Understanding Partial Differential Equations: Types and Solution Methods

Unlock the mysteries of Partial Differential Equations (PDEs) with this comprehensive presentation. Dive into the realm of 1st and 2nd order PDEs, exploring their classification into linear and non-linear forms.

In this enlightening slideshow, we delve into the intricacies of both linear and non-linear PDEs, shedding light on their significance in diverse fields such as physics, engineering, and mathematics. Discover the fundamental principles governing these equations and their varied applications.

One of the focal points of this presentation is the exploration of solution methods for 2nd order linear PDEs. Delve into techniques such as direct integration and variation of parameters, unraveling the step-by-step processes that lead to finding solutions to these complex equations.

Whether you're a seasoned mathematician, an aspiring physicist, or a curious learner, this slideshow is designed to demystify PDEs and equip you with the knowledge needed to tackle them confidently. Join us on this journey through the realm of Partial Differential Equations, where understanding meets application.

1. An Introductory Presentation on PARTIAL DIFFERENTIAL EQUATIONS By -Roshan Koirala

2. 1. Differential Equation Let x be independent variable. Derivatives of any function (say z 1 ), depend upon x is the rate of change z 1 as x changes slightly. The equation that contains one or more unknown functions of single independent variable and their derivatives is differential equation. When no. of independent variable is 2 or more(say x,y ) such that z ( x,y ) is function of x and y , then partial derivative of dependent variable with any one independent variable keeping other independent variable as constant. Ordinary Differential Equations (ODEs) deal with functions of one independent variable , while Partial Differential Equations (PDEs ) deal with functions of multiple independent variables . =2 , ( z x , z y ) + + z 2 = xy  

3. Taking z( x,y ), as a dependent and x,y as an independent variable, we will adopt the following notations through out the presentation. REMEMBER z x = z y = z xx = z xy = z yy =  

4. Introduction To PDE A partial differential equation (PDE ) is a mathematical equation that contains an unknown function ( z ) of two or more independent variables( x , y ), as well as the partial derivatives of that unknown function( z x , z y , z xx ) with respect to those variables . That unknown Function ( z ) is called dependent variables. PDEs can be classified into various types based on their order, linearity, and coefficients.

5. Examples :

6. Degree of PDE is degree of highest order partial derivative involved in that PDE (1 st degree) (2 nd degree) ( 1 st degree) (1 st degree) Order of PDE is order of highest order partial derivative involved in that PDE (1 st order) (3 rd order) (1 st order) (2 nd order) The concept of degree cannot be attributed to all PDE . For example , the given PDE doesnot have any degree. sin z x + e^(z y ) = 1

7. Non-Linear : z x 2 + z y 2 =0 z .z x + z .z y =z z x +z y +z xx +z xy +z yy =z 2 Linearity: Linear and Non Linear PDE : LINEAR :- If the dependent variable (z) and its partial derivatives ( z x , z y , z xx , z xy , z yy ) occurs in the first power only and are not multiplied. NON LINEAR :- Else EXAMPLES Linear: z x + z y =0 x.z x +y.z y =z z x +z y +z xx +z xy +z yy =z

8. First Order PDE (L)

9. The general form of first order PDE is of type f(x,y,z,p,q)= It can also be written as: A.z x +B.z y +C.z = D or A where is the function of independent variables and constants If D=0 , the PDE above becomes Homogeneous.  

10. Origin Of First Order Partial Differential Equation By the elimination of the arbitrary constants from a relation between x, y and z. By the elimination of arbitrary functions of these variables. g (x,y,z,a,b)=0 Differentiating g wrt . x and y partially, and from f, f x , and f y We get equation of the form f( x,y,z,p,q )=0 is required PDE f(u, v) = 0 , where u and v are function of x,y,z Differentiating f wrt . x and y, taking z as dependent variable We get equation of the form p P + q Q = R is required PDE Where P,Q,R are Lagrange’s linear equation. x 2 +y 2 = (z-c) 2 tan 2 α Algebric xq-yp=0 PDE z = xy + f(x 2 +y 2 ) Algebric py-y 2 = qx-x 2 PDE

11. Solution: Linear PDE of the First Order The PDE of the type p P + q Q = R , where P, Q, R are functions of (x , y, z), is called a linear PDE of the first order or Lagrange’s linear equation . Its solution is in the form of F(u, v) = 0, where F is arbitrary function and u(x, y, z) = c 1 and v(x, y, z) = c 2 Rules for solving p P + q Q = R Put the given PDE in the standard form p P + q Q = R . Write down Lagrange’s auxiliary equations = = Solve these equations Let u(x, y, z) = c1 and v(x, y, z) = c2 are two independent solutions. 4 . The general solution is then written one of the equivalent form F(u, v) =   PDE : y 2 p − xyq = x(z-2y ) Step1: P= y 2, Q= xy , R= x(z-2y) Step2 : x 2 +y 2 = c 1 , zy−y 2 = c 2 Step3 : F(c 1 ,c 2 ) = F(x 2 +y 2 , zy− y 2 )=0

