This document discusses methods for solving first order non-linear partial differential equations. It defines ordinary and partial differential equations, and describes four standard forms for first order partial differential equations: 1) equations not involving independent variables, 2) equations reducible to standard form through change of variables, 3) separable equations, and 4) Clairaut's form where the equation can be written as z = px + qy + f(p,q). Examples are provided for each method. Partial differential equations have applications in fields like fluid mechanics, heat transfer, and electromagnetism. The main applications discussed are the heat, wave, and Laplace equations.

1. ADVANCE ENGINEERING
MATHEMATICS
MADE BY:
MIHIR JAIN-36
AMIT JHALANI-37
ARNAV BHATT-38
ANMOL KHARE-39
SAJAL KHARE-40
PPT ON:
FIRST ORDER NON-LINEAR PARTIAL DIFFERENTIAL
EQUATION

2. Definition: A differential equation which
involves partial derivatives with respect to
two or more independent variables is called a
partial differential equation.
Ordinary Differential Equation:
Function has 1 independent variable.
Partial Differential Equation:
At least 2 independent variables.

3. 3
PDEs definitions
• General (implicit) form for one function u(x,y) :
• Highest derivative defines order of PDE
• Explicit PDE => We can resolve the equation
to the highest derivative of u.
• Linear PDE => PDE is linear in u(x,y) and
for all derivatives of u(x,y)
• Semi-linear PDEs are nonlinear PDEs, which
are linear in the highest order derivative.

4. • If the number of arbitrary constants to be
eliminated is equal to the number of
independent variables, the p.d.e formed is of
the first order
F(x, y, u, p, q) = 0

5. Methods of solving non-linear equations
of the first order: M-1
y
z
q
x
z
p






 Equations involving only p and q and no x, y, z : Such equation are of
form f(p,q)=0. Here :
 The complete solution is z = ax + by + c, where a&b are connected by
the relation f(a,b)=0.
 Solving for b, we get b=T(a).
 Hence the complete integral is z= ax + yT(a) + c, where a&c are
arbitary constants.

6. Example:

7. M-2:
Equations not involving the independent variable:
• Such equation are of form f(z,p,q)=0
• Assume that z=T(u), where u=x+ay, so that
• Subtitute the values of p and q in the equation.
• Solve the resulting ordinary differential equation
in the given equation in z and u.
• Replace u by x+ay
du
dz
aq
du
dz
p



8. Example:

10. M-3: Separable equations
• Such equations are of form f1(x,p)=f2(y,q)
• In such equations z is absent and the terms
involving x and p can be seperated from those
involving y and q.
• Assume that each side is equal to an arbitary
constant a. Then f1(x,p) = a = f2(y,q)
• Solving f1(x,p)=a, suppose we get p=F1(x) and
q=F2(y).
• Subtituting p anq in dz=p.dx +q.dy, we get
• Dz=pF1(x)dx + qF2(y)dy …….(1)
• Integrating (1), we get    bdyyFdxxFz )(2)(1

11. Example:

12. M-4: Equation reducible to standard form
• Many non-linear p.d.e of the first order do not
fall under any of the 4 standard forms.
However, it is possible to reduced a given
equation to any of the four forms by a change
of variable.

13. M-5: Clairauts’s form
• A first order p.d.e is said to be Clairaut’s form
if it can be written in the form
z = px + qy + f(p,q)
• The solution of this equation is :
z = ax + by + f(a,b), Where a and b are arbitary
constants.

14. Example:

15. Partial differential equations are fundamental
to :
• fluid mechanics
• heat transfer
• solid mechanics
• electrical engineering
• magnetism, relativity, planetary motion....
• Basis of many technical engineering jobs using e.g. CFD or
FEM software.
• New developments, (e.g. chaos, stochastic PDE’s for derivative
modelling).

16. There are various applications, but the
main three are:
• Heat equation:
• Wave equation:
• Laplace’s transform:
x xt UcU 2

x xtt UcU 2

0 yyxx UU