GENERATION OF MULTIVARIATE HERMITE INTERPOLATING POLYNOMIALS
CONTENTS I CONSTRAINED NUMBERS CONSTRAINED COORDINATE SYSTEM DE?NITION OPERATIONS WITH CONSTRAINED NUMBERS OPERATIONS WITH SETS OF CONSTRAINED NUMBERS GEOMETRIC REPRESENTATION CONTRAVARIANT COORDINATE REPRESENTATION COVARIANT COORDINATE REPRESENTATION (DUAL SYSTEM) VISUALIZATION OF THE COORDINATE AXES ORTHOGONAL REPRESENTATION OF THE COORDINATE AXES OBLIQUE REPRESENTATION OF THE COORDINATE AXES GEOMETRIC LOCATION OF THE CONSTRAINED NUMBERS ONE-DIMENSIONAL CONSTRAINED NUMBERS TWO-DIMENSIONAL CONSTRAINED NUMBERS TWO-DIMENSIONAL REPRESENTATION OF N-DIMENSIONAL COORDINATE AXES ZERO-DIMENSIONAL CONSTRAINED SPACE DE?NITIONS GEOMETRIC REPRESENTATION OF THE ZERO-DIMENSIONAL CON- STRAINED NUMBERS GENERATION OF THE COORDINATE SYSTEM GENERATION AS A CARTESIAN PRODUCT OF TWO SETS ?-DIMENSIONAL CONSTRAINED NUMBERS GENERATION OF THE ELEMENTS OF THE SET L? ?FOR THE UNDERLYING SET Z RULES FOR THE GENERATION RULES FOR THE PARTICULAR CASES ALGORITHM FOR THE GENERATION OF THE ELEMENTS GENERATION OF THE ELEMENTS OF THE SET L?(?) FOR THE UNDERLYING SET Z GENERATION OF THE ELEMENTS OF THE SET L? ? FOR THE UNDERLYING SET Z+ ALGORITHM FOR THE GENERATION OF THE ELEMENTS OF THE SET L? ?USING THE COORDINATE NUMBERS ALGORITHM FOR THE GENERATION OF THE ELEMENTS OF THE SET L? ?USING SUBLEVELS GENERATION OF THE ELEMENTS OF THE SET L?(?) FOR THE UNDERLYING SET Z+ ALGORITHM FOR THE GENERATION OF THE ELEMENTS OF THE SET L?(?) USING THE COORDINATE NUMBERS ALGORITHM FOR THE GENERATION OF THE ELEMENTS OF THE SET L?(?) USING SUBLEVELS EXAMPLES OF ZERO-DIMENSIONAL CONSTRAINED SPACE GENERATION OF THE ELEMENTS 0 AND L0(0) ELEMENTS OF THE SET L0(?) GEOMETRIC REPRESENTATION OF THE SET L0(?) EXAMPLES OF ONE-DIMENSIONAL CONSTRAINED SPACE 0 AND L1(0) 1 AND L1(1) 2 AND L1(2) 3 AND L1(3) EXAMPLES OF TWO-DIMENSIONAL CONSTRAINED SPACE 0 AND L2(0) 1 AND L2(1) 2 AND L2(2) 3 AND L2(3) EXAMPLES OF THREE-DIMENSIONAL CONSTRAINED SPACE 0 AND L3(0) 1 AND L3(1) 2 AND L3(2) 3 AND L3(3) 2 10 PROPERTIES OF SETS OF CONSTRAINED NUMBERS NATURAL COORDINATES ONE-DIMENSIONAL NATURAL COORDINATE NUMBERS LOCATION OF A POINT RELATION BETWEEN THE ONE-DIMENSIONAL NATURAL COORDI- NATE NUMBERS AND