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Introduction

This book is intended for students who have already studied basic
mathematics and need to study the methods of advanced mathematics. It
covers one content areas: Operational Calculus. This section belongs to typical
course of Advanced Mathematics for engineers. Our book contains guidelines for
solutions of different problems by methods of operational calculus.

Many useful examples and exercises are presented in the textbook.

Each sections contains basic mathematical conceptions and explains new

mathematical methods from the simplest to the most difficult problems.

Operational calculus, also known as operational analysis, is a technique by
which problems in analysis, in particular differential equations, are transformed
into algebraic problems, usually the problem of solving a polynomial equation.
The main idea of operational calculus is the thought of representing the

processes of calculus, derivation and integration, as operators. In particular, the
Laplace transformation reduces the problem of solving a differential equation to
an algebraic problem.

Laplace transform is powerful operational tool for solving constant coefficients
linear differential equations. The process of solution consists of three main steps:

• The given “hard” problem is transformed into a “simple” equation.

• This simple equation is solved by purely algebraic manipulations.

• The solution of the simple equation is transformed back to obtain the
solution of the given problem.

In the final section, we describe many applications of Laplace transform:

ordinary differential equations, systems of differential equations, integral
equations and boundary problems for partial differential equations.
Appendix contains a table of Laplace transform.

The authors welcome reader’s suggestions for improvement of future editions
of this book.

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