Computer Algebra Systems and its Computational Application to the Thermo-Elastic Problem

K. L. Verma¹, Rajesh Kumar²

¹Principal, Government College Sujanpur, Hamirpur

²Assistant Professor, Mathematics, Government Post Graduate College Hamirpur

*Corresponding Author: klverma@netscape.net

ABSTRACT:

What task should be played by symbolic mathematical computation facilities in Pure and applied Mathematics, scientiﬁc and engineering “problem solving environments”? Drawing upon standard facilities such as numerical and graphical libraries, symbolic computation should be useful for the creation and manipulation of mathematical models. Symbolic representation and manipulation can potentially play a central organizing role in PSEs since their more general object representation allows a program to deal with a wider range of computational issues. Symbolic processing power of software as MathCAD (MathCAD explorer version 8) is used to demonstrate the basics as well as its application dispersion of thermoelastic waves.

KEY WORDS:

1. INTRODUCTION:

The hunt for tools to make computers easier to use is as old as computers themselves. In the beginning years of the computer, and following the release of high level languages(HLL) ALGOL 60, Fortran, C, and C++, it was thought that the ready availability of compilers for such high-level languages would eliminate the need for professional programmers— instead, all educated persons, and certainly all scientists and researchers, would be able to write programs whenever needed.

While some aspects of programming have been automated, deriving programs from speciﬁcations or models remains a difficult step in computational sciences. Furthermore, as computers have gotten substantially more complicated it is more difficult to get the fastest performance from these systems. Advanced computer programs now depend for their efficiency not only on clever algorithms, but also on constraining patterns of memory access. Symbolic computation tools (including especially computer algebra systems) are now generally recognized as providing useful components in many scientiﬁc computing environments.

1.1 Symbolic Vs. Numeric

How a symbolic computing system distinct from a numeric one? For example, if one some one sought from a numeric program to “solve for x in the equation sin(x) = 0” it is possible that the answer will be some 32-bit quantity that we could print as 0.0. There is generally no way for such a program to give an answer “ ”. A program that could provide this more elaborate symbolic, non-numeric, parametric answer dominates the merely numerical from a mathematical perspective. The single numerical answer might be a suitable result for some purposes: it is simple, but it is a compromise. If the problem-solving environment requires computing that includes asking and answering questions about sets, functions, expressions (polynomials, algebraic expressions), geometric domains, derivations, theorems, or proofs, then it is plausible that the tools in a symbolic computing system will be of some use.

1.2 Computer Algebra Systems

Computer Algebra Systems (CAS) are software packages used in the manipulation of mathematical formulae in order to automate tedious and sometimes difficult algebraic tasks. The principal difference between a CAS and a traditional calculator is the ability of the CAS to deal with equations symbolically rather than numerically. Specific uses and capabilities of CAS vary greatly from one system to another, yet the purpose remains the same: manipulation of symbolic equations. In addition to performing mathematical operations, CAS often includes facilities for graphing equations and programming capabilities for user-defined procedures.

Computer Algebra Systems were originally conceived in the early 1970’s by researchers working in the area of artificial intelligence. The first popular systems were REDUCE, DERIVE and MACSYMA. Commercial versions of these programs are still available. The two most commercially successful CAS programs are Maple and Mathematica. Both programs have a rich set of routines for performing a wide range of problems found in engineering applications associated with research and teaching. Several other software packages, such as MATHCAD and MATLAB include a MAPLE kernal for performing symbolic-based calculation. In addition to these popular commercial CAS tools, a host of other less popular or more focused software tools are available, including AXIOM and MuPAD.

Advances in technology have led to the creation a new type of research environment, MathCAD is a powerful Computer Algebra System with a wide range of applications that make it an important addition to a scientists/researchers toolkit. Mathcad is the industry standard calculation software for technical professionals, educators, and college students. Mathcad is as versatile and powerful as programming languages, yet it’s as easy to learn as a spreadsheet. Plus, it is fully wired to take advantage of the Internet and other applications you use every day. Mathcad lets you type equations as you’re used to seeing them, expanded fully on your screen. In a programming language, equations look something like this: Although there are several excellent CAS programs available, MathCAD may be used in study for the following reasons:

Pages look like a textbook:

Since the math and text on the computer screen appear similar to what students see in a textbook, students find the worksheets easy to read. CAS is to mathematics as a word processor is to writing; the pages on the screen become an interactive math book.

Relatively easy to learn:

No special "programming syntax" is needed. Since CAS are by their nature very complex programs the learning curves for the leading CAS systems tend to be a bit extreme. By using Mathcad 8, we tried to avoid making the CAS another class on top of calculus.

