Our first live interactive session! The solution of the diffusion equation has wider uses, and for that reason, the subject will be of interest to people from very different backgrounds. It is hard to cover every possible use/approach to the problem in a video of this length, so we will be at hand to dispel any lingering doubts, and hopefully help connect the dots and our great community!
Thank you very much, I was able to grasp the concept more easily with your visual explanations, after looking for hours for an explanation, this is by far the best for me.
At 12:21. Is it really ok to put lambda = 1/sqrt(t) because the derivation of alternative function i.e. lambda * f( lambda * x, lambda^2 * t ) which satisfies the diffusion equation assumed that lambda is independent of x and t. I am saying this because we calculated d(u)/dx and d(v)/t assuming this
I didn't really get the step in 15:28 where you equated the two differentials of the tilda function. I suppose this is because the tilda function uses the parabolic transformation, but previously we supposed the λ to be constant. Now thow λ is dependent in time.
In my humble opinion as a practitioner, this playlist (full 7 videos) includes some the most important tools for IR and FX derivatives pricing. Btw, in terms of modelling playlists (SVol and IR), SABR and Cheyette could also be useful options to be added. Brilliant piece of QF work for one more time!
At 16:58 when you integrate dln(\bar{f}(z)) why do you integrate from 0 to z? Could you have integrated from any constant not equal to z to z? Your videos on diffusion are excellent. Thank you
Great question! You know that its differential equation is invariant under change of sign: z, -z. Then we know its total sum (integral) will be finite (total number of particles here or concentration), so it must go to zero at both ends: -\infty, +\infty. Hence easy to deduce that this must be an even function - symmetric around zero. So that's why we can integrate from zero to z, will work for both positive and negative values. Hope this helps!
Really great video again! Amazing work! I was following everything up until the last part (the integral representation). How would we go about solving the final integral to plot, for example, the distribution of particles for all x at some t given some initial condition psi(x)? Do we need to worry about z being dependent on x and t while we integrate (like if z was any variable we could take t and x out of the integral, but now z depends on t and x)? I know integral e^(-x^2) is the erf(x), but my calculus skills are quite rusty after many years of not using it a lot so not sure how to obtain the final answer. Thanks a lot for the great videos and sorry if my question is rather ignorant :)!
Many thanks for watching the video and providng feedback, and asking great question! Say we want to know the number of particles (or amount of heat!) at an arbitrary location x at some time t. We started the process at time zero with the initial distribution across the space, and then the particles/heat diffused across the space over time t according to the dynamics prescribed by the PDE. So the no. of particles that we will find at location x at a later time t could have originated from any location (which we now represent by z instead of x, so that we don't confuse the starting point, and the point of interest which is x). And the likelihood of particles moving from z to x depends on the distance between the points, the speed with which the particles are diffusing (co-efficient D), and the elapsed time. If the time interval (0 to t) is small, then the particles at x will have mostly come from the neighbouring location (z near x), and if the time interval is large, then they would have come from everywhere. So in summary we summed (aka integrate) the migration across the originating space. Hope this helps!
@@quantpie Thanks a lot for the great reply, this certainly helps to understand the physical meaning of the final equation :)! Maybe my question is more related to calculus. I always get afraid when I see an integral in the final solution (I would rather have just a function that you can easily compute, which would mean, I guess, that we need to solve the integral analytically). How would the final solution look like after solving the integral? The reason I am asking is that we wrote a computer code to compute the advection-diffusion equation (for multi-phase flow in porous media with chemical reactions) and want to compare it to the analytical solution (at least for single-phase flow without reactions, which you describe in your other video "Solution of Diffusion-Convection Equation"). Would we have to choose a specific initial condition and boundary condition for which solving the integral becomes easy? Like the point source that you describe in the other video? Thank you very much for your time :)!
I must say great video, but the last minute was so rushed, I didnt really get how you went from ~f to f, where the integral came from and some other questions.
Extremely underrated math channel!
Thanks for the kind words @The Toadman!
Our first live interactive session! The solution of the diffusion equation has wider uses, and for that reason, the subject will be of interest to people from very different backgrounds. It is hard to cover every possible use/approach to the problem in a video of this length, so we will be at hand to dispel any lingering doubts, and hopefully help connect the dots and our great community!
Thank you very much, I was able to grasp the concept more easily with your visual explanations, after looking for hours for an explanation, this is by far the best for me.
Thank you very much for distilling the complex concept in a way accessible to every one. Truly exceptional!