THE ONE-DIMENSIONAL CARTESIAN COOR- DINATE NUMBERS RELATION BETWEEN THE ONE-DIMENSIONAL NATURAL COORDI- NATE NUMBERS AND THE TWO-DIMENSIONAL CARTESIAN COOR- DINATE NUMBERS TWO-DIMENSIONAL NATURAL COORDINATE NUMBERS RELATION BETWEEN THE TWO-DIMENSIONAL NATURAL COORDI- NATE NUMBERS AND THE TWO-DIMENSIONAL CARTESIAN COOR- DINATE NUMBERS COORDINATES ORTHOGONAL TO THE SIDES OF THE TRIANGLE THREE-DIMENSIONAL NATURAL COORDINATE NUMBERS RELATION BETWEEN THE THREE-DIMENSIONAL NATURAL COORDI- NATE NUMBERS AND THE THREE-DIMENSIONAL CARTESIAN COOR- DINATE NUMBERS COORDINATES ORTHOGONAL TO THE FACES OF THE TETRAHEDRON ?-DIMENSIONAL NATURAL COORDINATE NUMBERS RELATION BETWEEN THE ?-DIMENSIONAL NATURAL COORDINATE NUMBERS AND THE ?-DIMENSIONAL CARTESIAN COORDINATE NUM- BERS COORDINATES ORTHOGONAL TO THE FACES OF THE ?-HEDRON COMPUTATION OF THE NUMBER OF ELEMENTS UNDERLYING SET Z UNDERLYING SET Z+ SUMMARY FOR THE ZERO-DIMENSIONAL SPACE ONE-DIMENSIONAL CONSTRAINED SPACE SUMMARY FOR THE ONE-DIMENSIONAL NUMBERS TWO-DIMENSIONAL SPACE SUMMARY FOR THE TWO-DIMENSIONAL NUMBERS THREE-DIMENSIONAL CONSTRAINED SPACE SUMMARY FOR THE THREE-DIMENSIONAL NUMBERS ?-DIMENSIONAL CONSTRAINED SPACE SUMMARY IF THE UNDERLYING SET IS Z IF THE UNDERLYING SET IS Z+ MATHEMATICAL TOOLS SUM OF THE TERMS OF AN ARITHMETIC PROGRESSION SUM OF SQUARES SUM OF CUBES SELECTED COMBINATORICS PROPERTIES POCHHAMMER’S SYMBOL AN ORDERING RELATION DE?NITION OF ORDER POSSIBLE DE?NITIONS ADOPTED DE?NITION INTEGER NUMBERS AS THE UNDERLYING SET TWO-DIMENSIONAL CONSTRAINED SPACE NONNEGATIVE INTEGER NUMBERS AS THE UNDERLYING SET GENERALIZATION FOR THE CONSTRAINED NUMBERS PROPERTIES OF THE ORDERING FUNCTION SUMMARY FOR THE UNDERLYING SET IS Z SUMMARY FOR THE UNDERLYING SET IS Z+ APPLICATION TO SYMBOLIC COMPUTATION OF DERIVATIVES POLYNOMIAL NOTATION EXAMPLES NOTATION FOR A COMPLETE POLYNOMIAL NOTATION FOR THE TERMS IN THE LEVEL ?OF COMPLETE POLY- NOMIALS PARTIAL DERIVATIVE NOTATION DERIVATIVE OF ORDER V OF A POLYNOMIAL POLYNOMIALS WHOSE POWERS ARE ELEMENTS IN THE SET Z+ POLYNOMIAL WHOSE POWERS ARE ELEMENTS IN THE SET Z ALGORITHM TO OBTAIN THE DERIVATIVE OF A POLYNOMIAL WHOSE POWERS ARE ELEMENTS IN THE SET Z+ ALGORITHM TO OBTAIN THE DERIVATIVE OF A