Advance engineering mathematics
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Transcript of Advance engineering mathematics
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