A
AAAS AASE Ablation ABSOLUTE EFFICIENCY ABSOLUTE PRESSURE ABSOLUTE TEMPERATURE ABSORPTIVITY ACCELERATION PRESSURE GRADIENT ACCIDENTS TO CHEMICAL PLANT Accommodation coefficient Acentric factor ACETIC ACID ACETONE ACETYLENE COMBUSTION ACHE'S ACID RAIN ACID VIOLET 19, MONOMETHYL ACKERMANN CORRECTION FACTOR ACOUSTIC CAVITATION ACOUSTIC FIELDS ACOUSTIC FLOWMETERS ACOUSTIC INSTABILITIES ACOUSTIC VIBRATION ACOUSTIC WAVES ACOUSTICS OF BONE ACRYLIC CATION RESINS, ACRS Activity coefficient ADAPTIVE GRIDS ADAPTIVE TWO-DIMENSIONAL MESH REFINEMENT METHOD Added mass ADDITIVES ADENOSINE TRIPHOSPHATE, ATP ADHESION BETWEEN LIQUIDS Adiabatic conditions ADIABATIC DISC TEMPERATURE ADIABATIC EXPONENT ADIABATIC PROCESSES ADIABATIC SATURATION TEMPERATURE ADIABATIC SHEAR BAND ADIABATIC THROTTLING Adiabatic wall temperature ADSORBATE ADSORBENT ADSORBERS Adsorption ADSORPTION OF GASES ADSORPTIVE BUBBLE TECHNIQUES ADSUBBLE TECHNIQUES ADVANCED BOILING WATER REACTOR, ABWR Advanced gas-cooled reactor AEA TECHNOLOGY AELOPILE OF HERO AERATION AERODYNAMIC COEFFICIENTS AERODYNAMIC EFFICIENCY AERODYNAMIC FLOW SPECTRUM AERODYNAMIC RESISTANCE OF ATMOSPHERE Aerodynamics AEROGELS AEROGENERATORS AEROSOL FILTRATION Aerosols AEROSOLS, CLIMATIC EFFECTS AFTERBURNING AGGLOMERATES Agglomerates and complex shape particles AGGLOMERATION OF PARTICLES AGITATED VESSEL HEAT TRANSFER AGITATED VESSEL MASS TRANSFER AGITATED VESSELS AGITATION CAVITATION NUMBER AGITATION DEVICES AGR AICHE AIR (PROPERTIES OF) AIR CARRYUNDER AIR CONDITIONING AIR COOLED CONDENSERS AIR COOLED HEAT EXCHANGERS AIR COOLERS AIR CUSHIONS AIR CYCLE HEAT PUMPS AIR CYCLE REFRIGERATION AIR EJECTOR AIR JET ENGINES AIR POLLUTANTS AIR POLLUTION AIR SPRAYS AIR-TO-AIR HEAT PUMPS AIR-TO-WATER HEAT PUMP AIR-WATER SYSTEM AIRCRAFT, AERODYNAMICS OF AIRCRAFT, PARABOLIC FLIGHTS AIRLESS DRYING AISI AL-SI PHASES FORMED FROM COMBUSTIBLE SOLID RESIDUES Albedo ALBEDO, OF EARTH ALCOHOL ALDEHYDES ALFVEN NUMBER ALFVEN WAVES ALGEBRA, FUNDAMENTAL THEOREM OF ALIGNED MAGNETIC FIELD ALIPHATIC HYDROCARBONS ALKANES ALLEN FLOW ALLOY SOLIDIFICATION ALLOY STEELS ALLOYS ALPHA PARTICLES ALTERNATIVE ENERGY SOURCES Alternative formulations ALUMINA ALUMINOSILICATE ZEOLITES ALUMINUM ALUMINUM COMBUSTION ALUMINUM OXIDE AMAGAT'S LAW AMERICAN ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE, AAAS AMERICAN INSTITUTE FOR CHEMICAL ENGINEERS, AICheE AMERICAN IRON AND STEEL INSTITUTE, AISI AMERICAN PETROLEUM INSTITUTE, API AMERICAN SOCIETY OF HEATING, REFRIGERATION AND AIR-CONDITIONING ENGINEERS, ASHRAE, INC AMERICAN SOCIETY OF MECHANICAL ENGINEERS, ASME AMMONIA COMBUSTION AMMONIUM NITRATE FERTILIZER AMMONIUM PERCHLORATE AMORPHOUS AND NANOSTRUCTURED SILICON FILMS AMPLIFIED SPECTRUM ANABOLISM ANALOGY BETWEEN HEAT AND MASS TRANSFER ANALYTICAL TREATMENT OF FINS WITH TEMPERATURE-DEPENDENT SURFACE HEAT FLUX ANEMOMETERS (LASER DOPPLER) ANEMOMETERS (PULSED THERMAL) ANEMOMETERS (VANE) ANEROID BAROMETER ANGLED TURBULATORS Angular discretization methods ANL ANNEALING ANNULAR DISPERSED FLOW ANNULAR FIXED BEDS Annular flow ANNULAR FLOW SYSTEM ANNULAR FLOW, IN LIQUID-METAL BOILING ANNULAR PIPES ANNULAR POROUS MEDIUM ANODE Anomalous diffraction ANTI-FREEZE ANTI-NEUTRINO ANTIDERIVATIVE FUNCTION ANTIGRAVITY ANZAAS API API GRAVITY Application to nongray media Application to rough surfaces Applications of inverse radiation analysis APPLICATORS Applied problems AQUEOUS SOLUTIONS WITH NONLINEAR SURFACE ENERGY AQUIFER ARC DYNAMIC ARCHIMEDES FORCE ARCHIMEDES NUMBER ARCHIMEDES PRINCIPLE ARGON ARGON-ION LASER ARGONNE NATIONAL LABORATORY, ANL ARMAND CORRELATIONS, FOR VOID FRACTION IN ANNULAR FLOW ARMORED VEHICLE CABIN AROMATIC CHEMICALS, OR AROMATICS AROMATIC HYDROCARBONS ARRHENIUS EQUATION ASCENDING SLUG FLOW IN A VERTICAL PIPE ASH FORMATION ASH LAYER MODEL ASHRAE ASME ASPECT RATIO ASSOCIATE CATALYSIS Association for Applied Solar Energy, AASE ASYMMETRIC HEATING ASYMPTOTE ASYMPTOTIC EXPANSION ASYMPTOTIC METHODS ATMOSPHERE ATMOSPHERE (PHYSICAL PROPERTIES OF) ATMOSPHERIC PRESSURE GLOW DISCHARGE ATOM ATOMIC ENERGY AUTHORITY ATOMIC HYDROGEN BEAM ATOMIC MEDIA ATOMIC NUMBER ATOMIC PORES ATOMIC SPECTROSCOPY ATOMIC SURFACE DIFFUSION ATOMIC WEIGHT ATOMISTIC DEFORMATION ATOMISTIC-CONTINUUM ADAPTIVITY ATOMISTIC-TO-CONTINUUM COUPLING ATOMIZATION ATOMIZATION TURBULENT ATOMIZERS ATOMIZING LIQUIDS ATOMIZING SPRAY ATTENUATION COEFFICIENT, PHOTON TRANSMISSION ATTENUATION, OPTICAL ATTRACTORS AUGMENTATION OF HEAT TRANSFER, SINGLE PHASE AUGMENTATION OF HEAT TRANSFER, TWO-PHASE AUSTENITE-MARTENSITE TRANSFORMATION Australian and New Zealand Association for the Advancement of Science, ANZAAS AUTO CORRELATION AUTO-IGNITION AUTOMATIC SAMPLING AUTOMOTIVE GAS TURBINES Available data for molecules of practical interest AVERAGE FILM FLOW RATE MEASUREMENT AVERAGE PHASE VELOCITY AVERAGE VOID FRACTION MEASUREMENT AVIATION FUEL-AIR REACTION AVOGADRO NUMBER AVOGADRO'S LAW AXIAL FLOW COMPRESSOR AXIAL FLOW FANS AXIAL TURBINE AXISYMMETRIC JET AXISYMMETRIC NARROWING AZEOTROPES D'ALEMBERT PARADOX
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ASYMPTOTIC EXPANSION

