Introducing Weird Differential Equations: Delay, Fractional, Integro, Stochastic!
Summary
TLDRThis educational video introduces viewers to four unusual types of differential equations: delay, integral, stochastic, and fractional. Delay differential equations (DDEs) are notable for their dependence on past values of the function. Integral differential equations combine derivatives and integrals of the function. Stochastic differential equations incorporate random fluctuations, while fractional differential equations involve non-integer order derivatives. The video briefly explains these concepts and mentions methods for solving them, such as the method of steps for DDEs and Laplace transforms for integral equations, without going into complex mathematical details.
Takeaways
- 📚 The lesson introduces four types of unusual differential equations not typically covered in early undergraduate courses.
- 🕒 Delay Differential Equations (DDEs) depend on the value of the dependent variable at a previous time, not just the present.
- 🔍 There are three types of DDEs: continuous delay, fixed and discrete delay, and scaled delay, with the latter including the pantograph equation as a special case.
- 📉 Solving DDEs requires specifying an initial function over a time interval, rather than a single initial condition, due to their dependence on past values.
- 👣 The method of steps is a special technique used to solve DDEs with discrete delays by iteratively solving over intervals.
- 🧩 Integral Differential Equations include both derivatives and integrals of the function of interest, and can be solved using integral transforms for simple cases.
- 🔄 Stochastic Differential Equations incorporate random fluctuations or 'white noise' terms, requiring techniques from stochastic calculus and numerical methods for solutions.
- 🎲 Fractional Differential Equations involve fractional derivatives, which are derived from fractional calculus allowing for non-integer orders of differentiation.
- 🎯 Fractional calculus extends the concept of differentiation to allow for taking derivatives of real or even complex numbers, not just integers.
- 👨🏫 The video is educational, aiming to introduce and explain these advanced types of differential equations to a presumably upper-level audience.
- 🌟 The video concludes with acknowledgments to patrons and an invitation for viewers to like and subscribe for more content.
Q & A
What are the four types of 'weird' differential equations mentioned in the lesson?
-The four types of 'weird' differential equations discussed are delay differential equations (DDEs), integral differential equations, fractional differential equations, and stochastic differential equations.
What is a delay differential equation (DDE)?
-A delay differential equation (DDE) is a differential equation that depends on the value of the dependent variable x at a previous time, denoted as x(t-tau), where tau represents the time delay.
Can you explain the three types of delay differential equations?
-The three types of DDEs are: 1) Continuous delay DDE, which depends on the sum of all possible past values of x. 2) Fixed and discrete delay DDE, where the delay is a fixed amount of time. 3) Scale delay DDE, which depends on a previous value of x corresponding to lambda times t, where lambda is a constant between 0 and 1.
What is the method of steps used for solving delay differential equations?
-The method of steps involves specifying an initial function from negative tau to 0, and then iteratively substituting the function into the DDE to solve for x(t) over successive intervals of tau units of time.
How does an integral differential equation differ from a regular differential equation?
-An integral differential equation contains both derivatives and integrals of the function of interest, unlike regular differential equations which only involve derivatives.
What is a stochastic differential equation?
-A stochastic differential equation is a differential equation that includes a random fluctuating or stochastic process, often represented by a term involving sigma(x, t) times epsilon(t), where epsilon is a white noise term.
What is fractional calculus and how does it relate to fractional differential equations?
-Fractional calculus is a field of study that allows for the taking of non-integer order derivatives, such as half or quarter derivatives. Fractional differential equations involve these fractional derivatives of an unknown function.
Why is it necessary to specify an initial function for a delay differential equation with a discrete delay?
-Specifying an initial function is necessary because to determine the value of dx/dt at a future time, one needs the value of x at previous times, which is not provided by just an initial condition at t=0.
How can integral transforms be used to solve integral differential equations?
-Integral transforms, such as the Laplace transform, can be used to convert integral differential equations into a form that is easier to solve analytically, especially for simple linear cases.
What techniques are used to solve stochastic differential equations?
-Solving stochastic differential equations typically involves using techniques from stochastic calculus in combination with numerical methods.
Outlines
📚 Introduction to Unusual Differential Equations
This paragraph introduces the topic of the video, which is about four unusual types of differential equations that are typically encountered in research and advanced applied mathematics courses. The instructor explains that these equations are not commonly studied in early undergraduate courses. The four types discussed are delay differential equations (DDEs), integral differential equations, stochastic differential equations, and fractional differential equations. The explanation begins with delay differential equations, emphasizing how they differ from standard differential equations by depending on the value of the dependent variable at a previous time, denoted as 'tau' seconds before the present time 't'. The paragraph also outlines the three types of DDEs: continuous delay, fixed and discrete delay, and scaled delay, with an example of the pantograph equation provided for the scaled delay type.
