Welcome to tutorial on heat equation. In this tutorial, we are going to solve a few problems associated to heat equation. The first problem is a problem in calculus which is integral of e power minus x square is root pi. We have used this information in our analysis of heat equation that is why I decided to do this proof. This involves a small trick I will explain that. So integral of e power minus x square on R is equal to 2 times integral on 0 to infinity of the same integral e power minus x square because it is an even function. Therefore, it is enough to compute 0 to infinity e power minus x square dx. Then the answer we are interested will be 2 times i. So to compute i, we look at i square. i square is i into i that is this quantity into the same quantity because I am writing the 2 times I have used x in one integral and y in the other integral. Anyway, this is so called dummy variable. So I can use any variable I want. Now what I do is that this quantity I take it inside. Actually, I am assuming that this is finite quantity and that is why everything is fine. So I take this inside and I get this. Now this is nothing but 0 to infinity 0 to infinity e power minus x square into e power minus y square is e power minus x square minus y square dx dy. And that is equal to in the polar coordinates R equal to square root of x square plus y square.  Now we have to find out the limits for the integration variables. Look at this integral. This integral is on 0 infinity cross 0 infinity that is the first quadrant. So the first quadrant the radius goes from 0 to infinity. But the part of the circle which is in the first quadrant corresponds to theta equal to 0 to theta equal to pi by 2. For example, these are the circles I am integrating on. So the angle is 0 to pi by 2 here. Pi by 2 is the angle here. So I square equal to this. This is what we have seen on the last slide. Now it is very easy to integrate. Put s equal to R square then inside integral with respect to R becomes with respect to s in this form. And this is easily integrable e power minus s integral is e power minus s by minus 1 limits 0 to infinity. So you get 1 from there. So total is half. So integral with respect to theta from 0 to pi by 2 of 1 by 2 that is pi by 4. So I square is pi by 4. Therefore I is root pi by 2. Therefore this integral which is 2 times I is root pi. This is a very simple trick that we use in calculus. To compute an integral of one variable we go through computation of integral in 2 variables. Luckily because of the form of the integrand we can convert that into polar coordinates and we could easily compute. Now let us solve the Cauchy problem for homogeneous heat equation. Of course we know a certain formula for the solution of the homogeneous heat equation and the corresponding Cauchy problem with a phi of x. We have an integral. But that integral is very integrable only in very special cases where we are very lucky namely you see pi of x we have taken e power 3x because the kernel the heat kernel is also exponential type. So therefore this exponential should helps us. That is why we are able to integrate. Otherwise if it is some arbitrary function of x which is continuous and bounded we do know that it gives rise to a solution. But then that is it. We cannot really compute. We have to be really lucky to be able to compute. So without worrying about uniqueness issues because we know that Cauchy problem solution is not unique. It is unique only when you are looking at special classes. Let us not worry about that and simply compute this integral and see what we get. So we will compute a solution. Maybe that is the solution that is the only solution in some class or maybe there are more solutions we do not know at because this may not belong to the class for which uniqueness theorem holds. In fact we have not discussed much about uniqueness theorems for which I have just given you the reference of book by Debenedetto on PDEs. You can look at your leisure time. But as far as this course is concerned we simply want to compute this integral with phi of y equal to e power 3y. Can we do it or not? So this formula we have derived in lecture 7.2. So now this is exponential into exponential therefore it is exponential of this plus this. So I have written down exactly that. So this is same as this. So I am only working in the power of this e exponential e to the power something. So that power only I am manipulating. Now I express that like this so called completion of squares. The advantage now is that I am going to set y minus x minus 60 by something equal to a some variable. So I am going to put y minus x minus 60 by root 40 as p. So then this term the first term by 40 that simply becomes p square e power minus p square and what remains is 12 x t by 40 36 t square by 40. So this is e power minus p square and what remains here is 12 x t by 40 is 3x 36 t square by 40 is 90 and dy 1 by root 40 is your dp from here. Dp is 1 by root 40 dy. So this is exactly same as this. Now if you see exponential of 3x plus 90 does not depend on p. So it just comes outside and what we have is 1 by root pi into integral over r of e power minus p square dp. We just found out this is root pi. So root pi by root pi that is 1. So what you have is exponential of 3x plus 90. So we are very lucky that we could actually compute the