POLYNOMIAL WHOSE POWERS ARE ELEMENTS IN THE SET Z PARTIAL INTEGRATION NOTATION INTEGRATION OF ORDER V OF A POLYNOMIAL ELEMENTS CHARACTERIZING A POLYNOMIAL ORDER OF DERIVATIVE DERIVATIVE OF THE PRODUCT OF FUNCTIONS IN ONE VARIABLE DERIVATIVE OF THE PRODUCT OF TWO FUNCTIONS USING CON- STRAINED MULTIPLICATION DERIVATIVE OF THE PRODUCT OF T FUNCTIONS USING CONSTRAINED MULTIPLICATION DERIVATIVE OF THE PRODUCT OF TWO FUNCTIONS IN ONE VARI- ABLE DERIVATIVE OF THE PRODUCT OF THREE FUNCTIONS IN ONE VARI- ABLE DERIVATIVE OF THE PRODUCT OF T FUNCTIONS IN ONE VARIABLE DERIVATIVE OF A PRODUCT OF FUNCTIONS OF SEVERAL VARIABLES DERIVATIVE OF A PRODUCT OF TWO FUNCTIONS OF TWO VARIABLES DERIVATIVE OF THE PRODUCT OF T FUNCTIONS OF D VARIABLES EXPANSION OF THE POWER OF A SUM OF FUNCTIONS EXPANSION OF THE POWER OF A SUM OF TWO FUNCTIONS EXPANSION OF THE POWER OF A SUM OF T FUNCTIONS 6 10 COMPUTATION OF THE DERIVATIVES OF THE FORM 1/(X ??)S 6 10 1 ONE FUNCTION AND ONE VARIABLE 6 10 2 PRODUCT OF T FUNCTIONS OF THE SAME VARIABLE X 6 10 3 COMPUTATION OF THE DERIVATIVES OF THE PRODUCT OF D FUNC- TIONS, WHICH ARE PRODUCT OF KJ FUNCTIONS OF THE SAME VARIABLE XJ 6 10 4 SUMMARY 6 11 COMPUTATION OF THE DERIVATIVES OF THE FORM 1/ (C0X0 + C1X1 + C2) 6 11 1 DERIVATIVE OF ONE FUNCTION 6 11 2 DERIVATIVE OF PRODUCT OF T FUNCTIONS II HERMITE INTERPOLATING POLYNOMIALS MULTIVARIATE HERMITE INTERPOLATING POLYNOMIAL POLYNOMIAL CONSTRUCTION THE POLYNOMIAL PE(X) THE POLYNOMIAL QE,N(X) THE POLYNOMIAL FE,N(A, X) COMPUTATION OF THE COE?CIENTS OF FE,N(A, X) HYPERSURFACE REGION CONSTRUCTION OF THE SYSTEM OF EQUATIONS COMPUTATION OF THE TERM [1/PE(X)](V) COMPUTATION OF [FE,N(A, X)](V) SOLUTION OF THE SYSTEM OF EQUATIONS PROPERTIES OF THE HERMITE INTERPOLATING POLYNOMIALS COMPUTATION OF THE COE?CIENTS OF FH,N(G, X) IN TERMS OF THE COE?CIENTS OF FE,N(A, X) IF THE REFERENCE NODES E AND H DI?ER ONLY FOR ONE COORDINATE NUMBER COMPUTATION OF THE COE?CIENTS OF FH,N(G, X) IN TERMS OF THE COE?CIENTS OF FE,N(A, X) IF THE REFERENCE NODES E AND H DI?ER FOR SEVERAL COORDINATE NUMBER RELATION BETWEEN THE POLYNOMIALS ?(N) E,N(X)|X=? AND ?(N) E,N(X)|X=? IF ?(N) E,N(X)|X=? = ??