🔍 Exploring Advanced Types of Differential Equations
The second paragraph delves into the specifics of solving delay differential equations, emphasizing the necessity of an initial function rather than a single initial condition due to the nature of delays in the equations. The method of steps is introduced as a technique for solving DDEs with discrete delays. The paragraph then transitions to integral differential equations, which involve both derivatives and integrals of the function of interest. It mentions the use of integral transforms like the Laplace transform for solving simpler cases. The discussion moves on to stochastic differential equations, which incorporate a random process represented by white noise. Solving these requires stochastic calculus and numerical methods. Lastly, the paragraph touches on fractional differential equations, providing a brief introduction to fractional calculus, which allows for the calculation of non-integer order derivatives. The video concludes with acknowledgments and a sign-off from the instructor.
Mindmap
Keywords
💡Differential Equations
💡Delay Differential Equations (DDEs)
💡Integral Differential Equations
💡Fractional Differential Equations
💡Stochastic Differential Equations
💡Dynamical Systems
💡Initial Function
💡Method of Steps
💡Fractional Calculus
💡White Noise
Highlights
Introduction to four unusual types of differential equations: delay, integral, fractional, and stochastic differential equations.
Delay differential equations (DDEs) depend on the value of the dependent variable at a previous time, not just the present.
Three types of DDEs: continuous delay, fixed and discrete delay, and scaled delay.
The method of steps is used to solve DDEs with discrete delays by specifying an initial function.
Integral differential equations involve both derivatives and integrals of the function of interest.
Integral transforms like the Laplace transform can be used to solve simple integral differential equations.
Stochastic differential equations incorporate random fluctuations or stochastic processes.
Solving stochastic differential equations requires techniques from stochastic calculus and numerical methods.
Fractional differential equations involve fractional derivatives, which are not limited to rational orders.
Fractional calculus allows for the computation of derivatives that are not whole numbers, such as half or quarter derivatives.
The video aims to introduce the concepts of these unusual differential equations without going into detailed solutions.
The importance of specifying an initial function for DDEs due to their dependence on past values.
The Pantograph equation as a special type of scaled delay differential equation.
Higher order integral and partial integral differential equations are also possible beyond first order.
The inclusion of white noise terms in stochastic differential equations to represent randomness.
The use of the gamma function in fractional calculus to compute fractional derivatives.
An acknowledgment of patrons supporting the educational content creation.
Transcripts
greetings students and welcome to my
lesson on weird types of differential
equations
you don't usually study these types of
differential equations in early or even
intermediate undergraduate courses
they tend to come up in areas of
research and upper level courses on
applied math
now by weird differential equations i'm
really going to talk about
four weird types of differential
equations delay differential equations
integral differential equations
fractional differential equations and
stochastic differential equations
we'll start by discussing delay
differential equations
when you look at dynamical systems
consisting of a single differential
equation
you conventionally think of an equation
like dx by dt
equals f of x comma t the differential
equation depends on the value of t
and the value of x at the present time
at the current time
however what if in addition to my
differential equation depending on
x at the present time it also depended
on the value of x
at a previous time let's say tau seconds
before the present time t
in that case we could write our
differential equation as dx by dt
equals a function f of x
at the time t x at the time t minus tau
and t where x of t minus tau represents
the value of x that occurred tau
seconds earlier this differential
equation which depends on the value of
the dependent variable x
occurring earlier is called a delay
differential equation
or dde a delay differential equation dx
by dt
is a differential equation that depends
on the value of x
sometime in the past now what i've
written up here and called the delay
differential equation
isn't the most general way to describe a
delay differential equation
in fact there's actually three types of
ddes the first type is a dde with
continuous delay
this dde depends on x t and the sum of
all possible past values of x and if you
want to sum so many values then in
general you can write that as an
integral
the integral from negative infinity to
zero of x of t
plus tau times d mu by d tau with
respect to tau
where mu is some function of the delay
tau
the second type of dde is the type we
wrote above where the delay is
fixed and discrete of course you're not
restricted to a single delay you can
have multiple delays in your system
tau 1 tau 2 all the way to tau n
the third type is one in which you have
a scale delay now what do i mean by this
basically your dde depends on x of t
the value of x at time t as well as a
previous
value of x corresponding to lambda times
t where lambda is a constant between 0
and 1.