integral. Now let us move on to the problem 3. For t positive let r be defined by this rectangle 0 pi cross 0 t. Let u be a solution to this problem which is initial bountiful problem homogeneous heat equation boundary conditions are 0 initial condition is sin square x. Show that 0 is less than or equal to u of x t is less than or equal to e power minus t sin x holds for x t in the rectangle. Hint is to use maximum principle because maximum principle allows us to compare certain things. So therefore the maximum principle using that is a hint here. So what we have to show is actually two inequalities. We have to show 0 is less than or equal to u x t and u x t is less than or equal to e power minus t into sin x. So let us prove the first one that is u x t is greater than or equal to 0. It follows from minimum principle. By minimum principle we know that minimum of u on r closure is actually minimum of u on the parabolic boundary of r. What is the parabolic boundary of r? This is r so parabolic boundary is this union of these three lines and u is 0 here, u is 0 here and u is sin square x here that is what is given to us. So minimum of u on the parabolic boundary is 0 because of this because minimum of sin square x on this 0 to pi is 0. So minimum is 0. Therefore minimum of u on r closure is 0 which means u is greater than or equal to 0 on r closure in particular on r. So this completes the proof of the first inequality. Now let us look at the second inequality. So we want to prove this. See if you want to apply maximum principle you can only apply that to solutions of heat equation. So therefore we naturally ask this question whether this is a solution of heat equation. Yes we have already seen when we were solving heat equation using separation of variables method that these are the things which are coming as the terms in the series. So is this a solution? Yes it is a solution. So what we do now is we define new function v which is u minus e power minus t sin x. This satisfies vt minus vxx equal to 0 because u is already a solution to the homogenous heat equation. This also solves homogenous heat equation. Therefore this equation being linear and homogeneous sum of two solutions or the difference of two solutions is a solution still. So v is a solution to this and what is that we want to show v of xt is less than or equal to 0. So showing this is same as showing v is less than or equal to 0. So in fact we will look at the problem that v satisfies. So homogenous heat equation v is a solution that we have already checked on r and v of 0t and v of pi t are also 0. v of x0 equal to sin square x minus sin x. Please check this it is very simple. So I am not going to check here. So this is the problem that v satisfies. Now by maximum principle applied to this v, maximum of v on our closure is nothing but maximum of v on the parabolic boundary. So therefore we will look at what is the maximum of v on the parabolic boundary. Parabolic boundary consists of three lines. One is the line x equal to 0, x equal to pi on which v is 0 and the other line is the x axis part between 0 to pi. So we need to look at what happens to this sin square x minus sin x. Sin square x minus sin x is always less than or equal to 0 on this interval. Why? Because sin x lies between 0 to 1 that is the reason we will see the details. So maximum of v on this is 0 this is because v of x0 is sin square x minus sin x that is always less than or equal to 0 and v of 0t and v of pi t both of them are 0. Therefore v of x t is less than or equal to 0 for every x t in r that implies u of x t is less than or equal to e power minus t sin x. This is what we wanted to show. So let us look at problem number 4. For t positive let RT be defined as this rectangle x between 0 to L and t between 0 to t equal to t. So this is what is called RT. So for example if you take t dash another line t equal to t dash this entire region will be RT dash. So let u be a solution to the homogenous heat equation in r cross 0 infinity and the boundary condition at x equal to 0 and x equal to L are 0 and u of x0 equal to phi x. Let m of t denote the maximum value of u on the rectangle RT and closure RT closure. Show that m of t is a constant function of t. So let t be less than t dash and RT is a subset of RT dash. RT dash is a bigger rectangle. Therefore RT closure also is contained in RT dash closure. Therefore m of t is less than or equal to m of t dash. m of t after all is a maximum of u on a smaller set. m of t dash is a maximum on a bigger set. Therefore this holds. So by maximum principle what is m of t? It is a maximum of u on the parabolic boundary of RT. Similarly m of t dash is maximum value of u on the parabolic boundary of RT dash. Parabolic boundary of RT is a subset of parabolic boundary of RT dash. u on both these sets takes the same values. Let us have just have a look at this picture. This is RT, this is RT dash. So this part, this part, this part is the parabolic boundary of RT. What is the parabolic boundary of RT dash? It is bigger than this. What is the new portion? It is only this much.  That is the boundary condition. Therefore and u is already 0 here. So u does not take any new value so that maximum increases. So therefore maximum is the same. So m of t equal to m of t dash. A similar statement can be made for the minimum also. Thank you.


