Contact Us

Popular posts from this blog

Accuracy Of Numbers

Image
              Accuracy Of Numbers 1.  Approximate Numbers :            There are two types of numbers exact and approximate . Exact numbers are 2,4,9,13,7,2,6.45....etc . But there are numbers such as 4/3 =1.3333....  ,          √(2)= 1.41413 and finite number of digits . These may be approximated by numbers 1.3333, 1.4141 , 3.1416 respectively . Such numbers which represent the given numbers to certain degree of accuracy are called approximate numbers .  2. Significant Figures :                     The digits used to express a number is called significant digits (figures) . Thus each of the numbers 7845, 3.589, 0.4758 contains four significant figures while the numbers 0.00389 , 0.000587 and 0.0000296 contain only three significant figures since zeros only help to fix the position of the decimal point . Similarly the numbers 45000 and 7300.00 have two significant figures only . 3. Rounding Off :                   There are numbers with
Read more

PFAFFIAN Differential Equations And It's Solutions

Image
PFAFFIAN Differential Equations And It's Solutions :-      The relation              n             Σ  Fᵢ(x₁,x₂,......xₙ) dxᵢ = 0            i=1   where Fᵢ's (i=1,2,....n) are functions of all or some of the n independent variables x₁,x₂,.....xₙ is called a Pfaffian differential equation . Here we shall study Pfaffian differential equation in three independent variables which is of the form              P dx + Q dy + R dz = 0    ...........(1)    where P, Q , R are functions of x , y and z . Setting -->                        -->               X = (P,Q,R)  and dr = (dx,dy,dz), equation (1) can be written in the vector notation as                   -->  -->                  X . dr    = 0                  ..............(2)  If there exists a function μ(x,y,z) such that μ(P dx + Q dy + R dz) is an exact differential dΦ , the equation is said to be integrable and to possess an integrating factor μ(x,y,z) . In such case(1) becomes
Read more

Derivation Of Composite Trapezoidal Rule

Image
Derivation Of Composite Trapezoidal Rule :                                                   b       Let us find the value of ∫ f(x) dx                                                  a numerically by composite trapezoidal rule. Solution :              Let [a,b] be divided into n equal sub intervals of length h each . Let                            a= x₀,x₁,x₂,.......xₙ=b  be the points of sub division .  i.e x₁ = x₀+ih  for i= 0,1,2,......n By Newton's Forward Difference Interpolation  f(x) = f(x₀)+uΔf(x₀) +u(u-1)/2!  Δ²f(x₀)+........        where x = x₀+uh b                 ∫ f(x) dx = a       xₙ          ∫ [f(x₀)+uΔf(x₀)+u(u-1)/2!  Δ²f(x₀)+...]dx           x₀              n        = h∫ [f(x₀)+uΔf(x₀)+u(u-1)/2  Δ²f(x₀)+...]dx             0     =h[uf(x₀) +u²/2 Δf(x₀) +                                                                                                       n                                  (u³/6 -u²/4)Δ²f
Read more