(N) E,N(X)|X=? RELATION BETWEEN [1/(N!PE(X))](?) AND [1/(N!PE(X)](µ) IF µ IS A PERMUTATION OF ? EQUALITY OF THE COE?CIENTS OF THE TERMS OF THE POLY- NOMIAL FE,N(A, X) THAT ARE IN THE SAME LEVEL AND WHICH INDICES ARE PERMUTATIONS COMPUTATION OF THE POLYNOMIALS RELATED TO ONE NODE IN TERMS OF THE POLYNOMIALS RELATED TO A SYMMETRIC NODE GENERATION OF THE HERMITE INTERPOLATING POLYNOMIALS POLYNOMIALS OF MINIMUM DEGREE THE DEGREE OF FE,N(A, X) IS KEPT CONSTANT THE DEGREE OF FE,N(A, X) IS ? HERMITE INTERPOLATING POLYNOMIALS: THE CLASSICAL AND PRESENT APPROACHES CLASSICAL APPROACH, ONE VARIABLE COMPUTATION OF A ONE-VARIABLE POLYNOMIAL GENERATING EXPRESSION CONSTRUCTION OF THE POLYNOMIAL ?A,0(X) CONSTRUCTION OF THE POLYNOMIAL ?A,1(X) CONSTRUCTION OF THE POLYNOMIAL ?A,2(X) GENERATION OF THE POLYNOMIALS RELATED TO THE REFERENCE NODE B = POLYNOMIAL ?B,0(X) POLYNOMIAL ?B,1(X) POLYNOMIAL ?B,2(X) COMPARISON BETWEEN THE TWO TECHNIQUES CLASSICAL APPROACH THE PRESENT APPROACH 10 NORMALIZED SYMMETRIC SQUARE DOMAIN 10 1 COMPUTATION OF A TWO-VARIABLE POLYNOMIAL 10 2 GENERATION OF THE POLYNOMIAL ?A0,00(X0, X1) RELATED TO THE REF- ERENCE NODE A0 = (?1, ?1) 10 2 1 ELEMENTS COMMON TO ALL POLYNOMIALS RELATED TO THE NODE A0 = (?1, ?1) 10 2 2 GENERATION OF THE POLYNOMIAL ?A0,00(X0, X1) 10 2 3 COMPUTATION OF THE COE?CIENTS OF THE POLYNOMIAL FA0,00(X) 10 3 GENERATION OF THE POLYNOMIAL ?A0,10(X0, X1) RELATED TO THE REF- ERENCE NODE A0 = (?1, ?1) 10 3 1 EXPRESSION FOR THE POLYNOMIAL ?A0,10(X0, X1) 10 3 2 COMPUTATION OF THE POLYNOMIAL FA0,N(X) 10 3 3 POLYNOMIAL ?A0,10(X0, X1) 10 3 4 SUMMARY OF THE POLYNOMIALS ?A0,N RELATED TO THE REFER- ENCE NODE A0 = (?1, ?1) 10 4 GENERATION OF THE POLYNOMIALS ?A1,N RELATED TO THE REFERENCE NODE (1, ?1) 10 4 1 GENERATION OF THE POLYNOMIALS ?A1,N 10 4 2 SUMMARY OF THE POLYNOMIALS ?A1,N RELATED TO THE REFER- ENCE NODE A1 = (1, ?1) 10 5 GENERATION OF THE POLYNOMIALS ?A2,N RELATED TO THE REFERENCE NODE (1, 1) 10 5 1 GENERATION OF THE POLYNOMIALS 10 5 2 SUMMARY OF THE POLYNOMIALS ?A2,N RELATED TO THE REFER- ENCE NODE A2 = (1, 1) 10 6 GENERATION OF THE POLYNOMIALS ?A3,N RELATED TO THE REFERENCE NODE A3 = (?1, 1) 10 6 1 GENERATION OF THE POLYNOMIALS 10 6 2 SUMMARY OF THE POLYNOMIALS ?A3,N RELATED TO THE REF- ERENCE NODE A3 = (?1, 1) 10 7 GENERATION OF THE