Hello and welcome to the session. Let us discuss the following question. It says, find equation of plane passing through the line of intersection of the planes vector r dot i cap plus j cap plus k cap is equal to 1 and vector r dot 2 i cap plus 3 j cap minus k cap plus 4 is equal to 0 and parallel to x axis. Now if we are given two planes vector r dot vector n 1 is equal to d 1 and the plane vector r dot vector n 2 is equal to d 2, then equation of plane through intersection of 1 and 2 is given by vector r dot vector n 1 minus d 1 plus lambda into vector r dot vector n 2 minus d 2 is equal to 0. So this knowledge will work as key idea. Let us now move on to the solution. We have to find equation of plane through the intersection of these two planes and parallel to x axis.   Now r is the position vector of any arbitrary point on the plane. So let us take vector r as x i cap plus y j cap plus z k cap.  Now we take the dot product of these two vectors. So we have x into 1 plus 2 lambda plus y into 1 plus 3 lambda plus z into 1 minus lambda minus 1 plus 4 lambda is equal to 0. Let us call this as 1. Now we are given that the plane is parallel to x axis. So this implies the point 1 0 0 satisfies the equation of the plane 0 into 1 plus 3 lambda plus 0 into 1 minus lambda is equal to 0. Solving this we get lambda to be equal to minus 1 upon 2. Now put lambda is equal to minus 1 upon 2 in 1.  So now this is equal to x into 1 minus 1 plus y into 2 minus 3 upon 2 plus z into 2 plus 1 upon 2 minus 1 minus 2 is equal to 0. Now this is equal to minus 1 by 2 y plus 3 by 2 z minus 3 is equal to 0. Taking LCM we have minus y plus 3 z minus 6 is equal to 0 and taking minus common we have y minus 3 z plus 6 is equal to 0 and this is the required equation. Hence the required equation of the plane is y minus 3 z plus 6 is equal to 0 and this completes the question. Bye for now. Take care. Have a good day.


Hi you guys my name is Brittany and welcome back to my youtube channel you guys in today's video I'm going to share with you my september homeschooling update So if any of you guys are new here to my channel, I'm Brittany I'm a homeschooling mom to three girls and I'm in my third year of homeschool So you guys we're going to go ahead and jump on into this update I really have a lot to share with you guys in this this month of september a lot of stuff has happened and um Yeah, so let's go ahead and get right on into this thing you guys So let's start off with the easy stuff Our reed aloud this month has been the secret zoo. It has definitely been a great reed aloud We have really been enjoying this Um, you guys I started using scribe again as far as our reed alouds goes We read the lemonade war which was our first reed aloud We read it out loud together But I'm finding I need that help when it comes to like being consistent with our reed aloud So we are back to using either scribe or audible It definitely has helped us out and we listen to our reed aloud while we are eating lunch And it definitely is going really well. This is a really really good story and we are really enjoying this Mystery fantasy. Um, I cannot wait till we finish it off and I can share with you guys Our review on it, but so far you guys our reed alouds this year has been like so good Um as far as leia goes let's start off with her since she's my younger one leia She is for we are doing our pre-k year this year And you guys she is absolutely doing very very well in our home school Uh the month of september we did our playing preschool apples unit and We had so much fun doing the apples unit and playing preschool And i'm going to make sure I put a few clips of like some of the activities that we got up to when we were doing our playing preschool Apple unit, but we had so much fun I stretched it out for the whole month of september We typically do our playing preschool messy art on fridays and it was so much fun Some of the books that uh my toddlers they were enjoying this month was an apples life And this is a nonfiction book talking about the apples life and I love how it has like Real life pictures in it and it's not like cartoony and they really enjoyed learning about the apple the apples life cycle I don't know if any of it really has stuck, but uh, this was one of our main books. We read Apple picking day was a fun one. This is actually one that we used last year and um, they definitely enjoyed this one And my youngest one alana. She has been enjoying apple farmer annie. I have to read this every single day to her She loves this one right now. Uh, we even uh listened to like the apple farmer annie on youtube They have like a rudal out and she loves me playing that on youtube for her as well So, um, that is what they did as far as like their fun unit that we are doing. Um We are really enjoying like this fall season you guys. Um as far as leah's core She is doing preschool math at home, which has been going so great I really have been incorporating a lot of hands on math activities for her and she's been loving it Um, one thing i'm learning about leah is she's definitely my hands on girl And I really need to limit the worksheets that i'm giving her and really just let her Have fun exploring with using like different manipulatives and things like that So with preschool math at home, I do have some of my fall manipulatives that i'm going to be using with her Uh this month