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Instructions for Converting POLYMATH Solutions to Excel Worksheets -Instructions for Converting POLYMATH Solutions to Excel Worksheets -IntroductionIntroduction
WHY EXCEL FOR NUMERICAL PROBLEM SOLVING?
SPREADSHEETS ARE THE COMPUTATIONAL TOOLS MOST WIDELY USED BYCHEMICAL ENGINEERS. PROVIDING THE CAPABILITY FOR NUMERICALPROBLEM SOLVING EXTENDS CONSIDERABLY THE COMPUTATIONALPOTENTIAL OF THE ENGINEER
WHY USE A POLYMATH PREPROCESSOR ?
THE MATHEMATICAL MODEL CAN BE MUCH EASIER AND FASTER CODED ANDDEBUGGED USING POLYMATH. THE POLYMATH MODEL SERVES AS BASIS FORTHE SPREADSHEET MODEL WHERE THE VARIABLE NAMES ARE REPLACED BYTHEIR ADDRESSES. IT ALSO SERVES AS AN EASY TO UNDERSTANDDOCUMENTATION OF THE MODEL
CONVERTING POLYMATH SOLUTIONS TO EXCEL WORKSHEETSCONVERTING POLYMATH SOLUTIONS TO EXCEL WORKSHEETS
TYPES OF PROBLEMS DISCUSSEDTYPES OF PROBLEMS DISCUSSED
1 ONE NONLINEAR ALGEBRAIC EQUATION – GOAL SEEK – SLIDES 3-9
2 SYSTEMS OF NONLINEAR ALGEBRAIC EQUATIONS – SOLVER – SLIDES 10-14
3 ODE – INITIAL VALUE PROBLEMS – 4TH ORDER EXPLICIT RK – SLIDES 15-24
4 ODE – BOUNDARY VALUE – EXPLICIT EULER + GOAL SEEK – SLIDES 25-29
5 DAE – INITIAL VALUE PROBLEMS – IMPLICIT EULER – SLIDES 30-36
6 PDE – INITIAL VALUE - METHOD OF LINES – EXPLICIT EULER – SLIDES 37-40
7 MULTIPLE LINEAR REGRESSION – LINEST – SLIDES 41-45
8 POLYNOMIAL REGRESSION – LINEST – SLIDES 46-49
9 MULTIPLE NONLINEAR REGRESSION – SOLVER – SLIDES 50-53
One Nonlinear Algebraic Equation Instructions for Conversion (1)One Nonlinear Algebraic Equation Instructions for Conversion (1)
To obtain a basic solution of a system containing one implicit nonlinear algebraic equation andseveral explicit equations the POLYMATH equations should be converted to Excel formulasand then the "Goal Seek" tool can be used. In order to obtain a well documented Excelworksheet, which can be easily modified for parametric runs it is recommended to carry out theconversion in the following steps:
1. Copy the implicit equation and the ordered explicit equations from the POLYMATHsolution report.
2. Paste the equations into an Excel worksheet; remove the text and the equation numbers.
3. Rearrange the equations in the order: constant definitions, functions of the constants,parameter definitions, unknown, explicit functions of the unknown and implicit functionof the unknown.
One Nonlinear Algebraic EquationOne Nonlinear Algebraic Equation
Instructions for Conversion (2)Instructions for Conversion (2)
4. Copy the right hand side of the equations into the adjacent cell and replace the variable namesby variable addresses. Note that "If" statements and some functions may require additionalrewriting and/or rearrangement. Use absolute addressing for the constants and the functions ofconstant and relative addressing for the unknown and its functions (Note that pressing F4converts selected reference from relative to absolute). In the cell adjacent to the unknown put itsinitial estimate.
5. Use the "Goal Seek" tool to set the value of the cell containing the implicit function of theunknown at zero while changing the value in the cell of the unknown..
One Nonlinear Algebraic EquationOne Nonlinear Algebraic Equation
Ordered POLYMATH FileOrdered POLYMATH File
The use of this procedure is demonstrated in reference to Demo 2.
Nonlinear equations [1] f(V) = (P+a/(V^2))*(V-b)-R*T = 0 Explicit equations [1] P = 56 [2] R = 0.08206 [3] T = 450 [4] Tc = 405.5 [5] Pc = 111.3 [6] Pr = P/Pc [7] a = 27*(R^2*Tc^2/Pc)/64 [8] b = R*Tc/(8*Pc) [9] Z = P*V/(R*T)
One Nonlinear Algebraic EquationOne Nonlinear Algebraic Equation
Excel FormulasExcel Formulas
CBA
=($C$5+$C$11/(C13^2))*(C13-$C$12)-$C$6*$C$7
f(V) = (P+a/(V^2))*(V-b)-R*T = 015
=$C$5*C13/($C$6*$C$7) Z = P*V/(R*T)Functions of theunknown
14
0.7 VUnknown13
=$C$6*$C$8/(8*$C$9) b = R*Tc/(8*Pc)12
=27*($C$6^2*$C$8^2/$C$9)/64 a = 27*(R^2*Tc^2/Pc)/6411
=$C$5/$C$9 Pr = P/PcFunctions of theconstants
10
=111.3 Pc = 111.39
=405.5 Tc = 405.58
=450 T = 4507
=0.08206 R = 0.082066
=56 P = 56Constants5
4
Equations3
One Nonlinear Algebraic Equation SolutionOne Nonlinear Algebraic Equation Solution
To solve the nonlinear equation in cell C15 "Goal Seek" is used to set the value in this cellat zero while changing the contents of cell C13.