2.1 Basics of MathCAD (MathCad explorer version 8)

MathCAD is a computer software program that allows you to enter and manipulate mathematical equations, perform calculations, analyze data, and plot data. This combination makes Mathcad an invaluable tool to Mathematician, Researchers and scientist.

2.2 The Workplace

After starting the Mathcad program, we will see a blank worksheet. Blank worksheet is used to write out our assignments and to practice working with MathCad. The worksheet is like a blank sheet of paper on which we can construct graphs, add pictures, write math formulas and add text just like the pages in our calculus book. One difference however is that Mathcad worksheets contain "live" mathematics that you can interact with in a dynamic manner.

2.3 Mathematical Expressions

Unlike other technical software, Mathcad does mathematics in the same we do. Mathematical expressions in MathCad look the way we see them in a text book or in reference books

2.4 Illustarion

The only difference is that Mathcad’s equations and graphs are live. Change any data, variable, or equation, and Mathcad recalculates the math and redraw the graphs.

3. Basics Simple Calculations (examples are shown as an icons)

Open up the Arithmetic Palette by double clicking on the calculator icon in the Math Palette. If the

Math Palette is not on your desktop then open it from the View Menu and enter the value square root symbol is on the arithmetic palette. On typing the equal sign MathCad will display the answer just like a regular calculator

Vector and Matrices

Solving Equations, Differential and integral Calculus

Basic Graphing

Open up the Graphing Palette by double clicking on the corresponding icon in the Math Palette.

Symbolic manipulation

Live Symbolics and keywords when transforming algebraic expressions. Live Symbolics will update if you change the definitions of any of the expressions on the left side. Expressions solved through the Symbolics menu must be re evaluated when the definitions change.

Programming The Programming toolbar buttons give you quick access to operators used in a program. To learn what a button does, move the cursor to the main worksheet window and allow the cursor to hover over the button until.

Graphics: The Graph toolbar buttons allow you to insert 2D and 3D graphs

4. Applications to Solid Mechanics

Thermoelastic Theories with thermal relaxations: The generalized coupled field equations governing dynamic thermoelastic processes for homogeneous heat conducting isotropic materials and in the absence of body forces and heat source can be written as

Thermoelastic theory without energy dissipation:

The fundamental equations for homogeneous isotropic materials and in the absence heat sources and body forces, in the context of generalized thermoelasticity developed by [1] in terms of del operator ( ) and the Laplace

To numerically investigate the effects of thermal relaxation time and coupling on the behavior of thermoelastic waves in generalized theories of thermoelasticities, choosing nickel material

Dispersion curves are drawn for thermal relaxation time 0.19ps, 1.9 ps, 19.0 ps and 190ps (1 pico seconds (1ps) = secs.), and it is shown that amount of attenuation or damping is shown to have large values when the slope of dispersion curves is steep.

REFERENCES:

1. Chandrasekharaiah, D. S., 1996, One dimensional wave propagation in the linear theory of thermoelasticity without energy dissipation. J. Thermal Stresses 19, 695- 710.

2. Chandrasekharaiah, D. S. 1996, A Uniqueness theorem in the theory of thermoelasticity without dissipation, J. Thermal Stresses 19(3), 267-272.

3. Green, A. E., and Lindsay, K. A., 1972, Thermoelasticity. J. Elasticity 2 1-7.

4. Green, A. E., and Naghdi, P.M. 1993, Thermoelasticity without Energy dissipation. J. Elasticity, 31, 189- 208.

5. Lord H. W., and Shulman, Y. A. 1967, A Generalized Dynamical Theory of Thermoelasticity. J. Mech. Phys. Solids 15, 299-309.

6. Nayfeh, A. H., and Nasser, S. N. 1972, Transient thermoelastic waves in a half-space with thermal relaxations, J. Appl. Maths, Physics 23, 50-67.

7. Suh, C. S., and Burger, C.P. 1998, Effects of Thermo-mechanical coupling and relaxation times on wave spectrum in dynamic theory of generalized thermoelasticity, J. Appl. Mech., 65, 605-612.

8. Verma, K. L. On the Thermo-mechanical coupling and Dispersion of Thermoelastic Waves with thermal relaxations, International Journal of Applied Mathematics & Statistics, Vol. 3, No. S05, 34-50, 2005.

9. http://www.mathsoft.com/mathcad/explorer/ (This free version of Mathcad is actually the full version, neutered so that it can't save any worksheets. It will run the Astro Utilities fine and allow you some file IO (user constants in the Utilities), through use of read/write components.)

Received on 23.12.2014 Accepted on 10.02.2015

Research J. Engineering and Tech. 6(1): Jan.-Mar. 2015 page 191-194

DOI: 10.5958/2321-581X.2015.00028.8