POLYNOMIAL ?A0,00(X0, X1) RELATED TO THE REF- ERENCE NODE A0 = (?1, ?1) WITH Q = 10 7 1 COMPUTATION OF THE COE?CIENTS OF THE POLYNOMIAL FA0,00(X) 10 7 2 SUMMARY OF THE POLYNOMIALS ?A0,N RELATED TO THE REFER- ENCE NODE A0 = (?1, ?1) AND Q = 6 11 RECTANGULAR NONSYMMETRIC DOMAIN 11 1 GENERATION OF A TWO-VARIABLE POLYNOMIAL IN A RECTANGULAR NON- SYMMETRIC DOMAIN 11 2 GENERATION OF THE POLYNOMIAL ?A3,00(X0, X1) RELATED TO THE REF- ERENCE NODE A3 = (2, 4) 11 2 1 INFORMATION ABOUT THE POLYNOMIAL 11 2 2 COMMON FACTORS RELATED TO THE REFERENCE NODE A3 = (2, 4) 11 2 3 EXPRESSION FOR THE POLYNOMIAL QA3,N(X) 11 2 4 EXPRESSION FOR THE POLYNOMIAL ?A3,N(X) 11 2 5 PROPERTIES OF THE POLYNOMIAL ?A3,00(X0, X1) 11 2 6 CONSTRUCTION OF THE CONSTANT TERM N!PA3(?) 11 2 7 CONSTRUCTION OF THE POLYNOMIAL QA3,N(X) 11 2 8 THE POLYNOMIAL ?A3,00(X0, X1) 11 2 9 COMPUTATION OF THE COE?CIENTS OF THE POLYNOMIAL FA3,00(X) 11 3 SUMMARY OF THE POLYNOMIALS 11 3 1 POLYNOMIALS ?E,I RELATED TO THE NODE A1 = (6, 12) 11 3 2 POLYNOMIALS ?E,I RELATED TO THE NODE A2 = (2, 12) 11 3 3 POLYNOMIALS ?E,N RELATED TO THE NODE A3 = (2, 4) 11 3 4 POLYNOMIALS ?E,I RELATED TO THE NODE A4 = (6, 4) 12 GENERIC DOMAINS 12 1 EXPRESSION FOR THE POLYNOMIAL 12 1 1 INFORMATION ABOUT THE POLYNOMIAL 12 1 2 PROPERTIES OF THE POLYNOMIAL ?A3,00(X0, X1) 12 2 EXPRESSION FOR THE POLYNOMIAL ?E,N(X) 12 2 1 COMMON FACTORS RELATED TO THE REFERENCE NODE A3 = (?5, 2) 12 2 2 EXPRESSION FOR THE POLYNOMIAL QA3,N(X) 12 2 3 CONSTRUCTION OF THE CONSTANT TERM N!PA3(?) 12 2 4 CONSTRUCTION OF THE POLYNOMIAL QA3,N(X) 12 2 5 THE POLYNOMIAL ?A3,00(X0, X1) 12 2 6 COMPUTATION OF THE COE?CIENTS OF THE POLYNOMIAL FA3,00(A, X) 12 3 TRIANGULAR DOMAIN 12 3 1 INFORMATION ABOUT THE POLYNOMIAL 12 3 2 COMMON FACTORS RELATED TO THE REFERENCE NODE A0 = (0, ?3/2) 12 3 3 EXPRESSION FOR THE POLYNOMIAL QA0,N(X) 12 3 4 EXPRESSION FOR THE POLYNOMIAL ?A0,N(X) 12 3 5 PROPERTIES OF THE POLYNOMIAL ?A0,00(X0, X1) 12 3 6 CONSTRUCTION OF THE CONSTANT TERM N!PA0(?) 