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Solution To Heat Equation by Separation of Variables and Eigenfunction and Expansion
We credit [2] for a second solution to the heat equation in a bounded domain x ∈ (0, l) for all time t > 0. If we are given initial conditions u(0, t) = u(l, t) = 0 and u(x, 0) = φ(x) then we can separate variables and write u as the product of one function of time only and one function of position only
Using the same method we used to solve the wave equation, we plug in this new function to the heat equation
and we find that both sides must equal a constant
These are ordinary differential equations whose solutions are given by
The boundary conditions imply D = 0 and 
We can find a general equation for u(x, t) by taking the in nite sum of all of the solutions
As in the case of the wave equation we can solve for the coeffcients by using our initial value function then using the same integration technique used in the solution to the wave equation.
Acknowledgments. It is a pleasure to thank my mentors, Jessica Lin and Yan Zhang, for their help and guidance.
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FAQs on Method of Separation of Variables for Wave Equation - CSIR-NET Mathematical Sciences - Mathematics for IIT JAM, GATE, CSIR NET, UGC NET
| 1. What is the wave equation? |  |
Ans. The wave equation is a second-order partial differential equation that describes the behavior of waves. It relates the second derivative of a function with respect to time to the second derivative of the same function with respect to position.
| 2. What is the method of separation of variables for the wave equation? | |
Ans. The method of separation of variables is a technique used to solve partial differential equations by assuming a solution that can be expressed as the product of two or more functions, each of which depends on only one variable. For the wave equation, this method involves assuming a solution of the form u(x,t) = X(x)T(t) and then substituting it into the equation to obtain two separate ordinary differential equations.
| 3. How does the method of separation of variables work for the wave equation? | |
Ans. In the method of separation of variables for the wave equation, we assume a solution of the form u(x,t) = X(x)T(t). By substituting this into the wave equation and dividing both sides by X(x)T(t), we obtain two separate ordinary differential equations: one involving only X(x) and the other involving only T(t). These equations can then be solved individually, and their solutions can be combined to obtain the general solution for u(x,t).
| 4. What are the boundary and initial conditions in the context of the wave equation? | |
Ans. In the context of the wave equation, boundary conditions refer to the conditions imposed on the solution u(x,t) at the boundaries of the domain. These conditions can specify the values of u(x,t), its derivatives, or a combination of both at the boundaries. Initial conditions, on the other hand, refer to the conditions imposed on the solution u(x,t) at a specific initial time, usually denoted as t=0. These conditions can specify the values of u(x,t), its derivatives, or a combination of both at different points in the domain at the initial time.
| 5. What are the main applications of the wave equation? | |
Ans. The wave equation has various applications in different fields. In physics, it is used to describe the behavior of waves in different mediums, such as sound waves, electromagnetic waves, and water waves. It is also used in engineering to analyze the vibrations of structures and the propagation of signals in communication systems. Additionally, the wave equation is employed in acoustics, seismology, optics, and many other areas where the study of wave phenomena is essential.
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