Partial Derivatives, Diffusion, and Waves
An educational company contracted us out to build a website to analyze wave propagation and heat transfer. The company provided us with two equations, the heat equation and the wave equation, both of which were partial differential equations. Along with these equations we were provided with a set of corresponding solutions as well as a set of constraints to prove these solutions are true. The constraints consisted of initial and boundary conditions. We asked to solve these partial differential equations in one and two dimensions. In this analysis we will prove that the solutions satisfy their respective partial differential equations and how the solutions can be manipulated to show the behavior of the functions.
Too begin this problem we had to evaluate the constraints on both the heat and the wave equations. To evaluate these we first tested the initial conditions and boundary conditions for both wave equations (1.1). We then needed to prove that the provided solutions held true for the given differential equation by taking second order partial derivatives for both x and t and making sure that they satisfied the wave partial differential equation (1.2). We followed similar steps to the heat solution provided, to ensure that its initial conditions were true and that its boundary conditions existed. From here we proved that both of the heat solutions fulfilled the heat partial differential equation. We then made plots of the initial conditions for all of the solutions provided. We did this to show how the functions changed with regard to position at time t=0, and with their respective constants a and c set equal to 1 (1.3). After examining the functions behavior with respect position we wanted to see how the functions would differ as the time changed. To accomplish this we used the animate function on both the heat and wave equations to show a movie of how the functions would change as time changed. This clearly illustrated how as time elapses the magnitude of the heat equation shrinks and the wave equation grows moves along the domain (1.4). After examining each plots we found that the answers would make sense. For the heat equation the longer that the time moves on, the heat distribution from end to end should become more uniform throughout. For the wave equation since waves moves through their medium then it makes sense that the waves would move along through the position of the graph. Now we will examine the two solutions more in depth.
For both the wave and the heat equation our client provided two solutions for each in one dimension. For the wave equations there were two solutions provided such that you can look at how the wave is moving from either the –x direction to the +x direction or from the +y direction to the –y direction. This is to show that the wave can be moving in either direction along the domain. As you change the different value of c the orientation of the of the waves change within the domain. The waves rotate from traveling in parallel to traveling almost in perpendicular to one another (1.5). This means that c represents the direction of the wave through the medium. For the two wave equations one solution was choose to represent the ends of a thin metal rod to be insulated u(x,t) = (1/2) e-at cos (x) + 1/2. The heat is distributed throughout the rod such that it is constant at the ends and high in the middle and then as time moves on the heat become uniformly distributed. The other heat equation, u(x,t) = (1/2) e-at sin (x) + 1/2 models the situation of ends of the metal rod held at constant temperature. The heat distribution throughout the medium is uniform in the shape of a Sin curve and to where as time goes on the distribution of the heat slowly becomes uniform throughout the medium. The size of a directly affects the magnitude of the graph (1.6). As a increases in size it makes the exponential function more negative, which causes the fraction to become larger and therefore shrink the magnitude. Meaning that a represents the heat capacity of different mediums in which this heat equation is applied. This is true because the higher the heat capacity of the medium the less heat change throughout the medium.
To prove the two dimesional equation the clients provided us we first examined the constraints of each individual equation. For the Wave equation we evaluated the constraints by plugging in t=0 and evaluating the function at that point (2.1). For the second constraint we took the first order partial derivative, and evaluated it at t=0. We then took this solution and proved that it satisfied the given constraints (2.2). After this, we proved that this solution fulfilled the Wave equation by taking second order partial derivatives for all of the variables (2.3). Next we had to prove that 2D solution for the heat equation was true. We followed the same procedure as the wave equation by evaluating the solution for all of the constraints and then testing to be sure the solution appeases the given heat partial differential. To prove the heat partial was true we had to take second order partials for x and y and a first order for t. This demonstrated that the provided solutions were true for these functions. Then we plotted all of the initial conditions of each function to graphically demonstrate the behavior of the function at these constraints (2.4). After this we wanted to find what the graphs would look like at different time intervals. The time interval was from Pi/8 to 7*Pi/8, in ¼ intervals of time. To do this we used the Mathmatica ContourPlot to find the level curves of the functions as the time changes. Level curves describe a region along the line that maintains the same value regardless of position. You can locate these same curves on the 3D plot by looking at the topography of the graph and seeing where the evaluations have the same value. We illustrated the changes through time by taking advantage of the manipulate and animate functions of Mathmatica. Both of these functions show how the graph changes as you vary the value of the time (2.5). After seeing how the level curves of the functions changed we made 3D plots of these functions and used the same Mathmatica functions, animate and manipulate, to illustrate how the function would change as you vary the value of time (2.6).
We looked back over our derived results and we saw that our result made sense for the heat equation because as a became larger then it forced the magnitude to become smaller in the 1D situation, and this follow suit in the 2D equation. This also makes sense as t gets bigger this makes the exponential fraction smaller, thus making the overall equation smaller and smaller. This also makes sense for the wave equation because as time elapses the wave will continue along its path.
Partial Derivatives and Airy’s Equation
Now that we have examined the heat and wave equations intensively we have noticed certain trends apparent in both 1D and 2D equations. There is a trend relating the number of initial conditions to the order of the final partial differential equation. Such that the wave equation was a second order partial and therefore had to have two constraints to be solvable. Likewise the heat equation was of the first order partial so it only had to have one constraint. We applied this tendency to the partial differential equation utttt = a uxx, and surmised that this must have 4 constraints in order for this function to be satisfied. In considering Airy’s equation we noticed that it was similar to the heat equation, to where it was a first order partial differential, subsequently we came to the conclusion that in order for the equation to be solvable we needed one initial condition. We used our deductions to prove the given solution with its given values was true. In our proof, we took a first order partial derivative with respect to t. Next we took a third order partial derivative with respect to x and multiplied it by –1 and found that these were equal so it must be a solution to Airy’s equation (3.1).
In our process we found that all the given initial conditions and solutions, satisfies both the heat and wave equations. We also found how the wave and the heat equations varied with time and how to interpret changes in functions using level curves. Within our analysis we came to the conclusion that a partial differential equation is only calculable when it has the same number of conditions as its order of partial differentials.