a special type of this equation is
called the pantograph equation given by
dx by dt equals a times x of t
plus b times x of lambda times t where a
and b are some real constants now how do
you solve
a delay differential equation i won't go
through a complete example here but i'll
briefly illustrate what the process
looks like
say i have a dde with a discrete delay
tau like the simple dde i described
earlier on in the video
i'll call this equation 1. to start
solving this dde
i need to first specify an initial
function from negative tau to 0
as opposed to an initial condition at t
equals 0 for instance
and why do i need to specify this
initial function let me explain this on
the side
say i had to solve this dde and i only
had the value of x
at time 0 as my initial condition
suppose also that my delay tau was
5. then in order to determine the value
of dx by dt and by extension
x of t at the time 1 i would need the
value of x
at time negative four according to this
dde
however i only have the value of x at
zero i don't have the value
at negative four so specifying x at zero
is simply
not enough to solve this dde i need to
specify what the entire function looks
like from negative 5 to 0
which is why i need an initial function
as my initial condition
instead of an initial point so given
this dde
an initial function what we can do is
substitute phi of t minus tau for
x of t minus tau into equation one with
t minus tau varying from negative tau to
zero
then with this function plugged in we
can solve for x of t
from t equals zero to t equals tau and
once we obtain
x of t from zero to tau we can then plug
that in as x
of t minus tau and then solve for x of t
from tau to two tau
then we can keep repeating these steps
plugging in our previous
x in place of the x of t minus tau and
using that to solve the differential
equation and obtain a full value of x
that's tau
units of time in the future this full
method by the way is given a special
name it's called
the method of steps another type of
weird differential equation we'll go
over is the integral differential
equation
as the name implies this type of
differential equation contains both the
derivatives of the function of interest
and the integrals of the function of
interest
for instance if i have a first order
differential equation in terms of dx by
dt
an integral differential equation could
look something like this where dx by dt
is f of x comma t plus an integral term
from t
naught to t of another function g of s
comma
x d s except for really simple cases
it's not that easy to solve integral
differential equations by hand
but in cases when you can determine the
solution analytically like
simple linear differential equations you
can use integral transforms like the
laplace
transform to find the answer i should
also note that we don't necessarily need
to have a first order integral
differential equation
we can just as well have second or
higher order or even partial
integral differential equations the next
type of weird differential equation
we'll discuss
is the stochastic differential equation
consider our typical differential
equation
dx by dt equals some function of x and t
if i add a term sigma of x and t times
an epsilon of t
where epsilon represents a white noise
term then i get something called a
stochastic differential equation
which is a differential equation that
depends on some random fluctuating or
stochastic process
solving stochastic differential
equations involves using techniques
developed in stochastic calculus in
combination with numerical methods
we won't go over those here since our
goal with this video is just to
introduce these
weird types of differential equations
and the last type of differential
equation i'll discuss is a fractional
differential equation
as the name suggests this is a
differential equation which involves the
fractional derivatives of an unknown
function
to see what this means let me give a
very brief pep talk on fractional
calculus
now in typical calculus you have a
function f of t
and you can take its derivative df by dt
or even a higher order derivative
like the d2f by dt squared which is the
second derivative
but what if i didn't want to fully
commit to taking a derivative
for instance what if i wanted to take a
fraction of a derivative
well i can do that using fractional
calculus
fractional calculus allows me to take a
half derivative or a quarter derivative
even if i wanted to
in general using a combination of clever
manipulations in the gamma function i
can compute
fractional derivatives like the alpha
derivative of f with respect to t
where alpha is some real number between
0 and 1.
in fact it could also be a complex
number so even though it says
fractional calculus i'm not necessarily
restricted to derivatives of a rational
order
so a fractional differential equation is
then a differential equation which
involves these fractional derivatives
anyway that should do it for this brief
video i'd like to thank the following
patrons for supporting me at the five
dollar level or higher and if you
enjoyed the video feel free to like and
subscribe
this is the faculty of khan signing out
浏览更多相关视频
Solving Delay Differential Equations With Julia | David Widmann | JuliaCon 2019
Overview of Differential Equations
LECTURE NOTES: AIRCRAFT AERODYNAMICS I, CHAPTER I, PART 1
Maxwell Denklemleri Ne Anlama Gelir?
Separation of Variables // Differential Equations
The Key Definitions of Differential Equations: ODE, order, solution, initial condition, IVP
5.0 / 5 (0 votes)