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

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

PDEs definitions• General (implicit) form for one function u(x,y) :

• Highest derivative defines order of PDE

• Explicit PDE => We can resolve the equationto 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, whichare linear in the highest order derivative.

• 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

Methods of solving non-linear equations of the first order: M-1

y

zq

x

zp

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.

Example:

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

dzaq

du

dzp

Example:

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 arbitaryconstant 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

Example:

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.

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 arbitaryconstants.

Example:

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).

There are various applications, but the main three are:

• Heat equation:

• Wave equation:

• Laplace’s transform:

xxt UcU 2

xxtt UcU 2

0 yyxx UU