## differential equations – NDSolve does’t solve nonlinear system of PDEs with forth spatial derivative

I am trying to solve a highly nonlinear system of coupled PDEs with NDSolveValue.
In the following you can find the system, the boundary conditions and the value of the numerical parameters.

and here is my code

``````Numerical parameters

e = 1.60217662*10^-19;
h = 6.62607004*10^-34;
(CurlyPhi)0 = h/(2*Pi*2*e);
(CapitalPhi)0 = h/(2*e);
Ic = 4*10^-6;
Lj0 = h/(2*Pi*2*e*Ic);
Cj = 32*10^-15;
Cgp = 64*10^-15;
Cgs = 62*10^-15;
L = 160*10^-12;
Ej0 = (h*Ic)/(2*e*2*Pi);
k = 0.9;
(CapitalPhi)dc = (2*Pi*(CurlyPhi)0)/3;
(Omega)p = 2*Pi*10^9*12;
(Omega)s = 2*Pi*10^9*8;
a = 30*10^-6;

s = NDSolveValue(
{
-Cj*(D(V(x, t) x, x, t, t) - Cgs/Cj*D(V(x, t), t, t))
+ (2*Ej0)/(CurlyPhi)0*(
Sin(1/(CurlyPhi)0*(D(V(x, t), x) + (1/2)*D(V(x, t), x, x)))*
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x)))) -
Sin(1/(CurlyPhi)0*(D(V(x, t), x) - (1/2)*D(V(x, t), x, x)))*
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x))))
) == 0,
Cgp*D(F(x, t), t, t) - 1/L*D(F(x, t), x, x) +
(k*Ej0)/(CurlyPhi)0*(
Cos(1/(CurlyPhi)0*(D(V(x, t), x) - (1/2)*D(V(x, t), x, x)))*
((1/
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x))))) Cos((CapitalPhi)dc/(
2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x)))*
Sin((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x))))
- Cos(1/(CurlyPhi)0*(D(V(x, t), x) + (1/2)*D(V(x, t), x, x)))*
((1/
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x))))) Cos((CapitalPhi)dc/(
2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x)))*
Sin((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x))))
) == 0,
V(0, t) == ((CapitalPhi)0/20)*Sin((Omega)s*t),
V(x, 0) == 0,
F(0, t) == ((CapitalPhi)0/2)*Sin((Omega)p*t),
F(x, 0) == 0
},
{F, V},
{x, 0.003},
{t, 10^-9}
)
``````

Now, every time I try to solve this system mathematica returns the following problem

Reading online I realized that this is a limit of mathematica 11.0 because my PDE contains a forth order derivative. For this reason I tried to change method of NDSolveValue using the following string

so the method of lines. But this time when I try to solve the system mathematica returns a different error message

This time I tried to change, add or remove some boundary conditions without really solving the problem.
Is there a different method I should try to solve this PDE system? Am I imposing wrong boundary conditions for the problem?
Thank you everybody for the help!

## differential equations – how to solve this wave edp with mathemamatica ? what is the best solver for this?

help me please to understand how to solve many Edp with Mathematica Please

In(67):= weqn=D(u(x,t),{t,2})==D(u(x,t),{x,2})

In(68):= Ivi={u(x,0)==1}

In(69):= Ibc={u(0,t)==0}

In(70):= Sol=DSolveValue({weqn,Ivi,Ibc},u(x,t),{x,t})
Out(70)= DSolveValue({(u^(0,2))(x,t)==(u^(2,0))(x,t),{u(x,0)==1},{u(0,t)==0}},u(x,t),{x,t})

## equation solving – Solve a linear system using the substitution method

I know that, for the substitution method, I have to isolate the first variable, plug the equation for that variable into the other equations, simplify the new equations, isolate the second variable, etc. I have been trying functions like `Reduce` and `Solve`, but they have failed to do so. I’m not sure how else to tackle this problem.

P.S. I know how to do it through `LinearSolve` and inverse matrix, but I need this one to be solved through substitution method.

