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**Jawaharlal Nehru Technological University Anantapur (JNTU) College of Engineering (CEP), Pulivendula, Pulivendula, Andhra Pradesh, India - JNTUACEP**- 2 Topics
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2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 4.4 4.5 4.6 Heat Equation Two Dimensional wave equation Laplacian in polar coordinates Circular Membrane ( Use of Fourier-Bessel Series) Laplace’s Equation in cylindrical and Spherical coordinates Solution of partial Differential equation by laplace Transform. Analytic function Cauchy –Reiman equation & Laplace equation Conformal mapping Line integral in complex plane Cauchy’s integral theorem Cauchy’s integral formula Derivatives of analytic function Power Series Taylor series Laurent Series Singularities, Pole, and Residue Residue Integral Evaluation of real integral Formation of Partial Differential Equations In practice, there are two methods to form a partial differential equation. (i) By the elimination of arbitrary constants. (ii) By the elimination of arbitrary functions. (i) Formation of Partial Differential Equations by the elimination of arbitrary constants method: * Let f(x,y,z,a,b)=0 be an equation containing 2 arbitrary constants ''a'' and ''b''. 3

* To eliminate 2 constants we require at least 3 equations hence we partially differentiate the above equation with respect to (w.r.t) ‘x’ and w.r.t ‘y’ to obtain 2 more equations. * From the three equations we can eliminate the constants ''a'' and ''b''. NOTE 1: If the number of arbitrary constants to be eliminated is equal to the number independent variables, elimination of constants gives a first order partial differential equation. But if the number of arbitrary constants to be eliminated is greater than the number of independent variables, then elimination of constants gives a second or higher order partial differential equation. NOTE 2: In this chapter we use the following notations: p = ∂z/∂x, q = ∂z/∂y, r = ∂2z/∂x2, s = ∂2z/(∂x∂y), t = ∂2z/∂y2 METHOD TO SOLVE PROBLEMS: Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’. Step 2: We know p = ∂z/∂x and q = ∂z/∂y. Step 3: Now find out a and b values in terms of p and q. Step 4: Substitute these values in the given equation. Step 5: Hence the final equation is in terms of p and q and free of arbitrary constants ''a'' and ''b'' which is the required partial differential equation. (ii) Formation of Partial Differential Equations by the elimination of arbitrary functions method: * Here it is the arbitrary function that gets eliminated instead of the arbitrary constants ''a'' and ''b''. NOTE: The elimination of 1 arbitrary function from a given partial differential equation gives a first order partial differential equation while the elimination of the 2 arbitrary functions from a given relation gives second or higher order partial differential equations. 4

METHOD TO SOLVE PROBLEMS: Step 1: Differentiate the given question first w.r.t ‘x’ and then w.r.t ‘y’. Step 2: We know p = ∂z/∂x and q = ∂z/∂y. Step 3: Now find out the value of the differentiated function (f'' ) from both the equations separately. [(f’’) =?] Step 4: Equate the other side of the differentiated function (f'' ) which is in terms of p in one equation and q in other. Step 5: Hence the final equation is in terms of p and q and free of the arbitrary function which is the required p.d.e. * Incase there are 2 arbitrary functions involved, then do single differentiation i.e. p = ∂z/∂x, q = ∂z/∂y, then also do double differentiation i.e. r = ∂2z/∂x2, t = ∂2z/∂y2 and then eliminate (f'' ) and (f'''' ) from these equations. Worked out Examples Elimination of arbitrary constants: Ex 1: 5

Elimination of arbitrary functions: Ex 2: ( ) ( ) Partial Differential Equations of First Order The general form of first order p.d.e is F(x, y,z,p,q)=0 .... (1) where x,y are the independent variables and z is dependent variable . Types of Solution of first order partial differential equations Complete Solution Any function f(x,y,z,a,b)=0 ........(2) involving arbitrary constants a and b satisfying p.d.e (1) is known as complete solution or complete integral of (1). General Solution 6

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