12 3 7 CONSTRUCTION OF THE POLYNOMIAL QA0,N(X) 12 3 8 THE POLYNOMIAL ?A0,00(X0, X1) 12 3 9 COMPUTATION OF THE COE?CIENTS OF THE POLYNOMIAL FA0,00(X) 13 EXTENSIONS OF THE CONSTRAINED NUMBERS 13 1 COMPARISON TO THE DIVIDED DI?ERENCE METHOD 13 1 1 COMPARISON 13 1 2 EXAMPLE 13 2 GENERALIZATION TO THE SET OF POSITIVE REAL NUMBERS 13 3 CONSTRAINED COORDINATE SYSTEM WITH MATRICES 14 FIELD OF THE COMPLEX NUMBERS 14 1 GENERALIZATION TO THE ?ELD OF THE COMPLEX NUMBERS 14 2 GENERATION OF THE POLYNOMIAL ?A0,0(X) RELATED TO THE REFERENCE NODE A0 = ?1 ?I 14 2 1 ELEMENTS COMMON TO ALL POLYNOMIALS RELATED TO THE NODE A0 = (?1 ?I) 14 2 2 GENERATION OF THE POLYNOMIAL ?A0,0(X) RELATED TO THE REFERENCE NODE A0 = ?1 ?I 14 2 3 COMPUTATION OF THE COE?CIENTS OF THE POLYNOMIAL FA0,00(X) 14 2 4 POLYNOMIALS RELATED TO THE NODE ?1 ?I 14 3 POLYNOMIAL ?A1,0(X) RELATED TO THE REFERENCE NODE A1 = 1 ?I 14 4 POLYNOMIAL ?A2,0(X) RELATED TO THE REFERENCE NODE A2 = ?1 + I 14 5 POLYNOMIAL ?A3,0(X) RELATED TO THE REFERENCE NODE A3 = 1 + I 15 ANALYSIS OF THE BEHAVIOR OF THE HERMITE INTERPOLATING POLY- NOMIALS 15 1 ONE DIMENSIONAL HERMITE INTERPOLATING POLYNOMIALS 15 1 1 THE SET OF FUNCTIONS 15 2 GRAPHICAL REPRESENTATION OF THE HERMITE INTERPOLATING POLYNOMIALS ?(µ) A,N(X) FOR ? = 15 2 1 GRAPHIC OF ?A,0(X) AND ITS DERIVATIVES 15 2 2 GRAPHIC OF ?A,1(X) AND ITS DERIVATIVES 15 2 3 GRAPHIC OF THE POLYNOMIAL ?A,2(X) AND ITS DERIVATIVES 15 2 4 GRAPHIC OF ?A,3(X) AND ITS DERIVATIVES 15 2 5 GRAPHIC OF THE POLYNOMIAL ?A,4(X) AND ITS DERIVATIVES 15 3 GRAPHIC OF POLYNOMIALS ?(µ) A,N(X), N = 0, 1, 2, 3, 15 3 1 GRAPHIC OF POLYNOMIALS ?(0) A,N(X), N = 0, 1, 2, 3, 4 15 3 2 GRAPHIC OF POLYNOMIALS ?(1) A,N(X), N = 0, 1, 2, 3, 4 15 3 3 GRAPHIC OF THE POLYNOMIALS ?(2) A,N(X), N = 0, 1, 2, 3, 4 15 3 4 GRAPHIC OF THE POLYNOMIALS ?(3) A,N(X), N = 0, 1, 2, 3, 4 15 3 5 GRAPHIC OF THE POLYNOMIALS ?(4) A,N(X), N = 0, 1, 2, 3, 4 15 3 6 GRAPHIC OF THE POLYNOMIALS ?A,I(X), I = 0, 3, , 27 15 3 7 GRAPHIC OF THE ?RST DERIVATIVES OF THE POLYNOMIALS ?(0) A,I(X), I = 3, 6, , 27 15 3 8 GRAPHIC OF THE SECOND DERIVATIVES OF THE POLYNOMIALS ?(0) A,I(X), I = 3, 6, , 27 15 4 GRAPHICAL REPRESENTATION OF THE HERMITE INTERPOLATING POLYNOMIALS ?(µ) A,N(X) AND ?(µ) B,N(X) FOR N = 0, 1, 2, 3, 15 4 1 GRAPHIC OF THE POLYNOMIALS ?(0) A,N(X) AND ?(0) B,N(X) FOR N = 0, 1, 2, 3, 4 15 4 2 GRAPHIC OF THE POLYNOMIALS ?(1) A,N(X) AND ?(1) B,N(X) FOR N = 0, 1, 2, 3, 4 15 4 3 GRAPHIC OF THE POLYNOMIALS ?