Application Laplace Transform: Solving Partial Differential Equation in the Second Derivative

Application Laplace Transform: Solving Partial Differential Equation in the Second Derivative. The main aim of the research work is to assess the application of the Laplace transform in solving partial differential equations in the second derivative.

Aim and Objectives of Study

The specific objectives of the study were:
I. to determine the exact solution to the problems stated above.
II. to determine whether PDEs can be verified using substitutions
III. To determine whether any particular solution of PDEs can solve a non-homogenous problem
IV. to investigate the factors affecting the use of the Laplace transform in solving differential equation

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SIGNIFICANCE OF STUDY
The study on the application of the Laplace transform in solving partial differential equation in the second derivative will be of immense benefit to the mathematics department as the study will serve as a repository of information for other researchers and students that wishes to carry out similar research on the above topic because the study will educate the students and researchers on how to apply Laplace transforms to PDEs in the second derivatives. Finally, the study will contribute to the body of existing literature and knowledge in this field of study and provide a basis for further research
SCOPE OF STUDY
The study on the application of the Laplace transform in solving partial differential equations in the second derivative will be limited to second-order PDEs. The study will cover how to apply Laplace transforms to PDEs in the second derivatives.
DEFINITION OF TERMS
PDEs: a differential equation that contains unknown multivariable functions and their partial derivatives

REVIEW OF RELATED LITERATURE
INTRODUCTION
This chapter gives an insight into various studies conducted by outstanding researchers, as well as explained terminologies about the application of the Laplace transform in solving partial differential equations in the second derivative. The chapter also gives a resume of the history and present status of the problem delineated by a concise review of previous studies into closely related problems
LAPLACE TRANSFORM
The Laplace transform can be used to solve differential equations. Besides being a different and efficient alternative to variation of parameters and undetermined coefficients, the Laplace method is particularly advantageous for input terms that are piecewise defined, periodic or impulsive. Definition 1
Given f, a function of time, with value f (t) at time t, the Laplace transform of f is denoted f˜ and it gives an average value of f taken over all positive values of t such that the value f˜(s) represents an average of f taken over all possible time intervals of length s.
Definition 2
L[f (t)] = f˜(s) = ∫ ∞ e−stf (t) dt, for s > 0. (2.1)
A short table of commonly encountered Laplace Transforms is given in Section 7.5. Note that this definition involves the integration of a product so it will involve frequent use of integration by parts—see Appendix Section for a reminder of the formula and of the definition of an infinite integral like (2.1).
This immediately raises the question of why to use such a procedure. The reason is strongly motivated by real engineering problems. There, typically we- counter models for the dynamics of phenomena that depend on rates of change of functions, eg velocities and accelerations of particles or points on rigid bodies, which prompts the use of ordinary differential equations (ODEs). We can use ordinary calculus to solve ODEs, provided that the functions nicely behave—which means continuous and with continuous derivatives. Unfortunately, there is much interest in engineering dynamical problems involving functions that input step change or spike impulses to systems—playing pool is one example. Now, there is an easy way to smooth out discontinuities in functions of time: simply take an average value over all time. But an ordinary average will replace the function with a constant, so we use a kind of moving average which takes continuous averages over all possible intervals of t. This very neatly deals with the discontinuities by encoding them as a smooth function of interval length s.

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The amazing thing about using Laplace Transforms is that we can convert a whole ODE initial value problem into a Laplace transformed version as functions of s, simplify the algebra, find the transformed solution f˜(s), then undo the transform to get back to the required solution f as a function of t.
Interestingly, it turns out that the transform of a derivative of a function is a simple combination of the transform of the function and its initial value. So a calculus problem is converted into an algebraic problem involving polynomial functions, which is easier.
There is one further point of great importance: calculus operations of differentiation and integration are linear. So the Laplace Transform of a sum of functions is the sum of their Laplace Transforms and multiplication of a function by a constant can be done before or after taking its transform.
In this course, we find some Laplace Transforms from first principles, ie from the definition (2.1), describe some theorems that help to find more transforms, then use Laplace Transforms to solve problems involving ODEs.

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