How to Solve Differential Equations in PYTHON 

Interesting, thank you AJ! I will take a look at these methods, may be interesting for a future video! And glad you're enjoying the channel :)

​@@MrPSolver (note: I already replied to you, but it didn't show ughhhh) Nice! I'm happy to hear that

Yes, you're right, but it's actually more convenient to do than slicing the 2d NumPy array. When it is transposed, index just by 0, 1, 2, etc, allows you to immediately access the entire time series without needing to slice the array. I think this is why in the newer solve_ivp function, the y values are outputted in rows instead of columns like in the older, classic odeint function.

Hi classmate

Let's say `dfdx` is an expression of some equation. Plug that expression into the script below: latex_script = sp.latex(sp.S(dfdx, evaluate=False)) print(f'{latex_script}') It printed this without the `$$! I put those symbols in for you to use the code as is inside a Jupyter Markdown cell. Just copy and paste to see what the differential of f(x) of some equation looks like. Perhaps you might then want to find the anti-derivative and then stick that expression of f(x) into sympy.latex(), to see what a monster the function really is: $$ - 2 a x e^{- a \sin{\left(x^{2} ight)}} \log{\left(\frac{c \sin^{2}{\left(x ight)}}{x} ight)} \sin{\left(b^{x} ight)} \cos{\left(x^{2} ight)} + b^{x} e^{- a \sin{\left(x^{2} ight)}} \log{\left(b ight)} \log{\left(\frac{c \sin^{2}{\left(x ight)}}{x} ight)} \cos{\left(b^{x} ight)} + \frac{x \left(\frac{2 c \sin{\left(x ight)} \cos{\left(x ight)}}{x} - \frac{c \sin^{2}{\left(x ight)}}{x^{2}} ight) e^{- a \sin{\left(x^{2} ight)}} \sin{\left(b^{x} ight)}}{c \sin^{2}{\left(x ight)}} $$ This is of course LaTex script of the expression `dfdx`, generated by the sympy.latex() method. I found it to be a very elegant way of writing LaTex inside Jupyter Notebook's cells set to 'Markdown', instead of `Code`, which is the default setting. sympy.latex() has thus far provided me with perfectly accurate LaTex Script for any mathematical expression. Cool eh? Hope that will give you a kick start with LaTex.

Yes, you are correct here! Should have dSdt(t, S) and not dSdx(x,S). The second one *technically* still gives the right solution in this case, but it is sloppy as it reuses the variable x and only confuses what's going on

@@MrPSolver You have a lot of very impressive videos on your channel! Nice!

Is Julia ripe enough for this? Does it have packages like scipy, sympy or IDE like jupyterlab?

@@aliexpress.official hi! Those are great questions! Julia actually has the DifferentialEquations.jl package for solving differential equations diffeq.sciml.ai/stable/ . It is much more extensive than SciPy with respect to ODEs because it has DOZENS more solvers for ODEs alone. You can also solve other kinds of Differential Equations with it such as Differential Algebraic Equations, Boundary-Value Problem ODEs, etc. There is also the ModelingToolkit.jl package that allows you to symbolically compose and model ODEs (and also Nonlinear System of Equations, PDEs, etc) then pass them on to DifferentialEquations.jl which will solve them numerically: mtk.sciml.ai/stable/ Julia has Symbolics.jl for its symbolic computing symbolics.juliasymbolics.org/stable/ . However, it's purpose is different than SymPy in that it's designed for symbolic-numeric computing symbolics.juliasymbolics.org/stable/comparison/ These packages I mentioned are part of the SciML packages that get updated very regularly like almost everyday.

@@AJ-et3vf thank you. I definitely need to give julia a shot

@@aliexpress.official VS Code and IJulia extension for JupyterLab/Notebook are your IDE options for Julia.

​@@aliexpress.official fun fact the "ju" jupyter stands for julia. JUlia PYThon and R

Machine learning for tumour segmentation in medical physics (specifically PET and CT scans).

@@MrPSolver just brilliant brother👏👏👏👌👌 solve cancer for us 😃😃😃😃😃😃😃

Watch tutorials in RU-vid, read the documentations for those packages, and get any math book preferably numerical methods book and solve the problems you find there.

@@AJ-et3vf Do you have book-suggestions?

Hi! That is actually a boundary-value problem ODE that you're saying, specifically one with a Neumann boundary condition at the right endpoint because the derivative is specified at the right instead of the function value. For such ODEs, the common methods for them are the shooting method and the finite difference method. In the shooting method, you make guesses for the initial slope and try to "shoot" for the boundary value, essentially turning the bvp ODE into an ivp ODE. This shooting for the boundary value can be treated as a root-finding problem and you can use the SciPy root_scalar function for this, using either bracketing methods or newton-type methods, like secant method. You can plot the shooting function vs your initial guesses so you can know what a good initial guess for the slope should be. Once you got the right initial slope, you now solve the ODE like any other. The finite difference method ( ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Upu3t8Ac2i8.html ) involves discretizing the 1st and 2nd derivatives using centered finite differences. You end up with a tridiagonal system if linear equations to solve. Note that this is doable only if the bvp ODE is linear. If it is nonlinear, you will have a system of nonlinear equations so you won't get the special tridiagonal system of linear equations from earlier. You can use the Gauss-Seidel method to iteratively solve the function values. You'll have to use Numba to speed up the for loop iterations An interesting method for solving bvp ODEs is the collocation method, applicable to both linear and nonlinear bvp ODEs: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-u8dVrzxTvSA.html . You approximate the solution with a polynomial at least quadratic or cubic order and solve for the coefficients by applying the boundary conditions and solving the ODE at "collocation points". Lastly, SciPy actually has a bvp ode solver named solve_bvp docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_bvp.html . It solves most bvp ODEs fine, but for bvp ODEs where either left or right boundary is actually a singularity due to either 1/x or 1/(x-b) being in the ODE, solve_bvp can end up failing because of the nature of the problem. The shooting method will also be very sensitive too for this, so this is where the finite difference method shines because it is more stable than the shooting method despite being more computationally expensive.

@@AJ-et3vf I know methods you cited but never tried to implement them. I'll check all the links you posted. It's very useful your reply. Thanks 🙏

@@nicolameoli I am very happy to be of great help! Here are more links and resources to help you man: Shooting Method ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-PDxYtycAUEQ.html ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-3hf2v39HJQE.html ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-0hwQEfAqsqM.html Orthogonal Collocation Method ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-jK2UtrrDO6U.html Finite Difference Method ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-qrS1L1VfP-k.html ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-OJ4SQuua-f0.html ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Dxi0gc8-f8c.html Gauss-Seidel Finite Difference Method (In here, Mr P Solver solves Laplace' equation which is an elliptic PDE using numba-accelerated for loops. This is what I'm referring to when I mentioned the gauss-seidel method for the finite difference method solution of ODE bvps. ODE bvps after all are elliptic PDEs with only one independent variable so you have one less for-loop here. The "mesh" that you're solving is only one dimensional, not two like in elliptic PDEs.) ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-dKCAVteveYc.html

Come join the discord server and ask there!

Thanks buddy!