(2) A,N(X) AND ?(2) B,N(X) FOR N = 0, 1, 2, 3, 4 15 4 4 GRAPHIC OF THE POLYNOMIALS ?(3) A,N(X) AND ?(3) B,N(X) FOR N = 0, 1, 2, 3, 4 15 4 5 GRAPHIC OF THE POLYNOMIALS ?(4) A,N(X) AND ?(4) B,N(X) FOR N = 0, 1, 2, 3, 4 15 5 ONE-DIMENSIONAL MODI?ED HERMITE INTERPOLATING POLYNOMIALS 15 5 1 THE SET OF FUNCTIONS 15 6 ANALYSIS OF THE ERROR FOR SELECTED DEGREES OF FE,N(A, X) 15 6 1 APPROXIMATION TO A LINE 15 7 SELECTED EXAMPLES OF HERMITE INTERPOLATING POLYNOMIALS 15 7 1 ONE-DIMENSIONAL HERMITE INTERPOLATING POLYNOMIAL 15 7 2 TWO-DIMENSIONAL HERMITE INTERPOLATING POLYNOMIAL 15 7 3 THREE-DIMENSIONAL HERMITE INTERPOLATING POLYNOMIAL 15 7 4 SUMMARY OF THE POLYNOMIALS ?(N0,N1,N2) E,N RELATED TO THE NODE AE = (?1, ?1, ?1) 15 7 5 SUMMARY OF THE POLYNOMIALS ?(N0,NN,N2) E,N RELATED TO THE NODE AE = (?1, ?1, ?1) AND ? = 15 7 6 FOUR-DIMENSIONAL HERMITE INTERPOLATING POLYNOMIAL 15 7 7 SUMMARY OF THE POLYNOMIALS ?(N0,N1,N2,N3) E,N RELATED TO THE NODE AE = (?1, ?1, ?1, ?1) AND ? = 3 III SELECTED APPLICATIONS 16 CONSTRUCTION OF THE APPROXIMATE SOLUTION 16 1 ONE-DIMENSIONAL PROBLEMS 16 1 1 SOLUTION EXPANDED IN TERMS OF THE HERMITE INTERPOLATING POLYNOMIALS 16 1 2 COORDINATE TRANSFORMATION 16 1 3 ONE-DIMENSIONAL BOUNDARY CONDITIONS 16 2 HIGHER-DIMENSIONAL PROBLEMS 16 2 1 ANALYSIS OF THE BOUNDARY CONDITIONS 16 2 2 REPRESENTATION OF A TWO-VARIABLE POLYNOMIAL ON ANY BA- SIS 16 2 3 REPRESENTATION OF A THREE-VARIABLE POLYNOMIAL ON ANY BASIS 16 2 4 EXAMPLE OF A THREE-VARIABLE POLYNOMIAL ON A CHEBYSHEV BASIS 16 2 5 EXAMPLE OF THE TWO-DIMENSIONAL SINE FUNCTION 16 2 6 THE POLYNOMIAL APPROXIMATION 16 2 7 COMPUTATION OF THE COE?CIENTS OF THE EXPANSION OF THE SOLUTION 16 3 RESIDUAL ANALYSIS 16 3 1 RESIDUAL ANALYSIS OF THE GOVERNING EQUATION 16 3 2 RESIDUAL ANALYSIS OF THE SOLUTION 16 3 3 RESIDUAL ANALYSIS IN THE DOMAIN 16 4 COMMENTS ABOUT THE COMPUTER PROGRAM 17 ONE-DIMENSIONAL TWO-POINT BOUNDARY VALUE PROBLEMS 17 1 LINEAR DI?ERENTIAL EQUATION 17 1 1 APPROXIMATE SOLUTION IN THE DOMAIN [?0 5, 0 5] 17 1 2 APPROXIMATE SOLUTION IN THE DOMAIN [?1, 1] 17 1 3 ERROR ANALYSIS 17 2 NONLINEAR DI?ERENTIAL EQUATION 17 2 1 MODELING VISCOELASTIC ?OWS: CASE I 17 2 2 MODELING VISCOELASTIC ?OWS: CASE II 17 2 3 ANALYSIS OF THE SOLUTION IN THE