Abstract
With a few exceptions, the chapters of this handbook are concerned with mathematical methods useful in the quantitative analysis of problems in science and engineering.† An important and challenging aspect of any quantitative study of a real-life phenomenon is the formulation of mathematical problems which are relevant to a better understanding of the phenomenon and to which these mathematical methods can be applied. Real-life phenomena are usually too complex to be analyzed quantitatively without idealization and simplification. It is just not feasible or practical to follow the individual motions of trillions of molecules in a cubic centimeter of air or the evolution of billions of stars in a typical galaxy. For many practical purposes, however, information about a body of matter (or a galaxy) can be obtained by treating the collection of molecules (or stars) in that body as a “continuous medium” having properties, such as density, velocity, etc., that vary smoothly throughout the body. In this continuum model, the equilibrium or motion of the body under external forces and torques, for example, may be taken as a consequence of Euler’s laws of mechanics for continuous media. The mathematical methods described in this handbook may now be used to deduce from Euler’s law an initial/boundary-value problem for differential equations that governs the mechanical behavior of the continuum. Section 4.7 of this handbook gives a sample derivation of some relevant differential equations of this mathematical model, widely known as continuum mechanics, for the study of the mechanics of deformable bodies of matter.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Thompson, M. D., et al., Case Studies in Applied Mathematics, MAA Publication ( Comm. on the Undergrad. Program in Math. ), 1976.
von Kârmân, T., and Biot, M. A., Mathematical Methods in Engineering, McGraw-Hill, New York, 1940.
Genin, J., and Maybee, J. S., Introduction to Applied Mathematics, vol. 1, Holt, Rinehart and Winston, New York, 1970.
Pollard, H., Applied Mathematics: An Introduction, Addison-Wesley, Reading, Mass., 1972.
Lin, C. C., and Segel, L. A., Mathematics Applied to Deterministic Problems in the Natural Sciences, Macmillan, New York, 1974.
Lancaster, P., Mathematics: Models of the Real World, Prentice-Hall, Englewood Cliffs, N.J., 1976.
Haberman, R., Mathematical Models, Prentice-Hall, Englewood Cliffs, N.J., 1977.
Bluman, G. W., “Dimensional Analysis, Symmetry and Modelling,” Appl. Math. Notes, 6, 122–135, 1981.
Barenblatt, G. I., Similarity, Self-Similarity, and Intermediate Asymptotics, Consultants Bureau (Div. Plenum ), New York, 1979.
Bluman, G. W., and Cole, J. D., Similarity Methods for Differential Equations, Springer-Verlag, New York-Heidelberg-Berlin, 1974.
Birkhoff, G., Hydrodynamics ( 2nd ed. ), Princeton Univ. Press, Princeton, N.J., 1960.
Bridgman, P. W., Dimensional Analysis (rev, ed.), Yale Univ. Press, New Haven, Conn., 1931 (paperback ed., 1963 ).
Sedov, L. J., Similarity and Dimensional Methods in Mechanics ( 4th ed. ), Academic Press, New York, 1959.
de Jong, F. J., Dimensional Analysis for Economists, North-Holland Publishing, Amsterdam, 1967.
Becker, H. A., Dimensionless Parameters. Theory and Method, Halsted Press (div. Wiley ), New York, 1976.
Kurth, R., Dimensional Analysis and Group Theory in Astrophysics, Pergamon Press, Oxford-New York, 1972.
Taylor, G. I., “The Formation of a Blast Wave by a Very Intense Explosion, I.: The Atomic Explosion of 1945,” Proc. Roy. Soc. A, 201, 175, 1950.
Baker, W. E., Explosions in Air, Univ. of Texas Press, Austin, Tex., 1973.
Mills, E. S., Urban Economics, Scott, Foresman, Glenview, Ill., 1972.
Bartholomew, H., Land Uses in American Cities, Harvard Univ. Press, Cambridge, Mass., 1955.
Haig, R. M., “Toward an Understanding of the Metropolis,” Quart. J. Econ., 40, 421–423, 1926.
Alonso, W., Location and Land Use, Harvard Univ. Press, Cambridge, Mass., 1964.
Luce, R. D., and Raiffa, H., Games and Decisions, Wiley, New York, 1957.
Solow, R. M., “Congestion Cost and the U se of Land for Streets,” Bell J. Econ. Manag. Sci., 4, 602–618, 1973.
Arnott, R. J., and MacKinnon, J. G., “Market and Shadow Land Rents with Congestion,” Am. Econ. Rev., 68, 588–600, 1978.
Solow, R. M., “Congestion, Density and the Use of Land in Transportation,” Swedish J. Econ., 74, 161–173, 1972.
Wan, F. Y. M., “Perturbation and Asymptotic Solutions for Problems in the Theory of Urban Land Rent,” Studies Appl. Math., 56, 219–239, 1977.
Kanemoto, Y., Theories of Urban Externalities North-Holland, Amsterdam-New York, 1980. (Also, “Cost-Benefit Analysis and the Second Best Land Use for Transportation,” J. Urban Econ. 4, 483–503, 1977.)
