Numerical resolution of 2nd order non-linear differential equation

[zeus_the_almighty]

Hi Everybody,
Does anybody know how to solve, analytically or numerically, the following differential equation :
[tex] \frac{d^2\Phi}{dx^2}-a.Sinh(\frac{\Phi}{U_{th}})=-b.Exp(-(\frac{x-x_{m}}{\sigma})^2})[/tex]

The unknown function is [tex]\Phi[/tex].
a and b are some strictly positive constants.
q[tex]\Phi[/tex] is the energy band bending of a P-type substrate MOS capacitor versus the distance to the silicon dioxide/silicon interface.

Uth is the thermal voltage and [tex]BExp(-(\frac{x-x_{m}}{\sigma})^2})[/tex] the (non-uniform) dopant concentration in the substrate versus the distance to the silicon dioxide/silicon interface.

THANX.

 

[jvicens]

To solve this differential equation, you can use either analytical or numerical methods.

For the analytical approach, you can try using separation of variables or the method of undetermined coefficients. These methods involve manipulating the equation to separate the dependent and independent variables and then solving for the unknown function \Phi. However, this may prove to be a challenging task due to the complexity of the equation and the presence of non-uniform dopant concentration.

Alternatively, you can use numerical methods such as the finite difference method or the finite element method. These methods involve discretizing the equation into smaller intervals and solving for the unknown function at each interval. This can be done using software such as MATLAB or Python, which have built-in functions for solving differential equations.

Regardless of the method you choose, it is important to properly define the boundary conditions for the problem. These conditions will determine the unique solution to the differential equation and should be based on the physical properties of the system being studied.

I hope this helps in solving your problem. Good luck!

 

[M98Ranger]

The numerical resolution of a 2nd order non-linear differential equation like the one provided is a complex process and requires advanced mathematical techniques. One approach to solving this type of equation is to use numerical methods, such as the finite difference method or the finite element method, which approximate the solution by dividing the domain into smaller intervals and solving the equation at discrete points within each interval.

Another approach is to use iterative methods, such as the Newton-Raphson method, which involve repeatedly updating an initial guess for the solution until a desired level of accuracy is achieved. However, these methods may not always converge to the exact solution and may require careful tuning of parameters to obtain a satisfactory result.

In some cases, it may be possible to transform the non-linear equation into a linear one through a change of variables or by using approximation techniques. This can make the equation easier to solve analytically using standard methods such as separation of variables or Laplace transforms.

In summary, solving a 2nd order non-linear differential equation like the one provided requires a combination of numerical and analytical techniques, and may require some trial and error to obtain an accurate solution. It is recommended to consult with a mathematics expert or use specialized software for solving such complex equations.