INTERVAL [?1, 1] 17 3 FLOW OF NON-NEWTONIAN ?UIDS: POWELL-EYRING MODEL 17 3 1 PRANDTL-EYRING MODEL: CASE I 17 3 2 POWELL-EYRING MODEL: CASE II 17 4 TWO-POINT BOUNDARY VALUE PROBLEM 17 4 1 SINGULAR PERTURBATION: PROBLEM I 17 4 2 SINGULAR PERTURBATION: PROBLEM II 17 4 3 SINGULAR PERTURBATION: PROBLEM III 18 APPLICATION TO PROBLEMS WITH SEVERAL VARIABLES 18 1 APPLICATION TO HEAT PROBLEMS 18 1 1 ONE-DIMENSION HEAT ?OW EQUATION 18 1 2 TWO-DIMENSION HEAT ?OW EQUATION 18 1 3 TWO-DIMENSION HEAT ?OW EQUATION WITH LATERAL HEAT LOSS 18 2 APPLICATION TO THE WAVE EQUATION 18 2 1 HOMOGENEOUS ONE-DIMENSION WAVE EQUATION 18 2 2 INHOMOGENEOUS ONE-DIMENSION WAVE EQUATION 18 2 3 HOMOGENEOUS TWO-DIMENSION WAVE EQUATION 18 2 4 INHOMOGENEOUS TWO-DIMENSION WAVE EQUATION 18 3 NONLINEAR PARTIAL DI?ERENTIAL EQUATION 18 4 INHOMOGENEOUS ADVECTION EQUATION 19 THERMAL ANALYSIS OF THE SURFACE OF THE SPACE SHUTTLE 19 1 PHYSICAL MODEL 19 1 1 THERMAL PROPERTIES 19 1 2 MATERIAL MODELING 19 1 3 PROPERTIES OF THE REMAINING LAYERS 19 1 4 HEAT GENERATED AT THE EXTERNAL SURFACE 19 1 5 GEOMETRY 19 1 6 TEMPERATURES 19 2 MATHEMATICAL MODEL 19 2 1 NONLINEAR GOVERNING DI?ERENTIAL EQUATIONS 19 2 2 LINEAR GOVERNING DI?ERENTIAL EQUATIONS 19 2 3 SUMMARY OF GOVERNING EQUATIONS 19 2 4 BOUNDARY CONDITIONS 19 3 CONSTRUCTION OF THE APPROXIMATE SOLUTION 19 3 1 TYPES OF SOLUTIONS 19 3 2 SECOND-ORDER DI?ERENTIAL EQUATION FORMULATION 19 3 3 FIRST-ORDER DI?ERENTIAL EQUATION FORMULATION 19 3 4 COMPUTATION OF THE RESIDUAL OF THE DI?ERENTIAL EQUATION 19 4 NUMERICAL COMPUTATION 19 5 ANALYSIS OF THE SOLUTIONS 19 5 1 GOVERNING EQUATION RESIDUAL ANALYSIS 19 5 2 TWO-TEMPERATURE BOUNDARY CONDITION APPROXIMATE SO- LUTION 19 5 3 HEAT-?UX AND TEMPERATURE BOUNDARY CONDITION 19 5 4 FIRST-ORDER DI?ERENTIAL GOVERNING EQUATION 19 6 TEMPERATURE DISTRIBUTION 19 7 COE?CIENTS FOR THE APPROXIMATE SOLUTION 19 7 1 POWER SERIES, CASE 1 DEGREE 15 19 7 2 CHEBYSHEV POLYNOMIAL, CASE 1 DEGREE 15 19 7 3 POWER SERIES, CASE 7 DEGREE 15 19 7 4 CHEBYSHEV POLYNOMIAL, CASE 7 DEGREE 15 19 7 5 POWER SERIES, CASE 8 DEGREE 15 19 7 6 CHEBYSHEV POLYNOMIAL, CASE 8 DEGREE 15 19 7 7 POWER SERIES, CASE 9 DEGREE 15 19 7 8 CHEBYSHEV POLYNOMIAL, CASE 9 DEGREE 15 19 8 CONCLUSION REFERENCES