Using Higher Mathematics

 With the type of  application   software  available now a days,    you can investigate different situations using higher mathematics.

 You can experiment with and learn more about mathematical functions we read in higher mathematics and apply to different situations easily.  You can use your own algorithm . The functions and procedures to implement an algorithm ,like   conditional selections ,summations , other elementary and  special functions and environment for doing iterations are found  available in these software.

 For example , software like ms-excel, ms-access for manipulation of huge amount of data with graphical presentation facilities, special application software in different fields   are available  which provide environment for manipulating data your way. (ex: ‘ofm’ in petroleum engineering applications).

You can use   following  steps.

 1)Analyze the problem-

 Define the problem clearly by understanding what output are required and what inputs will be needed.

To accomplish this you must have an understanding of how inputs can be used to produce the desired output.

 2)Develop a solution-

 We determine and select a solution for the problem. An acceptable and complete solution is  refined from the preliminary solution identified in the analysis step.

 Sometime solution is simple sometimes complex.

 3)Computerize the solution-

 Consists of translating the solution into a usable form in your computer. Constructing a graphical user interface and providing the necessary computer instruction using the facilities/environment available in the software you are using.

 4)Test and correct the application.

 It is impossible to construct a successful application for a problem that is not fully understood.

 Following  paragraphs discuss the mathematics you come across while dealing with your problem.

 1)In arithmetic(and in algebra) one deals mainly with constant quantities (which characterize states), in analysis one deals with variable quantities(which describe processes).

If a relation between two real variables y and x is such that when x is given y is determined then y is said to be a function of x.

The range of values of x for which y is defined as a function of x is called the interval of definition.

 2)Algebraic function – Any relation of the type p(x)y^m  +q(x)y^m-1 +     +u(x)y + v(x)=0

 Where p(x),q(x),…v(x) are polynomials of any degree in x, defines y as an algebraic function of x.

 3)Transcendental functions- Certain functions like sinx, e^x, logx are not definable in terms of a finite number of polynomials and are not therefore of algebraic type.

 4) Elementary functions are algebraic and transcendental(or nonalgebraic) function occurring in calculus.

All others that can be constructed from these by adding, subtracting, multiplying, dividing, or forming a function of a function are elementary function.

Beyond the elementary functions there are functions  called the special function which arise while solving different problems influencing special way.

 Few special functions are so rich in meaning and influence( Gamma function, Rieman zeta function, Elliptic function) that studied individually.

 5)The starting point of any discussion:  physical principles involved

 For example it has been observed that radioactive materials decay at a rate proportional to the amount of material present, that heat passes from a warmer to a cooler body at a rate proportional to the temperature difference,That objects move about in accordance with Newton’s laws of motion and that isolated insect populations grow at a rate proportional to the current population.

 Each of these statements involves a rate of change (derivative) and connected to the variable involved with one another by means of the basic scientific principles that govern the process, consequently, when expressed mathematically, takes the form of a differential equation.

 6)Differential equations is the most important part of mathematics for understanding the physical sciences. It is the source of most of the ideas and theories which constitute higher analysis: Power series, Fourier series, the Gamma function and other special functions ( Legendre, Lebesigue,  Bessel ,Hermite, Laguerre ,Dirac delta function and many more), integral equations etc.

Special functions vary  widely with respect to their origin, nature, and applications.

  A partial differential equation is one involving more than one independent variable, so that the derivative occurring in it are partial derivatives (usually on time t and on one or several space variables).

Only slmplest physical systems can be modelled by ordinary differential equations where as most problems in fluid mechanics, elasticity, heat transfer, electromagnetic theory and other areas lead to partial differential equations.

The mathematical rigor is to use different style to suit the situation. The taste differs with the person using the function and approach to finding solution. Few prefer to study and use large classes of functions where as few others like  using  outstanding individuals like special functions.

 7) Basic differential calculus-

As stated earlier differential calculus forms the mathematical basis of describing many processes. In many engineering applications knowledge of a function’s value is insufficient to solve a given problem. Additional information such as the rate at which the function changes is often required.

 Calculus text books give standard formula for the first derivative of a function. You can keep a table listing several common function with their first derivatives.

Taylor series expansion and simple applications-

One of the most useful tools in numerical analysis in the Tayler series expansion which provides a means for converting most well behaved functions into simple polynomials.

Ex:   Taylor series expansion of e^x = e^x0 + (x-x₀)/1! e ^x0 +---

Ex:   Find the 2^nd term approximation to the derivative of an arbitrary function f(x).

f(x) = f(x₀) + (x- x₀)/1!  df/dx  + R2 ----

 df/dx yields    = (f(x) –f(x₀))/(x-x₀)

8)Solution of differential equation-

The solution of a differential equation is a function that does not contain any derivatives with respect to the independent variables, that satisfies any specified conditions and that satisfies the relationship defined by the differential equation.

Initial and boundary conditions.

In solving second-order  differential equations, two additional pieces of information are required to specify the constants and obtain a unique solution to the differential equation. In general , each derivative in the equation requires one integration constant and one additional piece of information.

 The additional information required to obtain a unique solution to a given diff eqtn are the initial and/or boundary conditions of the problem. These conditions are so called because they can only be known either initially or on the boundaries.

 The solution to differential equation is found using different methods.

9) It is normally a simple method to verify that a given function y=y(x) is a solution of an equation like F(x,y,dy/dx,d²y/dx²,…,d²y/dx²)=0

 All that is necessary is to compute the derivatives of y(x) and to show that y(x) and these derivatives , when substituted in the equation, reduce it to an identity in x.

 For example  you can use simple formula for differentiation to see that y=e^2x and y=e^3x are both solution of the second order equation y”-5y’+6y=0 and more generally that

 Y=c1e^2x+c2e^3x is also a solution for every choice of the constant c1 and c2.

 In most cases procedures of this kind are easy to apply to a suspected solution of a given differential equation.

 There are certain standard types of first order equation for which routine methods of solution are available. One such example is use of a known solution to find another.

 10)  By numerical method we understand the methods of solving problems which are reduced to arithmetic and certain logical operation on numbers, that is to the operation performed by a computer.

  For example you can use Taylor series expansion described earlier to find first order derivative and second order derivative from tabular data(ex: data from an experiment). You can use Excel for this computation.

 In many situations it is often convenient to express the equations  in curvilinear coordinate system to solve the physical problem.

 11)Basic linear algebra

 This branch of mathematics deals with vector, matrices and linear transformations , solution of linear equations.

 Vectors, unit vectors are introduced and used because they simplify derivations or permit a clearer, more geometric or improved analytic treatment of physical phenomena.

 A matrix is an array of quantities, usually numbers, but sometimes functions, in m rows and n columns. Systems of linear simultaneous equations can be expressed in matrix form and then manipulated as a single entity.

 Following simple application shows using matrix algebra in Excel.

 You can use excel chart wizard to see the effect pictorially by creating a two dimensional figure using a range of data and then changing/transforming the data range by operation like changing scale to adjust the size and by using different scale factors in the x- and y- directions to adjust its shape and translating a basic shape to a different position in the plane.

Use the option create an xy(scatter) chart with connecting lines. Create a chart using a data series. Then use modify existing chart by modifying range of data used by the chart.

  Ex:   Fill columns x and y by choosing  value of x and  y. Now scale the axis differently in another range of data, x’=k_x x and y’=k_y y . When you modify and select the new data range in the chart wizard the shape of the figure changes. The translation of the shape is accomplished by means of the linear transformation x'=a+x , y'= b+y by selecting another range of data.