## 2 Badges

4 years, 245 days

## Numerical solutions of PDE...

Maple

Dear experts

I am interested in solving the following differential equation numerically

```diff(eta(k,t),t,t) + gamma*diff(eta(k,t),t) + omega^2*eta(k,t) + (3/4)*k*omega*eta(k,t)*eta(k,t) + (3/2)*k*eta(k,t)*(omega*eta(k,t) + (3/4)*k*eta(k,t)*eta(k,t)) + (omega*gamma*eta(k,t) + (3/2)*k*eta(k,t)*(gamma*eta(k,t)))*(diff(eta(k,t),t) + gamma*eta(k,t))/(omega*eta(k,t) + (3/4)*k*eta(k,t)*eta(k,t)) = 0;
```

where eta(x,t) is the Fourier transform of zeta(x,t) and plot of zeta(x,t) is the goal. I think FFT method can help but I failed in running and applying this method on the equation above.
Initial conditions are

```f= 0.28
C = 1/L*0.2/f*tanh(L/f)
L=20
amp = 0.5e-2
zeta0 = amp/cosh(f*x)/cosh(f*x)-C

and u0 as initial velocity can be 0 or an arbitrary function such as
u0 =2*amp/cosh(f*x)/cosh(f*x)*f*tanh(f*x)```

I would appreciate it if one could help me.

## numerical output and plot integral...

Maple

Dear experts

how can I numerically plot the following integral and have output as csv file.

in this relation, there is a list of omega1 and omega2 for each k1 and k2. for example,

k1 = [1,2,3,4,5]

omega11=[1,2,3,4,5], omega12= [1,2,3,4,5], omgega21= [1,2,3,4,5],omega22= [1,2,3,4,5],

all other coefficients would be calculated based on k values and corresponding omegas

thanks in advance

## solve non linear DE numerically...

Maple

hi Maplers
I want to solve the following differential equation numerically. but i get error

```restart;
eqn := (diff(H(K),K))^3+4*K^4*H(K)^4*diff(H(K),K)+8*K^4*H(K)^5=0;
ics := H(0)=1/2;
sol := dsolve({eqn,ics},numeric);```
```Error, (in DEtools/convertsys) unable to convert to an explicit first-order system
```

## solve ODE with boundary conditions...

Maple

dear all
I would like to solve the following differential equation in Maple Numerically.

```restart;
pde := I*10*diff(u(k, t), t) + (diff(u(k, t), t, t)) -16* u(k, t) = 0;
ics := u(k, 0) = sqrt(Pi/2)*csch(Pi/2*k), (D[2](u))(k, 0) = -(1/4)*sqrt(2)*Pi^(3/2)*csch((1/2)*Pi*k)*coth((1/2)*Pi*k)
bcs := u(0, t) = 0, u(100, t) = 0;
sol := pdsolve(pde, {ics, bcs}, numeric, range = 0 .. 1, output = listprocedure);
sol;
```

here the sech(x)^2 in real space and corresponding fourier transformaation is the initial condition

the Gaussian function as initial condition is also an alternative.

How can I solve it?

## Solve ODE in fourier space numerically a...

Maple 2018

Dear all,

consider two lists of complex values :

list1 := [l1,l2,l3,l4,l5]

list2 := [s1,s2,s3,s4,s5].

There is a set of second order differential equation

d^2u(k)/dt^2+I*A*du/dt-B*u=0

where A is sum of elements of list1 and list2 and B is multiplication of their element. Therefore,

d^2u[1](k)/dt^2+I*(l1+s1)*du[1]/dt-(l1*s1)*u[1]=0

d^2u[2](k)/dt^2+I*(l2+s2)*du[2]/dt-(l2*s2)*u[2]=0

d^2u[3](k)/dt^2+I*(l3+s3)*du[3]/dt-(l3*s3)*u[3]=0

d^2u[4](k)/dt^2+I*(l4+s4)*du[4]/dt-(l4*s4)*u[4]=0

d^2u[5](k)/dt^2+I*(l5+s5)*du[5]/dt-(l5*s5)*u[5]=0

How can I create a set of differential equations and initial conditions based on nops(list1), then solve this system of differential equations numerically in Maple.

since u[i] are function of k, next step is to transforme them to real space by inverse fourier transform.

finally save the results and plot them.

Note that for simplisity I wrote a linear equation but it is not. so, because of nonlinear terms it is not possible to use superposition of the solution. I have to take them as coupled system of equations.

====

for example

```list1 := [ [0., -5.496799068*10^(-15)-0.*I], [.1, 5.201897725*10^(-16)-1.188994754*I], [.2, 6.924043163*10^(-17)-4.747763855*I], [.3, 2.297497722*10^(-17)-10.66272177*I], [.4, 1.159126178*10^(-17)-18.96299588*I] ]

list2 :=[ [0., -8.634351786*10^(-7)-67.81404036*I], [.1, -0.7387644021e-5-67.76491234*I], [.2, -0.1433025271e-4-67.59922295*I], [.3, -0.2231598645e-4-67.25152449*I], [.4, -0.3280855430e-4-66.56357035*I] ]```

where first element is k and the second value is l_i and s_i

the differential equation is

`ode_u[i]:= diff(u[i](t),t\$2)+I*(list1[i][2]+list2[i][2])*diff(u[i](t),t)-list1[1][2]*list2[2][2]*u[i](t)=0;`

eta is in fourier space where k values are in list1[i][1].

We laso know that f(-k)= - f*(k) where f=list[i][2]

and u[i] as function of k, initially has a Gaussian shape at t=0 in fourier space..

Thanks in advance for your help

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