Robson, A. J., “Cost-Benefit Analysis and the Use of Urban Land for Transportation,” J. Urban Econ., 3, 180–191, 1976.
Wan, F. Y. M. M., “Accurate Solutions for the Second Best Land Use Problem, I: Absentee Ownership,” I.A.M.S. Tech. Report 79–30, Univ. of British Columbia, July 1979.
Wan, F. Y. M. M., “Accurate Solutions for the Second Best Land Use Problem, II: Public Ownership,” I.A.M.S. Tech. Report 83–20, Univ. of British Columbia, 1983.
Pearse, P., “The Optimum Forest Rotation,” Forestry Chron., 2, 178–195, 1967.
Fisher, I., The Theory of Interest, Macmillan, New York, 1930.
Clark, C. W., Mathematical Bioeconomics, Wiley, New York, 1976.
Samuelson, P. A., “Economics of Forestry in an Evolving Society,” Econ. Inqu., XIV, 466–492, 1976.
Faustmann, M., “Berechnung des Werthes, welchen Weldboden sowie nach nicht haubare Holzbestande fur die Weldwirtschaft besitzen,” Allgemeine Forst und Jagd Zeitung, 25, 441, 1849.
Faustmann, M., “On the Determination of the Value Which Forest Land and Immature Stands Pose for Forestry,” in Martin Faustmann and the Evolution of Discounted Cash Flow M. Gane (ed.)., Oxford Institute Paper No. 42, 1968, Oxford.
Heaps, T., and Neher, P. A., “The Economics of Forestry When the Rate of Harvest Is Constrained,” J. Environ. Econ. Manage., 6, 297–319, 1979.
Davidson, R., and Hellsten, M, M., “Optimal Forest Rotation With Costly Planting and Harvesting,” presented at the Fifth Canadian Conference on Economic Theory in Vancouver, B.C., May 1980.
Heaps, T, T., “The Forestry Maximum Principle,” presented at the Can. Appl. Math. Soc. Annual Meeting (Montreal), May 1981.
Wan, F. Y. M., and Anderson, K., “Ordered Site Access and Age Distribution in Harvesting Once-and-for-all Forests,” Stud. Appl. Math., 65, 1983.
Anderson, K., and Wan, F. Y. M. M., “Finite and Infinite Sequences Harvests of Ongoing Forests,” I.A.M.S. Tech. Rep. 81–14, Univ. British Columbia, 1981.
Flugge, W., and Riplog, P. M. M., “A Large Deformation Theory of Shell Membranes,” Tech. Rep. No. 102, Div. of Eng. Mech., Stanford Univ., Sept. 1956.
Johnson, M. W., “On the Dynamics of Shallow Elastic Membranes,” Theory of Thin Elastic Shells, (ed. W. T. Koiter ), Proc. IUTAM Symp. Delft, 1959.
Simmonds, J. G., “The Finite Deflection of a Normally Loaded Spinning Elastic Membrane,” J. Aerospace Sci., 29, 1180–1189, 1962.
Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity ( 4th ed. ), Dover Publications, New York, 1944.
Reissner, E., “Symmetric Bending of Shallow Shells of Revolution,” J. Math. Mech., 7, 121–140, 1958.
Clark, R. A., “On the Theory of Thin Elastic Toroidal Shells,” J. Math. Phys., 29, 146–178, 1950.
Clark, R. A., “Asymptotic Solutions of Elastic Shell Problems,” Asymp. Soln. of ODE & Appl. (ed. C. H. Wilcox ), Wiley, New York, 1964, 185–209.
Seaman, W. J., and Wan, F. Y. M., “Lateral Bending and Twisting of Thin-Walled Curved Tubes,” Stud. Appl. Math., 53, 73–89: 1974.
Reissner, E., and Wan, F. Y. M., “Rotating Shallow Elastic Shells of Revolution,” J. SIAM 13, 333–352, 1965. (Also, T. V. Karman in Memoriam SIAM Publication, Philadelphia, 1965, 159–178.)
Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells ( 2nd ed. ), McGraw-Hill, New York, 1959.
Larkin, P. A., “Scientific Technology Needs for Canadian Shelf-seas Fisheries,” Interim Report, Fisheries Research Board of Canada, Ottawa, Feb. 1975.
Ludwig, D., “Some Mathematical Problems in the Management of Biological Resources,” Appl. Math. Notes, 2, 39–56, 1976.
Aronson, D. G., and Weinberger, H. F., “Nonlinear Diffusion in Population in Mathematics, vol. 446 (Partial Differential Equations and Related Topics), Springer, Berlin, 1975, 5–49.
Wan, F. Y. M., “Bifurcation Theory and the Two Hundred Mile Fishing Limit,” Appl. Math. Notes, 4, 74–87, 1977.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Van Nostrand Reinhold
About this chapter
Cite this chapter
Wan, F.Y.M. (1990). Mathematical Models and Their Formulation. In: Pearson, C.E. (eds) Handbook of Applied Mathematics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-1423-3_19
Download citation
DOI: https://doi.org/10.1007/978-1-4684-1423-3_19
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-442-00521-4
Online ISBN: 978-1-4684-1423-3
eBook Packages: Springer Book Archive