when we are doing like our math skills and things like that And I just picked up these pumpkins from the dollar tree. Um another thing that I have added in for leah If you guys see my last video, we are doing all about reading pre-reading for her phonics instruction and We are only two lessons in but so far it's going so well. She loves ziggie the zebra She loves the posters the phonics cards We are really enjoying the activities that goes along with uh all about reading pre-reading. Um She hasn't quite grasped the concept of the rhyming games just yet, but we're only two lessons in So I definitely will continue to update you guys on it If you want to know like more about this all about reading pre-reading I will go ahead and link the video that I uh did my unboxing for it But so far it's going so good and i'm so happy i'm starting off with this all about reading from the ground up I was kind of nervous about just giving her a really good solid like basic phonemic awareness skill before we really get You know head deep into like reading instruction and um, I was really nervous you guys and I really feel like All about reading this pre-reading program is really holding my hand the way that I needed to To give her phonics instructions and for me to feel confident as like homeschooling mom so Uh, definitely thumbs up and I will continue to like update you guys on all about reading as we go throughout the program So that is really like all for my uh four-year-old leah and of course my two-year-old alana She tags along our plans for the next upcoming weeks is we are going to use our purely preschool fall unit We did this unit last year and uh since I already did the apples unit from the playing preschool I'm going to start off with the pumpkin unit We're going to do the leaves unit and the squirrels unit and finish it off in the playing preschool. So Hopefully they will enjoy us continuing like this fall type of theme fall books. Um and everything like that So that is like my plan for my uh four-year-old She's really easy you guys I definitely see why a lot of people love homeschooling in the younger ages because there is really really fun I'm really having a lot of fun with them So as far as briel you guys we have like a big big update with briel But i'm going to go ahead and start off with like the simple stuff first and foremost I have incorporated a student planner for briel in our homeschool because i'm noticing briel really wants that sense of independence and you know since we started our homeschooling journey she was in the third grade and You know, I feel like i'm so used to her needing me more than she does now And I really need to let go as mom and let briel take the reins in like her education in her schooling And this simple uh planner really has been helping her She's able to see everything she needs to get done and here goes the layout. Hopefully you guys can see that But it's just a simple student planner and this was a free printable online I think I've seen this uh planner originally mentioned from I forgot her name, but her channel is mama of five She is using this with her kiddos and it was a free printable So I went ahead and printed this off to use it with briel. It's been going really well What I do is the day before so like if we are on monday, uh, I will Make sure we write out everything she's going to be doing for tuesday on friday I uh have her come and I tell her everything she's going to do for monday So she's always like a day ahead of what I want her to do what I want her to accomplish Any type of like independent work she's going to be doing like uh independent read-alouds any type of like things like that I make sure she writes it down and it's really given her a sense of accountability and I think she loves uh it And I can't believe she's like a planner girl So that's definitely something that i'm incorporating in our homeschool especially as briel is progressing in her academics You guys this is like her last year of Technically elementary school So I really need to let go the reins I really need to give her more responsibility and allow her to take more ownership over her education And she definitely is loving me like taking this step back. Um, I no longer like set out her books for her anymore I really just let her you know pull out her homeschooling cart pull out her materials She needs and I'm really like taking a step back So, um, it's kind of sad my baby's growing up, but I really think that um, this is the best uh for hearts for her to be able to grow And you know, she doesn't she's not always gonna need me you guys So, um, as far as like our struggles, I really can say for Brie It definitely has been adding in science you guys Um, I really have been trying to figure out like our schedule when it comes to like history and science As far as like our math and our, uh, grammar writing all of that is going really really well As far as our schedule and rhythm goes it's just um trying to figure out like our history and science at first I was trying to do history and science for Four days out the week every single day together and just doing like little bits like 20 minutes of history 20 minutes of science, but I was finding that that wasn't working because You know when it came to a topic that Brielle wanted to like deep dive in either science or history We kind of felt like we had to rush it to