12. Second Order PDE (L)

13. A second order PDE involves second-order partial derivatives of an unknown function(z) with respect to one or more independent variables. General Second order Linear PDE (with 1D/2I variable) is: A z xx + B z xy + C z yy + D z x + E z y + F z = G ------( i ) when, G=0, equation ( i ) is homogeneous. Depending on the coefficients , second-order PDEs can be classified as: Parabolic if B 2 -4AC = 0 Elliptic if B 2 -4AC < 0 Hyperbolic if B 2 -4AC > Solutions (BCs and ICs) : Second-order PDEs often require boundary conditions for elliptic and hyperbolic equations, while parabolic equations typically require initial conditions along with boundary conditions.

14. Solution method for second order L . PDE (Direct Integration) Direct integration (PDEs ) is a method where both sides of a PDE are integrated with respect to one of the independent variables to eliminate one derivative from the equation. And the process continues till all Partial derivatives are removed. Example : where f(t) and g(t) are unknown function using ICs, Final Sol. n

15. Why Separation of Variable? Reduction to ODEs : Separate solution into functions of single variables , simplifying PDE into manageable ODEs. Homogeneous Boundary Conditions : Effective for PDEs with homogeneous boundary conditions, facilitating solution combination. Orthogonality of Solutions : Solutions may form orthogonal sets, easing coefficient determination for boundary value problems . Versatility : Applicable to various linear second-order PDEs , including heat, wave, and Laplace's equations. Physical Interpretation : Provides clear physical interpretations , aiding understanding in fields like mathematical physics and engineering. When the equation is more complex or doesn't lend itself well to direct integration. In such cases , separation of variables becomes a valuable alternative.

16. Solution method for second order L. PDE (Separation of Variable) If we have a second order PDE u t =α u xx -------------( i ) (where u is dependent variable depend upon x and t ) To get solution we assume product 2 different function X ( x ) and T ( t ) of x and t respectively, be the solution of PDE above. ie ’. u ( x,t )= X ( x ). T ( t ) -----(ii) Solving ( i ) and (ii), we get = k (Separation Constant) According to the value of k (k <,=,> 0) we can found 3 pair of distinct solution for X and T and final Solution will be u ( x,t )= X ( x ). T ( t ), where, 3 different solutions can be found. Finally, using initial and boundary condition , we can found one out of 3 as a non trivial solution.  

17. Example: PDE: u t =3.u xx , 0<x<2, t>0 BCs: u(0,t)=0, u(2,t)=0, t>0 IC: u(x,0)=x We get the non-trivial solution from 2 nd solution u( x,t )= . exp { }   Example 2: PDE: u t = a .u xx , 0<x<L, t>0 BCs: u(0,t)=0, u x ( L,t )=0, t>0 IC: u(x,0)=x We get the non-trivial solution from 2 nd solution u( x,t )= . exp { }   Three Different solution for above equation( u t = α u xx ) is found as : 1. u( x,t )=( A.e + B.e )e αλ ^2.t , when k= λ 2 for λ >0 2. u( x,t )=( C.cos + B.sin ) e - αλ ^2.t , when k =- λ 2 for λ > 3. u( x,t )= E.x+F , when k=0  

18. Other Solution Methods: Fourier and Laplace Transforms : Transforming the PDE into a different domain (frequency or Laplace space ) can simplify the problem , allowing for easier solution. Numerical Methods : When analytical solutions are not feasible, numerical methods such as finite difference, finite element, or spectral methods are employed to approximate solutions.

19. One Dimension Heat Equation: W e need to find the temperature distribution 𝑢(𝑥,𝑡 ) along the medium over time. This involves determining how the temperature varies with both position 𝑥 along the medium and time 𝑡. It is a classic example of a parabolic PDE and is used to model heat conduction processes . Homogeneous Medium ( ρ , s:const ) Heat flows in direction of decreasing Temperature Heat flow rate (Q) across an area (A) is proportional to A and temperature gradient. ( ρ as proportionality Constant) Quantity of heat gained and lost by body is proportional to mass of body ‘m’ and change in temperature ‘ dT ’ ((‘ s’ as proportionality Constant) ASSUMPTIONS

20. Q 1 = -kA x Q 2 = -kA x+ Δ x Δ Q = Q 2 -Q 1 =kA x+ Δ x - kA x Δ Q = (A. Δ x. ρ ) . S . u t =a 2 u xx