Jidnyasa Academy

  Partial differential equation, its types, its solution and fundamental theorem of superposition of solutions are discussed in the following video. Also method of solving pde by separation of variables is explained with examples.  You can read the details of the concepts / proofs  explained  in the video here - 

The vibrations of an elastic string are governed by the partial differential equation which is known as wave equation. The details of the derivation are -  Solution of wave equation subject to boundary conditions and initial conditions is discussed here -  The details are -  The significance of solution of wave equation, normal modes, are discussed here. How the deflection function of vibrating string is expressed as superposition of two functions when initial velocity is zero, is explained with one example.  Another two examples with solutions - 

Heat equation governs the temperature 'u' in a body in space. The following physical assumptions are to considered - 1. The specific heat  𝛔 and the density  ρ of the material of the body are constant. No heat is produced or disappears in the body. 2. Heat flows in the direction of decreasing temperature and the rate of flow is proportional  to the gradient of 'u'. 3. The thermal conductivity  k is constant. The boundary conditions i.e conditions on two ends of a thin bar can be different. For different boundary conditions, solution of heat equation is also different. In the following video, solution of heat equation is discussed where both the ends are kept at temperature 0. These boundary conditions are known as homogeneous boundary conditions.  The details of solution are -  In the following video, the solution of one dimensional heat equation where both the ends are insulated is discussed.  Det...

The solution of heat equation depends on the boundary conditions of the rod. Following are two standard sets of boundary conditions : 1. Both the ends of rod are kept at 0 degree Celsius. - Also known as Homogeneous boundary Conditions. 2. Both the ends are insulated. Two problems based on these conditions are discussed in the following two videos - There are many combinations of boundary conditions. Most of the times, we reduce such problems to the problem with  either one of the above two sets of boundary conditions. This technique is discussed in the following video.  These are some solved problems based on one dimensional heat equation - 

The basic concepts regarding steady state solution of two dimensional heat equation are discussed in the following video. An example is discussed in the following video -  Solutions of some problems on two dimensional heat equation are -