Tag: equations

strongly non-linear algebraic equations

I can not solve the system.Can any one help me with that? Thanks.

ClearAll("Global`*");
f1(S_, e_, I1_, I2_, R_) := 
b*(S + R) - d*S - g*S*(S + e + I1 + I2 + R)/K - bcd*S*(I1 + I2);
f2(S_, e_, I1_, I2_, R_) := -d*e - g*e*(S + e + I1 + I2 + R)/K + 
bcd*S*(I1 + I2) - scd1*e;
f3(S_, e_, I1_, I2_, R_) := -d*I1 - 
g*I1*(S + e + I1 + I2 + R)/K + (1 - f)*scd1*e - scd2*I1;

f4(S_, e_, I1_, I2_, R_) := -d*I2 - g*I2*(S + e + I1 + I2 + R)/K + 
f*scd1*e - acd*I2;

f5(S_, e_, I1_, I2_, R_) := -d*R - g*R*(S + e + I1 + I2 + R)/K + 
scd2*I1;
Expand({f1(S, e, I1, I2, R) == 0, f2(S, e, I1, I2, R) == 0, 
f3(S, e, I1, I2, R) == 0, f4(S, e, I1, I2, R) == 0, 
f5(S, e, I1, I2, R) == 0})
{b R + b S - d S - bcd I1 S - bcd I2 S - (e g S)/K - (g I1 S)/K - (
g I2 S)/K - (g R S)/K - (g S^2)/K == 
0, -d e - (e^2 g)/K - (e g I1)/K - (e g I2)/K - (e g R)/K + 
bcd I1 S + bcd I2 S - (e g S)/K - e scd1 == 
0, -d I1 - (e g I1)/K - (g I1^2)/K - (g I1 I2)/K - (g I1 R)/K - (
g I1 S)/K + e scd1 - e f scd1 - I1 scd2 == 
0, -acd I2 - d I2 - (e g I2)/K - (g I1 I2)/K - (g I2^2)/K - (
g I2 R)/K - (g I2 S)/K + e f scd1 == 
0, -d R - (e g R)/K - (g I1 R)/K - (g I2 R)/K - (g R^2)/K - (g R S)/
K + I1 scd2 == 0}
NSolve(%, {S, e, I1, I2, R}, Reals)`

differential equations – Basic problem using NDSolve

I am trying to solving a system of three equations. However each time I try to use NDSolve as follows

test={-4+(Tau) (14.10532228364108` +(Tau) (17.506023005076905` +5.327608594407096` (Tau)))+17.506023005076905` x((Tau))+y((Tau)) (-10.453361863256365`+(Tau) (17.506023005076905` +7.9914128916106435` (Tau))+(4.376505751269226` +3.9957064458053217` (Tau)+0.665951074300887` y((Tau))) y((Tau)))==2 (1+4.376505751269226` (1.` (Tau)+0.5` y((Tau)))^2)^2 (x^(Prime))((Tau)),-4.`-1.6480542736638855` (0.` -4 (x((Tau))-z((Tau)))^2+8 (x((Tau))-2 y((Tau))+z((Tau))))==2 (1+1.6480542736638855` (x((Tau))-1.` z((Tau)))^2)^2 (y^(Prime))((Tau)),-5.92690787297127`+y((Tau)) (-19.967971976722627`+(0.3807993054639045` +0.665951074300887` y((Tau))) y((Tau)))+17.506023005076905` z((Tau))==2 (1+1.0941264378173066` (-2.`+y((Tau)))^2)^2 (z^(Prime))((Tau))};
Inittest={x((Tau))==1,y((Tau))==1,z((Tau))==1};
Unkntest={x,y,z};'''

NDSolve(Join(test, Inittest), Unkntest, {(Tau), 0, 10})

Although I am (correctly?) feeding NDSolve with three invitational conditions and the three functions, I get the following error

NDSolve::overdet: There are fewer dependent variables, {x((Tau)),y((Tau)),z((Tau))}, than equations, so the system is overdetermined.

Thanks a lot !

Solving a system of simultaneous equations with modified multiplication rules

Hello and thanks in advance for your help. Is there a way to solve a system of simultaneous equations in Mathematica where because of the nature of the system being modeled, the rules for multiplying variables is non-standard such that during algebraic manipulation, a variable $x_i^n$ is always equal to $x_i$. For example under these modified multiplication rules, $5x^2y^9z^7$ would simplify to $5xyz$.

It is known that solutions to the system always result in variable values equal or greater than zero, and thus odd power exponents will never change the sign of a term. The equations in the system are all polynomials of the first order, with terms that may include any number or combination of system variables and one constant and only involve multiplication. One of the terms in an equation may be a constant. Each polynomial is always equal to zero. For example if there are ten variables and ten equations, one of the polynomial example terms could be $3x_3x_4$ or $7x_1x_5x_8x_9$ or in a more extreme example, $5x_1x_2x_3x_4x_5x_6x_7x_8x_9$.

