Mathquery

Sturm - Liouville Problem:   The Sturm - Liouville Problem is a special class of linear homogeneous boundary value problem given in the following definition.  Definition :      Consider a second order homogeneous linear differential equation of the form   d/dx [p(x) dy/dx ] + [q(x) + λ γ(x)]y=0, a≤x≤b                                                          ...............(1)   where p, q and γ are real - valued continuous functions on [a,b] and λ  is real parameter . Further consider the following two sets of prescribed conditions        a1 y(a) + a2 y'(a) = 0}                                            }...................(2)   ...

Laplace Transform Of Periodic Functions : Definition :            A function f(t) is said to be periodic , with a period ' l ' , if it satisfies the functional equation                   f(t±l) = f(t)     So, if t>0 , a periodic function f(t) can be written as            f(t) = f(t + nl) , n= 0,1,2.. ......(1)      For example sin t = sin(t+2πn) , n=0,1,2.... is a periodic function with period 2π .       In case a function f(t) is periodic , the Laplace transform can be expressed as an integral over one cycle of the function instead of an integral over an infinite range.            The transform of equation(1) is                       ∞           f̅(p) = ∫ e⁻ᵖᵗ f...

Unit Impulse Function                 The above figure shows a function which has a zero value when t is negative , rises instantaneously to a value 1/t₀ when 0<t<t₀ , and is zero thereafter.                  If we let t₀--> 0 , this function  tends towards infinity as t₀--> 0, but width decreases inversely with the magnitude ; hence area under the curve remains finite . The unit impulse function is represented by δ(t) .  Among Physicists, the unit impulse function is referred to as the 'Dirac δ-function ' after the name of great Physicist Dirac , who is the first to use this function in systematic manner .  The Dirac δ - function at the point t=a as in above figure is represented by δ(t-a) . Thus                       {0 for t < a         δ(t-a) ={ 1/t₀ for a...

Unit Step Function :         A unit step function is a function which has a zero value for t≤t₀ and then rises instantaneously to a sustained unity value.  This is shown in the figure below. In electric circuit , and in mechanical system it could represent a force suddenly impressed on the system .            This unit step function starting at zero time will be designated by u(t) , and that starting at time t₀ is u(t-t₀) . Thus                           { 0 for 0<t≤t₀            u(t-t₀) ={                           { 1 for t>t₀               .....(1)    The Laplace transform of unit step function is                       ...

The Shifting Theorem :            If L{f(t)}= f̅(p), then           L{eᵃᵗf(t)} = f̅(p-a) , p>a Proof :     By definition                         ∞             f̅(p) = ∫ e⁻ᵖᵗ f(t) dt                       0                                    ∞ Therefore, f̅(p-a) = ∫ e^-(p-a)t f(t) dt                                   0                                  ∞                               = ∫ e⁻ᵖᵗ[eᵃᵗ f(t)] dt...

Existence And Property Of Laplace transforms : Existence Of Laplace Transforms :      In order to discuss the theorem on existence of transform we need the following definition . Definition :         A function f(t) is said to be piecewise continuous in a finite range a≤t≤b , if it is possible to divide the range into a finite number of sub-intervals such that f(t) is continuous inside each sub-interval and approaches finite values as t approaches either end of any interval from the interior. Theorem :        Let f(t) be a function which is piecewise continuous on every finite interval in the range t≥0 and satisfies          |f(t)|≤Me^αt , for all t≥0        .........(1)    and for some constants α and M . Then the Laplace transform of f(t) exists for all p>α . Proof :           Since f(t)...