Thanks @Viswanadha Reddy for the kind words!
At 12:21. Is it really ok to put lambda = 1/sqrt(t) because the derivation of alternative function i.e. lambda * f( lambda * x, lambda^2 * t ) which satisfies the diffusion equation assumed that lambda is independent of x and t. I am saying this because we calculated d(u)/dx and d(v)/t assuming this
I didn't really get the step in 15:28 where you equated the two differentials of the tilda function. I suppose this is because the tilda function uses the parabolic transformation, but previously we supposed the λ to be constant. Now thow λ is dependent in time.
Hello, and many thanks for the question! Are they not equal by definition? We are just transforming the variables.
In my humble opinion as a practitioner, this playlist (full 7 videos) includes some the most important tools for IR and FX derivatives pricing. Btw, in terms of modelling playlists (SVol and IR), SABR and Cheyette could also be useful options to be added. Brilliant piece of QF work for one more time!
Thanks for that! SABR is on the list, and we need to do bit more on interest rates as well!
Incredibly helpful! Thank you!
Thanks @Kavin Kaufman for the kind words!
Thank you very much you saved my life
Very nice presentation
can you show the same for stokes problem in fluid dynamics
At 16:58 when you integrate dln(\bar{f}(z)) why do you integrate from 0 to z? Could you have integrated from any constant not equal to z to z? Your videos on diffusion are excellent. Thank you
Great question! You know that its differential equation is invariant under change of sign: z, -z. Then we know its total sum (integral) will be finite (total number of particles here or concentration), so it must go to zero at both ends: -\infty, +\infty. Hence easy to deduce that this must be an even function - symmetric around zero. So that's why we can integrate from zero to z, will work for both positive and negative values. Hope this helps!
A bit different take on solving the diffusion equation, but quite nice and clear explanation. Thank you!
Glad you liked it! thank you!
I want to ask is the (t) means summation of all time of the particle or just delta (t=t0_t1) for example.
Hello and many thanks for the question! It means the number of particles at time t - we start at zero so probably your delta means the same thing.
Really great video again! Amazing work! I was following everything up until the last part (the integral representation). How would we go about solving the final integral to plot, for example, the distribution of particles for all x at some t given some initial condition psi(x)? Do we need to worry about z being dependent on x and t while we integrate (like if z was any variable we could take t and x out of the integral, but now z depends on t and x)? I know integral e^(-x^2) is the erf(x), but my calculus skills are quite rusty after many years of not using it a lot so not sure how to obtain the final answer. Thanks a lot for the great videos and sorry if my question is rather ignorant :)!
Many thanks for watching the video and providng feedback, and asking great question! Say we want to know the number of particles (or amount of heat!) at an arbitrary location x at some time t. We started the process at time zero with the initial distribution across the space, and then the particles/heat diffused across the space over time t according to the dynamics prescribed by the PDE. So the no. of particles that we will find at location x at a later time t could have originated from any location (which we now represent by z instead of x, so that we don't confuse the starting point, and the point of interest which is x). And the likelihood of particles moving from z to x depends on the distance between the points, the speed with which the particles are diffusing (co-efficient D), and the elapsed time. If the time interval (0 to t) is small, then the particles at x will have mostly come from the neighbouring location (z near x), and if the time interval is large, then they would have come from everywhere. So in summary we summed (aka integrate) the migration across the originating space. Hope this helps!
@@quantpie Thanks a lot for the great reply, this certainly helps to understand the physical meaning of the final equation :)! Maybe my question is more related to calculus. I always get afraid when I see an integral in the final solution (I would rather have just a function that you can easily compute, which would mean, I guess, that we need to solve the integral analytically). How would the final solution look like after solving the integral? The reason I am asking is that we wrote a computer code to compute the advection-diffusion equation (for multi-phase flow in porous media with chemical reactions) and want to compare it to the analytical solution (at least for single-phase flow without reactions, which you describe in your other video "Solution of Diffusion-Convection Equation"). Would we have to choose a specific initial condition and boundary condition for which solving the integral becomes easy? Like the point source that you describe in the other video? Thank you very much for your time :)!
ah ok thanks! something like this: th-cam.com/video/irudcwJyo3o/w-d-xo.html
Thank you, I can get how to derive the diffusion equation with this method
Excellent! thanks!
I must say great video, but the last minute was so rushed, I didnt really get how you went from ~f to f, where the integral came from and some other questions.