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Asymptotic expansion is one of the methods of approximating functions by their asymptotic expressions. The function g(x) is an asymptotic expression of the function f(x) for x → a , if

Hence it follows that

i.e., a relative error of replacing the function by its asymptotic expression for x → a tends to zero.

An asymptotic expansion is a special kind of an asymptotic expression, in which the function f(x) is approximated by partial sums of some convergent or divergent series

so that partial sums Sk = φ0(x) + φ1(x) + ... + φk(x) are the asymptotic expressions of the function f(x) and

i.e., each successive partial sum is the best asymptotic expression for the function f(x).

If the form of the function φi(x) is unknown beforehand, there exists a broad range of possibilities for choosing particular asymptotic expansions.

In the theory of fluid dynamics and heat mass transfer, and also in other branches of mechanics of continuous media the method of perturbation has found extensive application, in which approximate solutions of the problem, which is defined by a single or a system of (integro) differential equations and by the corresponding boundary conditions, are found through asymptotic expansions of dependent variables when one or several parameters of the problem (for instance, numbers M, Re, Pr, Sc etc., or their complexes) are small or large (in this latter case reciprocal values of the parameters are considered). Usually, it suffices to only define the first, or more rarely the first and the second approximations; the rest of the approximations serve for adjusting the first one. As a rule, the application of perturbation theory brings about a solution of more simple differential equations, of a smaller dimension or order, and also for different equations in partial derivatives, of more simple equations of another type (for instance, parabolic equations of a boundary layer approximation instead of Navier-Stokes elliptic equations in problems of dynamics of viscous liquid for large Reynolds numbers).

If the asymptotic expansion in terms of a small parameter is uniformly exact in the entire domain of definition of independent variables, a problem of regular perturbations holds; if it is nonuniformly exact, the problem is of singular perturbations. Among the methods of solution of the last problem the methods of deformed coordinates, of joint asymptotic expansions and of many scales are widely employed.

REFERENCES

Nayfeh, (1973) Perturbation Methods. J. Wiley & Sons, New York. London, Sydney, Toronto.

Van Dyke, M. (1964) Perturbation Methods in Fluid Mechanics. Acad. Press, New York, London.

Cole, J. D. (1968) Perturbation Methods in Applied Mathematics. Blaisdell Pull. Comp., Toronto, London.

References

  1. Nayfeh, (1973) Perturbation Methods. J. Wiley & Sons, New York. London, Sydney, Toronto.
  2. Van Dyke, M. (1964) Perturbation Methods in Fluid Mechanics. Acad. Press, New York, London.
  3. Cole, J. D. (1968) Perturbation Methods in Applied Mathematics. Blaisdell Pull. Comp., Toronto, London.

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