1])), x = X[1]))

-.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196437*A[3]+0.2943035529371538572764190161291686939566489048254142676062694413579691965959198426857178194767357150e-1*A[16]-0.5518191617571634823932856552421913011687166965476517517617552025461922436173497050357209115188794656e-2*A[14]

 (16)

NULL

eq[8] := evalf(eval(lambda[C]^2*(diff(S[2], x)+diff(V[2], x, x))+lambda[C]*(S[2]+diff(V[2], x))-alpha*beta*lambda[C]*(diff(V[2], x, x, x)-lambda[C]^2*(diff(V[2], x, x, x, x, x)))-(1-alpha)*beta*(int(exp((x-z)/lambda[C])*(diff(V[2], x, x, x)-lambda[C]^2*(diff(V[2], x, x, x, x, x))), z = X[1] .. X[2])), x = X[1]))

2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427*A[2]+.2174625462767236188288229977082129998205797674959967659973574102179261304282838075657105742820133142*A[9]+0.551819161757163482393285655242191301168716696547651751761755202546192243617349705035720911518879466e-2*A[8]

 (17)

# D: Variationally Consistent Conditions:

 

eq[9] := evalf(eval(V[1], x = X[0]))

A[11]+A[15]+A[16]

 (18)

NULL

NULL

eq[10] := evalf(eval(V[2], x = X[2]))

A[5]+A[6]+A[7]+A[8]+7.389056098930650227230427460575007813180315570551847324087127822522573796079057763384312485079121795*A[9]+.1353352832366126918939994949724844034076315459095758814681588726540733741014876899370981224906570488*A[10]

 (19)

 

eq[11] := evalf(eval(V[1]-V[2], x = X[1]))

A[11]+.5*A[12]+.25*A[13]+.125*A[14]+2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427*A[15]+.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196437*A[16]-1.*A[5]-.5*A[6]-.25*A[7]-.125*A[8]-2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427*A[9]-.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196437*A[10]

 (20)

 

eq[12] := evalf(eval(S[1]-S[2], x = X[1]))

2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427*A[4]+.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196437*A[3]-1.*A[12]-1.0*A[13]-.6900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*A[14]-5.219101110641366851891751944997111995693914419903922383936577845230227130278811381577053782768319540*A[15]+.7063285270491692574634056387100048654959573715809942422550466592591260718302076224457227667441657159*A[16]-2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427*A[2]-.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196437*A[1]+A[6]+1.0*A[7]+.6900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000*A[8]+5.219101110641366851891751944997111995693914419903922383936577845230227130278811381577053782768319540*A[9]-.7063285270491692574634056387100048654959573715809942422550466592591260718302076224457227667441657159*A[10]

 (21)

 

eq[13] := evalf(eval(lambda[C]^2*(diff(V[1], x, x, x, x))-(diff(V[1], x, x)), x = X[0]))

-2.*A[13]

 (22)

NULL

eq[14] := evalf(eval(lambda[C]^2*(diff(V[2], x, x, x, x))-(diff(V[2], x, x)), x = X[2]))

-2.*A[7]-6.*A[8]

 (23)

NULL````

eq[15] := evalf(eval(diff(V[1], x, x)-(diff(V[2], x, x))-lambda[C]^2*(diff(V[1], x, x, x, x)-(diff(V[2], x, x, x, x))), x = X[1]))

-2.*A[7]-3.0*A[8]+2.*A[13]+3.0*A[14]

 (24)

NULL

eq[16] := evalf(eval(diff(V[1], x, x, x)-(diff(V[2], x, x, x))-lambda[C]^2*(diff(V[1], x, x, x, x, x)-(diff(V[2], x, x, x, x, x)))-1, x = X[1]))

6.*A[14]-1.-6.*A[8]

 (25)

 

> 

equations := [seq(eq[i], i = 1 .. 16)]:
unknowns := [seq(A[i], i = 1 .. 16)]:  

assign( solve(equations, unknowns)):
> 

 

> 

 

> 

 

> 

display(plot(V[1],x=X[0]..X[1]))

Warning, expecting only range variable x in expression A[11]+A[12]*x+A[13]*x^2+A[14]*x^3+A[15]*exp(2*x)+A[16]*exp(-2*x) to be plotted but found names [A[11], A[12], A[13], A[14], A[15], A[16]]

  

 

eval(V[1], x = .5)

A[11]+.5*A[12]+.25*A[13]+.125*A[14]+2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427*A[15]+.3678794411714423215955237701614608674458111310317678345078368016974614957448998033571472743459196437*A[16]

 (26)

 

Hi.

I have faced a problem in solving a system of differential equations with 16 boundary conditions with maple. Unfortunately, maple does not solve it and I could not find out what the problem is. I share the maple file here and I will be grateful for any 

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