get to the next subject So what I'm doing now and I'm trying out this new schedule and hopefully it works is on monday and tuesdays It's going to be our history days and on wednesdays and thursdays It's going to be our science days and whenever we are not able to get to any of those lessons I always overflow it to fridays Where if we miss the science lesson, we can make it up on fridays or if we miss the history lesson We can make that up on fridays as well. That's how I kind of like utilize my fridays, but um science you guys Other than us trying to really get a flow with everything as far as science and history It's been really going well. I'm going to show you guys some snippets of brielle's um Science notebook. We really have been enjoying doing our notebooking. This is our schoolness notebook science notebook and here are some of her pages We are studying space right now. So she is really enjoying Just making her space journal right now. You guys the other day, um brielle and my husband They stayed up late to see jupiter And it was so crazy because when we first started off our space unit A breeze she wasn't really interested into it until they stayed up late at night and it was like a fun experience for her and for my husband It really was able like the bond over it and um, she had so much fun the next day you guys She had the encyclopedia out and she just couldn't stop talking about space. So I'm really uh finding that I really need to make sure I still incorporate those like real life experiences, especially when it comes to studying like specific types of topics and things like that, but um overall Hopefully I this new schedule with like our history and science will work out like a lot better and another thing you guys like Um as far as scheduling and rhythm goes like I really had to take the pressure off of myself as far as like timeframes It's so crazy like in my neighborhood the school bus comes around at 245 and For the past two years every single time that school bus comes around my coda sack I am always like, oh, we're not done with school yet The school bus is already here and that was kind of like a marker if I was Successful at the end of the day when I heard that school bus coming because The bus stop for the public school is like right in front of our house And it's just so crazy how I had to tell myself this year like that school bus is not my timer Like we are on our own time our own flow And uh, I really have just let that go and I really have found peace in it. Um, I definitely have been prioritizing like outside time with my uh kids just in general In georgia, you guys the summers are so hot and we were right cooped up in the house because like we could barely breathe outside That's how hot it is In georgia, um, we're able to sneak out in the mornings, but after that you guys like it's miserable out there So now that the weather is like cooling down It's beautiful. It's windy. We can fly kites throughout the homeschooling day I really take a big chunk from our morning time um and like our in between like our morning time and lunch we take like a whole hour before coming back in me putting my toddlers down for a nap and um, Brielle and me starting like either history or science now that's like our new schedule, but um Yeah, I really have been prioritizing that time and I don't mind you guys if we're not done with our homeschool days Like typically right now since we are taking that bigger gap We're not done with homeschool until like, you know, uh, three o'clock But i'm really okay with that because uh, I really want to prioritize that outside time with the girls As far as like a curriculum goes you guys Brielle we have finished saxon six five and I am so proud of her So um in my video you guys I was trying to debate on if I was going to go ahead and move her to saxon seven Six or if we were going to do some review But you guys she passed the saxon six five Um The last test with like flying colors She only missed one problem in like the last test And honestly, she really didn't miss it because when I told her brielle once you look at that She found her mistake like right away So she did so well all the gaps that I was worried about in saxon six five When I took you guys's advice with saxon and I used the supplemental practice in the back I slowed down those lessons And we really uh reviewed on those concepts that she needs to review when we went back to the lessons The last like 15 lessons of saxon six five She just like breezed through them and i'm so happy You guys put those comments in my video when I was like having problems with uh those particular gaps And i'm so happy that I stopped and I took the time to do it Um because now she's fully prepared to go ahead and start saxon seven six So like i'm so proud of this girl. Um, so we're going to go ahead and start saxon seven six and We're going to just take it easy One thing I love about saxon is the first like 40 lessons of saxon is typically a review from the previous books So the first a few lessons is going to review Saxon six five so she's going to have that opportunity to get a second chance of review But so far she is thriving. She's doing so well and i'm happy to like continue on with saxon I'm so proud of this girl you guys So that is uh, what brielle is doing for math as far as like grammar and english goal She guys brielle is definitely um very