GOVERNMENT ENGINEERING COLLEGE BHARUCH,014 SUBJECT- ADVANCED ENGINEERING MATHEMATICS

APPLICATION OF PARTIAL DIFFERNTIAL EQUATION TOPIC APPLICATION OF PARTIAL DIFFERNTIAL EQUATION

Partial differential equation One of the classical partial differential equation of mathematical physics is the equation describing the conduction of heat in a solid body (Originated in the 18th century). And a modern one is the space vehicle reentry problem: Analysis of transfer and dissipation of heat generated by the friction with earth’s atmosphere.

Initial and Boundary conditions; Initial condition- If time t is one of the independent variables, then variable or its derivatives prescribed at t=o. Boundary conditions- when assumes the values of dependent variables on the boundaries of a region. Boundary conditions; 1) Homogenous 2) Nonhomogenous

BVP AND IBVP When a partial differential equation is solved satisfying some conditions known as boundary conditions the problem is called as Boundary valued problem(BVP). Or if specified with initial and boundary conditions the problem is called as Initial Boundary valued problem(IBVP).

Classification of partial differential equations; Partial differential equation can be classified as heat equation, wave equation, Laplace equation, etc. Heat equation is parabolic. Wave equation is hyperbolic. Laplace equation is elliptic.

Classification of PDEs

Classification of PDEs;

HEAT EQUATION

One Dimensional Heat Equation

The method of separation of variables Introducing solution of the form u(x,t) = X(x) T(t) . Substituting into the I.V.P, we obtain:

Boundary Conditions Imply that we are looking for a non-trivial solution X(x), satisfying: We shall consider 3 cases: k = 0, k > 0 and k < 0.

Case (i): k = 0. In this case we have X(x) = 0, trivial solution Case (ii): k > 0. Let k = 2, then the D.E gives X  - 2 X = 0. The fundamental solution set is: { e x, e -x }. A general solution is given by: X(x) = c1 e x + c2 e -x X(0) = 0  c1 + c2 = 0, and X(L) = 0  c1 e L + c2 e -L = 0 , hence c1 (e 2L -1) = 0  c1 = 0 and so c2 = 0 . Again we have trivial solution X(x)  0 .

Finally Case (iii) when k < 0. We again let k = - 2 ,  > 0. The D.E. becomes: X   (x) + 2 X(x) = 0, The auxiliary equation is r2 + 2 = 0, or r = ±  i . The general solution: X(x) = c1 e ix + c2 e -ix or we prefer to write: X(x) = c1 cos  x + c2 sin  x. Now the boundary conditions X(0) = X(L) = 0 imply: c1 = 0 and c2 sin  L= 0, for this to happen, we need  L = n , i.e.  = n /L or k = - (n /L ) 2. We set Xn(x) = an sin (n /L)x, n = 1, 2, 3, ...

u(x,t) =  un(x,t), over all n. More precisely,

Consider the heat flow problem:

Solution Since the boundary condition forces us to consider sine waves, we shall expand f(x) into its Fourier Sine Series with T = . Thus

With the solution

In this video heat equation is explain in detail

WAVE EQUATION

In the study of vibrating string such as piano wire or guitar string.

Motion of a string Imagine that a stretched string is vibrating. The wave equation says that, at any position on the string, acceleration in the direction perpendicular to the string is proportional to the curvature of the string.

The one-dimensional wave equation Let x = position on the string t = time u(x, t) = displacement of the string at position x and time t. T = tension (parameter)  = mass per unit length (parameter) Then Equivalently,

Solving the one-dimensional WE First, we make a wild assumption: suppose that u is a product of a function of x and a function of t: Then the wave equation becomes So, The only way for this equation to be true for all x and for all t is for both sides to be constant; that is, T’’(t)/a2T(t) =  and X”(x)/X(x) = .

Separation of variables Now means that If  < 0, T(t) and X(x) are both trig functions.* *If  = 0, they’re linear, and if  > 0, they’re exponential. But let’s not worry about that now. (Explanation on next slide.)

Clarification: the sign of  There are three types of solutions to the equation if  = 0, then X(x) is linear (ax + b), which won't satisfy the boundary conditions; if  > 0, X(x) is exponential (ket), which also won't satisfy the boundary conditions; and if  < 0, then X(x) is a sinusoid that satisfies the boundary conditions. This is why in slide 8.

Example: f(x) = 6 sin 2x + 9 sin 7x - sin 10x , and g(x) = 11 sin 9x - 14 sin 15x. The solution is of the form:

In this video wave equation is explain clearly with the help of figures.

LAPLACE EQUATION

It is clear that if the heat flow is steady, i. e It is clear that if the heat flow is steady, i.e. time independent, then ∂u/ ∂t=0 so the temperature u(x,y)is solution of is the two-dimensional Laplace equation.

Some areas in which Laplace’s equation arises are; Electrostatics Gravitation (c) Steady state flow of in viscid fluids (d) Steady state heat conduction

In the following video the derivative of Laplace equation is explain.

BHANUSHALI JEET .L. Er no- 130140105004 Prepared by; BHANUSHALI JEET .L. Er no- 130140105004

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