Unless I am mistaken, equations of this kind define hyperplanes, and the solution of the system, should it exist, is a hyperpoint representing the intersection of the hyperplanes. Thanks in advance for any help. Inherent in this question is whether it is possible in Mathematica to change the rules by which a basic operator functions, in this case the multiplication operator. Also, what is the best Mathematica function to use in solving a system of equations of this kind, putting aside the unusual multiplication situation? Thanks again for any help.

nonlinear – System of Non Linear Equations

I have a system with 7 equations and 7 unknowns with 11 parameters. I wish to get closed form solutions for each of those 7 equations. However, the system is non linear in the parameters which makes the solution even more complicated. Is there any way to solve this? Or should I attempt to linearize the system of equations? If linearizing is the option how should I do it in Mathematica?

differential equations – How to implement limit boundary condition in solving PDE

I have to solve a partial differential equation for a function $F(x,t)$ where one of the boundary condition is formulated in terms of a limit:

$lim_{xrightarrow +infty} e^x partial_xF(x,t)=0$

Is it possible to implement this as a boundary condition in NDSolve (or possibly NDSolve`FiniteDifferenceDerivative)?

differential equations – Unable to find the plots of NDSolve

My code is given below

I1(E1_, z_) := NIntegrate((-Log(u))*Exp(-E1*f1(z)/u), {u, 0, 1});
I2(E1_, E2_, z_) := NIntegrate(((-Log(u))^3)/u*Exp(-(Log(u))^2/(f2(z))^2)*Exp(-E1*f1(z)*
      Exp((Log(u))^2/(2*(f1(z))^2)))/(1 + {1 + (E2*Exp(-(Log(u))^2/(f2(z))^2))/(f2(z))^2}^(2/5)) * (E1/f1(z)*Exp((Log(u))^2/(2*(f1(z))^2)) - (4*E2*Exp(-(Log(u))^2/(f2(z))^2)*{1 + (E2*Exp(-(Log(u))^2/(f2(z))^2))/(f2(z))^2}^(-3/5))/(5* (f2(z))^4*(1 + {1 + (E2* Exp(-(Log(u))^2/(f2(z))^2))/(f2(z))^2}^(2/5)))), {u, 0,1});
js(l_, p_, E1_, E2_, den_, z_) := NDSolve({f1''(z) + 1/f1(z)*(f1'(z))^2 == 1/(f1(z))^3 + 2*E1*den*I1(E1, z), f2''(z) + 1/f2(z)*(f2'(z))^2 == 1/(f2(z))^3 + 2*den*I2(E1, E2, z)*1/(f2(z))^3, f1'(0) == 0, f1(0) == 1, f2'(0) == 0, f2(0) == 1}, {f1, f2}, {z, 0, 5});
ja1(z_) = js(0, 0, 0.05, 2.5, 6, z) // Quiet
ja2(z_) = js(0, 0, 0.05, 2.5, 6, z) // Quiet
ja3(z_) = js(0, 0, 0.05, 2.5, 6, z) // Quiet
Plot(Evaluate(f1(z) /. {ja1(z), ja2(z), ja3(z)}), {z, 0, 5})
Plot(Evaluate(f2(z) /. {ja1(z), ja2(z), ja3(z)}), {z, 0, 5})

I have checked it many times but unable to find my mistake. Kindly help me in getting my plot.

Solve Troubles Simultaneous equations – Mathematica Stack Exchange

Code copied below. (I know it is not pretty but still learning.) I am trying to solve a system of two equations for two unkowns. In the upper code I solve one equation for one variable and then substitute into the second equation. This method works and produces a result. (Note I need a numeric solution here not an exact solution.) However, when I try to solve the equations simultaneously Solve returns the result D0->0. Same issue using NSolve. I am not sure what is going on or how to get “fix” Solve to work with both equations simultaneously.

CIn(101):= (*Problem 5.24*)
Clear("Global`*"); Clear(Derivative);
D1(Q_, T_, D0_, k_) = D0*Exp(-Q/(k*T));
EQ1 = D1(Q, 650 + 273.15, D0, 9.62*10^-5) == 5.5*10^-16;
Temp = Solve({EQ1}, {Q});
EQ2 = D1(Q /. Temp((1)), 900 + 273.15, D0, 9.62*10^-5) == 1.3*10^-13;
SolD0 = Solve(EQ2)
EQ3 = D1(Q, 650 + 273.15, D0 /. SolD0((1)), 9.62*10^-5) == 5.5*10^-16;
Solve(EQ3)

Solve({D1(Q, 650 + 273.15, D0, 9.62*10^-5) == 5.5*10^-16, 
  D1(Q, 650 + 273.15, D0, 9.62*10^-5) == 1.3*10^-13}, {D0, Q})
NSolve({D1(Q, 650 + 273.15, D0, 9.62*10^-5) == 5.5*10^-16, 
  D1(Q, 650 + 273.15, D0, 9.62*10^-5) == 1.3*10^-13}, {D0, Q})


During evaluation of In(101):= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

During evaluation of In(101):= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out(106)= {{D0 -> 0.0000756184}}

During evaluation of In(101):= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out(108)= {{Q -> 2.27762}}

During evaluation of In(101):= Solve::svars: Equations may not give solutions for all "solve" variables.

