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 z1), depend upon x is the rate of change z1 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.
•
dz1
dx
•
d2z1
dx2 +
dz1
dx
+ z1=2
•
𝜕𝑧
𝜕𝑥
,
𝜕𝑧
𝜕𝑦
(zx , zy)
•
𝜕𝑧
𝜕𝑥
+
𝜕𝑧
𝜕𝑦
+ z2 = 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
zx=
𝜕𝑧
𝜕𝑥
zy =
𝜕𝑧
𝜕𝑦
zxx =
𝜕2
𝑧
𝜕𝑥.𝜕𝑥
zxy=
𝜕2
𝑧
𝜕𝑥.𝜕𝑦
zyy=
𝜕2
𝑧
𝜕𝑦.𝜕𝑦

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

7. Non-Linear:
• zx
2+zy
2=0
• z.zx+z.zy=z
• zx+zy+zxx+zxy+zyy=z2
Linearity: Linear and Non Linear PDE:
• LINEAR :- If the dependent variable (z) and its partial derivatives (zx, zy, zxx, zxy, zyy)
occurs in the first power only and are not multiplied.
• NON LINEAR :- Else
EXAMPLES
Linear:
• zx +zy =0
• x.zx+y.zy=z
• zx+zy+zxx+zxy+zyy=z

8. First Order PDE (L)

9. • The general form of first order PDE is of type f(x,y,z,p,q)=0
• It can also be written as:
A.zx+B.zy+C.z = D or A
𝝏𝒛
𝝏𝒙
+B.
𝝏𝒛
𝝏𝒚
+C.z = D
where D 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, fx, and fy
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
pP + qQ = R
is required PDE
Where P,Q,R are Lagrange’s linear equation.
x2+y2 = (z-c)2 tan2α Algebric
xq-yp=0 PDE
z = xy + f(x2+y2) Algebric
py-y2 = qx-x2 PDE

11. Solution: Linear PDE of the First Order
• The PDE of the type pP + qQ =
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) = c1 and
v(x, y, z) = c2
Rules for solving pP + qQ = R
• Put the given PDE in the standard form pP + qQ = R.
• Write down Lagrange’s auxiliary equations
𝒅𝒙
𝑷
=
𝒅𝒚
𝑸
=
𝒅𝒛
𝑹
• 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) = 0
• PDE : y2p − xyq = x(z-2y)
• Step1: P= y2, Q= xy, R= x(z-2y)
• Step2: x2+y2 = c1, zy−y2 = c2
• Step3: F(c1,c2) = F(x2+y2, zy−y2)=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:
Azxx + Bzxy + Czyy + Dzx + Ezy + Fz = G ------(i)
when, G=0, equation (i) is homogeneous.
Depending on the coefficients, second-
order PDEs can be classified as:
Parabolic if B2-4AC = 0
Elliptic if B2-4AC < 0
Hyperbolic if B2-4AC > 0
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
ut=αuxx -------------(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
𝑋"
𝑋
=
1
α
𝑇’
𝑇
= 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: ut=3.uxx , 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 2nd solution
u(x,t)= 𝑖=1
∞ 4. −1 𝑛
+
1
𝑛π
sin
𝑛π𝑥
2
. exp{
−3𝑛2
𝜋2
𝑡
4
}
Example 2:
PDE: ut=a.uxx , 0<x<L, t>0
BCs: u(0,t)=0, ux(L,t)=0, t>0
IC: u(x,0)=x
We get the non-trivial solution from 2nd solution
u(x,t)= 𝑖=1
∞ 4. −1 𝑛
+
1
𝑛π
sin
𝑛π𝑥
2
. exp{
−3𝑛2
𝜋2
𝑡
4
}
Three Different solution for above equation(ut=αuxx ) is found as:
1. u(x,t)=(A.eλ𝑥+B.e
−λ𝑥)eαλ^2.t , when k=λ2 for λ >0
2. u(x,t)=(C.cosλ𝑥+B.sinλ𝑥)e-αλ^2.t , when k=-λ2 for λ >0
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:
• We 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.
1. Homogeneous Medium (ρ,s:const)
2. Heat flows in direction of decreasing Temperature
3. Heat flow rate (Q) across an area (A) is proportional to A
and temperature gradient. (ρ as proportionality Constant)
4. 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. • Q1= -kA
𝝏𝒖
𝝏𝒙 x
• Q2= -kA
𝝏𝒖
𝝏𝒙 x+Δx
• ΔQ= Q2-Q1
=kA
𝝏𝒖
𝝏𝒙 x+Δx -kA
𝝏𝒖
𝝏𝒙 x
• ΔQ= (A.Δx.ρ) . S .
𝝏𝒖
𝝏𝒕
•
𝒌
𝒔ρ
.
𝝏𝟐
𝒖
𝝏𝒙𝟐 =
𝝏𝒖
𝝏𝒕
• ut=a2uxx