## recurrence relations – How to solve \$Tleft( nright) = 4Tleft({nover 2}right) + 2^{nover2}\$ using Akra-Bazzi method?

$$Tleft( nright) = 4Tleft({nover 2}right) + 2^{nover2}$$

Let $$g(n) = 2^{nover2}$$, we can see that $$|g'(n)|$$ is not bounded by a polynomial. Therefore Akra-Bazzi method cannot be used to find bounds of the above problem. How come the above problem can still be solved using Master theorem which is just a corollary of Akra-Bazzi method.

## algorithms – How can I create the table and solve the error, why did the error happened?

import world_population.csv
population = Table.with_columns(
“Population”, population_amounts,
“Year”, years
)
population

No module named ‘world_population’

## complexity theory – Failing to solve a recurrence by induction

My question is strongly related to the one asked here:

How do I show T(n) = 2T(n-1) + k is O(2^n)?

$$T(n)=2T(n-1)+1$$

Going with the steps, I reached the point where:

$$c*2^{n}geq c*2^{n}+1$$
which implies
$$0geq 1$$
which is false for all possible values of $$c$$ and thus the claim $$T(n)=O(2^{n})$$ should be incorrect.

However, most answers to the question just mention that in such case, one should try alternative methods to find/prove the upper bound. I don’t understand how should I be able to justify that, is this a scenario in which “induction fails” ? because I’ve never heard of such one.

## Solve this problem of Kronecker Product and Prove the given statement true or not

let A,B,C,D are four complex matrices.let
P=Kronecker Profuct (A,B)
Q=Kronecker Profuct (C,D)

Is it true that P.Q= Kronecker Product ((A.C),(B.D))

## Want to solve Laplace equation for two mediums separated by an interface

I want to solve the Laplace equation for electrostatics in two different regions separated by an interface. I can’t understand how to impose a continuity condition for the electric field on the interface. Can anyone help me to figure out the problem?

## manipulate – System of equations does not solve with Solve

``````Clear("Global`*")

y(a_, k_, l_, d_) := a*k^d*l^(1 - d);
mpl(a_, k_, l_, d_) := (1 - d)*a*k^d*l^(-d);
lsupply(b_, m_, l_) := b + m*l;
``````

`l` should not be an argument to `intersectlabor`. Use `NSolve`

``````intersectlabor(a_?NumericQ, k_?NumericQ, d_?NumericQ, b_?NumericQ,
m_?NumericQ) := {l, mpl(a, k, l, d)} /.
NSolve({mpl(a, k, l, d) == lsupply(b, m, l), a > 0, b > 0, m > 0, l > 0,
k > 0, d > 0}, l)((1))

Manipulate(
Plot({mpl(a, k, l, d), lsupply(b, m, l)}, {l, 0, 100},
PlotRange -> {25, 1000},
AxesLabel -> {"L", "MPL, w"},
LabelStyle -> Black,
Epilog -> {Blue, PointSize@Large,
Point@intersectlabor(a, k, d, b, m)}), {{a, 80, "A"}, 1, 100, 1,
Appearance -> "Labeled"},
{{k, 350, "K"}, 200, 1000, 5, Appearance -> "Labeled"},
{{d, 0.3}, 0.01, 0.99, 0.01, Appearance -> "Labeled"},
{{b, 4}, 2, 200, 2, Appearance -> "Labeled"},
{{m, 5}, 0.1, 10, 0.1, Appearance -> "Labeled"})
``````

## plotting – Solve complex equation in mathematica

How can I solve the following equation in Mathematica?

the parameter tao is as follows:

The value of the other parameters are dh/R=90000, ln A=-235.5, Tf=117, beta=0.85 and x=0.6. All parameters are consistent with the units.

Question:

After solving the equation I would like to make the following plot of Tf-Ta vs Log ta (log ta is simply log t and goes from -2 to 6. Ta is simply T and goes from 75 to 107. Tf was defined above with a value of 117):

Note: For reference, the equation above is called the Tool-Narayanaswamy-Moynihan (TNM) equation