Out(109)= {{D0 -> 0.}}

During evaluation of In(101):= NSolve::svars: Equations may not give solutions for all "solve" variables.

Out(110)= {{D0 -> 0.}}

I have tried using Reduce but it just returns False.

Any help appreciated.

ordinary differential equations – What does this notation of a differentiability class: $C(J, mathbb{R}^2)$ mean?

I found out from here that $C(a, b)$ means the space of all continuous functions on the interval $(a, b)$.
I also searched the wikipedia page about differentiability classes but didn’t see that notation being used.

I stumbled upon this notation in a lecture on ordinary differential equations. A similar one I found is $C(J, mathbb{R}^star)$

The context for the notation in the question is $g in C(J, mathbb{R}^2)$ and it is said to be a space

differential equations – Unable to solve nonlinear PDE with NDSolve

Lately, I’ve been trying to solve the following PDE:
begin{equation} -v_0 |nabla F| + {bf f}cdot nabla F +Dnabla^2F = -1
end{equation}
inside a 2D region between two disks both centered in the origin with radii $r=1$ and $l=5$ respectively. The boundary conditions are $F=0$ on the inner circle and $hat{n}cdotnabla F=0$ on the external one. Here is the code I have written:

v0 = 1.; D1 = 0.01; f = 0.7;
r2 := 1; l2 := 5; cell2 := 0.001;
[CapitalOmega]2 = 
  RegionDifference[Disk[{0, 0}, l2], Disk[{0, 0}, r2]];
pde2 = D1 Laplacian[FF[x, y], {x, y}] + f D[FF[x, y], x] - 
   v0 Sqrt[D[FF[x, y], x]^2 + D[FF[x, y], y]^2] ;
dcond2 = DirichletCondition[FF[x, y] == 0, x^2 + y^2 == r2^2];

Fsol = NDSolveValue[{pde2 == -1 + NeumannValue[0., x^2 + y^2 == l2^2],
     dcond2}, {FF[x, y]}, {x, y} [Element] [CapitalOmega]2, 
   Method -> {"PDEDiscretization" -> {"FiniteElement", 
       "MeshOptions" -> {"MaxCellMeasure" -> cell2}}}];

enter image description here
However, as you can see I am getting some error messages but I don’t know why nor how to deal with them.
I actually think this is coming from the sqrt in the $|nabla F|$ term, but I can’t get rid of that. Hope it is clear enough and thanks in advance for your help!

mathematical physics – Unitary representations of the universal cover of the Poincaré group in terms of solutions to wave equations

In high-energy Physics we study the unitary irreducible representations of the universal cover of the Poincaré group to classify the possible Hilbert spaces of quantum states of relativistic particles. These representations are classified first by a number $min (0,+infty)$ that in Physics we identify with the mass of the particle. Then if $m > 0$ the representation is further classified by one unitary irreducible representation of ${rm SU}(2)$ labelled by some spin $sinfrac{1}{2}mathbb{Z}_+$.

In the standard presentation, detailed e.g. in Weinberg’s The Quantum Theory of Fields, chapter 2, these representations are described in terms of certain functions defined the space of the possible momenta. More precisely, let $$H_m^+={p in mathbb{R}^{1,3} | p^2=-m^2, p^0>0}tag{1}$$
then the representation labelled by $(m,s)in (0,+infty)times frac{1}{2}mathbb{Z}_+$ is defined on the space $L^2(H_m^+,dOmega)$ where the measure $dOmega(p)=frac{1}{2p^0}d^3p$.

My question here is: can these unitary representations also be described in terms of solutions to classical wave equations in Minkowski spacetime $mathbb{R}^{1,3}$?

For example. Take the Klein-Gordon equation $(Box -m^2)phi=0$. Suppose we study the space of solutions to this equation. By Fourier transforming the equation we find the solutions $e^{pm ipcdot x}$ where $pin H_m^+$. We can therefore write a general solution in terms of a Fourier expansion in these plane waves. Moreover, there is a natural inner product on the space of solutions, the Klein-Gordon inner product. So it would seem to be natural to ask if we can realize the representation $(m,0)$ of the Poincaré group in the space of such solutions to the Klein-Gordon equation.

More generally, the possible classical fields obeying wave equations can be classified as well. In that case we look to the $(A,B)$ representations of ${rm SL}(2,mathbb{C})$ where $A,Bin frac{1}{2}mathbb{Z}_+$ label ${rm SU}(2)$ representations. The scalar field $phi(x)$ obeying the Klein-Gordon equation would just be the $(0,0)$ case.

The question then would be: can the space of classical fields transforming in some $(A,B)$ representation of ${rm SL}(2,mathbb{C})$ carry one of the unitary irreducible representations of the Poincaré group constructed by Wigner’s method and described above? In other words: can the unitary irreducible representations of the Poincaré group be described in terms of functions on spacetime instead of functions in momentum space? And if affirmative what would be the mapping between the two descriptions?