advanced when it comes to like her grammar and her english and writing skills And you guys i'm not like bragging on my baby girl like it's just you know What it is some kids like accelerate in different uh subjects and english is definitely the subject where she is Accelerated in and I really feel like I have been holding brielle back, especially when it comes to like english and things like that um But you guys like after talking with brielle and she was expressing like a lot of boredom With the levels that I was giving her as far as like her english and grammar instruction I went ahead and I decided to bump up brielle in our grammar in our english to more of a sixth grade level And you guys this was like the first time that brielle actually like she's starting to really have to think when it comes to Uh her english because before she was just always getting all the answers right You know and you know, she never really had to take the time and think or really It'd be something she has to actually learn now. She's actually learning and i'm finding that i'm probably at the right level when it comes to her Uh grammar english all of that goes So i'm definitely going to make sure that I continuously evaluate um My homeschool and I don't like put a great category On my daughter as far as okay. She's in the fifth grade. We have to do fifth grade level work I really need to continuously look at her skill level Not necessarily her age and I need to stop like holding her back you guys. Um, But so far uh as she has been doing rodent staff six is definitely a better fit for her We're also doing 180 days and she is on the sixth grade book as far as the spelling It definitely is more of a challenge for her. It's right where she needs to be at and um, I'm so happy as far as that with iow guys iow is going good. Um at first I was kind of worried that iow was too easy for her But now you guys she is finally slowing down on iow. So I really think that this was the perfect level for her We are on week eight and now she is writing like three paragraph essays and you guys, uh, she's finally had like slow down So I know this is the right level for her at first She was doing like the whole week's work of iow in two days But now we are really taking that full week doing iow. So that has been going great fix a grammar She's almost finished with book two. So we're going to go ahead and move to fix a grammar book three Robin Hood So that is like briel's total like lineup one piece of curriculum that I am dumping and I really feel like I just was adding too Much on my plate and I just honestly don't really like it that much Is wordly wise you guys don't kill me in the comment section below, but I'm not gonna lie I really don't like this wordly wise curriculum. Um, my daughter She was just brain dumping the vocabulary words at the end of the week And it wasn't an effective way for her to really master vocabulary I'm finding I really enjoy 180 days of spelling and word study because it really focuses on Those uh prefixes and root words and she's able to retain the vocabulary so much better With a fix a grammar and with the 180 days of spelling than wordly wise. Um, so you guys this right here is You know, it's a flop for me. Um, I really don't like it that much I really think that um, when I'm looking for a vocabulary program for her in the future I'm definitely going to look for a program that really focuses on those root words those prefixes and suffixes because If she knows those words, she will be able to find the meaning of any word Instead of just learning, uh, just the definitions and dumping them at the end of the day So this is being tossed and um, it is what it is. It's just a supplemental workbook I really don't feel bad tossing this and honestly, um, she is getting vocabulary in so many other places Now you guys we did another big dump as far as curriculum goes and I'm going to make a separate video all about that Don't worry you guys.  It is our history curriculum So you guys um, really this is my september homeschooling update. Nothing really big has happened. Well, at least I don't think so Um, but it's everything still has been going great. I really feel blessed. Um, we're just taking this homeschooling journey We're taking the ups and the downs and you know, we're just riding away as you guys over here I hope everyone's homeschool and your schedule and your flow is going well If you're having some hiccups, it's okay. You know, we're going to get into the swing of this thing Thank you guys so much for watching today's video and I look forward to seeing everybody in my next one Bye


Almost everybody left me. My mother stayed, but I could have blown up the entire world and she still would have stuck by me. That's what mothers do. My wife stayed, though I deserved her to leave me too. But she loved me even through this. But you know who else stayed with me? God did. Paul says something that I can agree with. He said that he was up and that he was also down. But what he's learned is that no matter what state he's in, he's learned to be content. Guys, when I learned to be content and when I just fully trusted in him and having him in my life was all that mattered. That's when he opened the prison doors. I was released from prison five years early, not from some clemency or anything, but by the move of God. That's what trusting in God will do.