fractional calculus and applied analysis

Any proper rational function can be expressed as sum of rational fractions, each having a factor of Q(x). Each such fraction is known as Partial fraction..

Before using this technique of integration, let us recall what we have learnt about partial fraction..

Rational function: If P(x) and Q(x) are two polynomials in x, then the ratio of two polynomials, P(x) / Q(x) is called a rational function, where Q(x) is not equal to zero. Proper rational function: If the degree of the numerator of the rational function is less than that of the den..

Differential calculus can be considered as mathematics of motion, growth and change where there is a motion, growth, change. Whenever there is variable forces producing acceleration, differential calculus is the right mathematics to apply..

Let us began this chapter with the following statement: Often a physician may want to test how small changes in dosage can affect the body's response to a particular drug. An economist may want to study how investment changes with variation in interest rates. How the velocity of a heavy meteorite e..

..

If the degree of the numerator is greater than the degree of the denominator in a rational fraction, then the rational function is called improper rational function. Like the case of improper fractions reducible to an integer added to a proper fraction, improper ration..

is a proper rational function. This can be expressed as are the partial fractions. \ The given proper rational function is resolved into two simpler rational fractions. Note that the denominators of the rational functions are resolved into facto..

(1) Let be rational function. If is improper, divide P(x) by Q(x). Let T(x) be the quotient and P 1 (x) be the remainder, then Where T(x) is a polynomial and is a proper rational function. (2) Resolve the proper rational function in to partial fractions. (3) Write as sum of partial f..

The degree of a differential equation is the highest power of the highest order derivative after making the equation free from radicals and fractional indices as far as the derivatives are concerned. In other words, the degree of a differential equation whose terms are polynomial in the ..