8.4999E-075.85576f(V) = (P+a/(V^2))*(V-b)-R*T = 015
0.871831.06155 Z = P*V/(R*T)14
0.574890.7V13
0.037370.03737 b = R*Tc/(8*Pc)12
4.196954.19695 a = 27*(R^2*Tc^2/Pc)/6411
0.503140.50314 Pr = P/Pc10
111.3111.3 Pc = 111.39
405.5405.5 Tc = 405.58
450450 T = 4507
0.082060.08206 R = 0.082066
5656 P = 565
4
SolutionInitial values 3
One Nonlinear Algebraic EquationOne Nonlinear Algebraic Equation
Modifying the Equation SetModifying the Equation Set
The example is next solved for Pr = 1, 2, 4, 10 and 20. To achieve this, the parameter Tr and itsfunction P=Pr*Pc are added to the equation set and the cells containing the unknown and itsfunctions are copied and modified as necessary.
CBA
=(C24+$C$11/(C25^2))*(C25-$C$12)-$C$6*$C$7
f(V) = (P+a/(V^2))*(V-b)-R*T = 0 27
=C24*C25/($C$6*$C$7) Z = P*V/(R*T)Functions of theunknown
26
0.233508696752435VUnknown25
=C23*$C$9P = Pr*PcFunction of theparameter
24
1PrParameter23
One Nonlinear Algebraic EquationOne Nonlinear Algebraic Equation
Complete solution setComplete solution set
To obtain the solution for other values of Pr cells 24 – 27 of column C are copied and thevalue of Pr entered in row 23. "Goal Seek" is applied separately to every column containinga different Pr value.
6.962E-096.184E-082.208E-067.604E-073.940E-06f(V) = (P+a/(V^2))*(V-b)-R*T = 0
2.783481.533410.731260.465780.70381 Z = P*V/(R*T)
0.046180.050880.060650.077270.23351V
22261113445.2222.6111.3P = Pr*Pc
2010421Pr
Systems of Nonlinear Algebraic EquationsSystems of Nonlinear Algebraic Equations
Instructions for Conversion (1)Instructions for Conversion (1)
To obtain a basic solution of a system containing several implicit nonlinear algebraic equationsthe POLYMATH equations are converted to Excel formulas and then the "Solver" tool isused. The recommended steps for conversion are:
1. Copy the implicit equations and the ordered explicit equations from the POLYMATHsolution report.
2. Paste the equations into an Excel worksheet; remove the text and the equation numbers.
3. Rearrange the equations in the order: constant definitions, functions of the constants,parameter definitions, unknowns, explicit functions of the unknowns and implicit functionsof the unknowns.
4. Add an equation with the sum of squares of the implicit functions.
Systems of Nonlinear Algebraic EquationsSystems of Nonlinear Algebraic Equations
Instructions for Conversion (2)Instructions for Conversion (2)
5. Copy the right hand side of the equations into the adjacent cell and replace the variablenames by variable addresses. Use absolute addressing for the constants and the functions ofconstant and relative addressing for the unknowns and functions of the unknowns. In the celladjacent to the unknowns put initial estimates.
6. Use the "Solver" tool to set the value of the cell containing the sum of squares of theimplicit functions of the unknowns at zero (or minimizing its value) while changing the valuesin the cells of the unknowns.
Systems of Nonlinear Algebraic EquationsSystems of Nonlinear Algebraic Equations
Ordered POLYMATH FileOrdered POLYMATH File
The use of this procedure is demonstrated in reference to Demo 5
Nonlinear equations [1] f(CD) = CC*CD-KC1*CA*CB = 0 [2] f(CX) = CX*CY-KC2*CB*CC = 0 [3] f(CZ) = CZ-KC3*CA*CX = 0 Explicit equations [1] KC1 = 1.06 [2] CY = CX+CZ [3] KC2 = 2.63 [4] KC3 = 5 [5] CA0 = 1.5 [6] CB0 = 1.5 [7] CC = CD-CY [8] CA = CA0-CD-CZ [9] CB = CB0-CD-CY
Systems of Nonlinear Algebraic Equations Excel FormulasSystems of Nonlinear Algebraic Equations Excel Formulas
=C12-$C$9*C15*C11f(CZ) = CZ-KC3*CA*CX = 0 19
=C17^2+C18^2+C19^2sum=f(CD)^2+f(CX)^2+f(CZ)^2Sum of squares of errors20
=C11*C13-$C$8*C16*C14f(CX) = CX*CY-KC2*CB*CC = 0 18
=C14*C10-$C$7*C15*C16f(CD) = CC*CD-KC1*CA*CB = 0 17
=$C$6-C10-C13CB = CB0-CD-CY 16
=$C$5-C10-C12CA = CA0-CD-CZ 15
=C10-C13CC = CD-CY 14
=C11+C12CY = CX+CZFunctions of the unknowns13
0CZ 12
0CX 11
0CDUnknowns10
5KC3 = 5 9
2.63KC2 = 2.63 8
1.06KC1 = 1.06 7
1.5CB0 = 1.5 6
1.5CA0 = 1.5Constants5
Equations 4
CBA
Systems of Nonlinear Algebraic Equations Excel FormulasSystems of Nonlinear Algebraic Equations Excel Formulas
The "Solver" tool is used to minimize the sum of squares of errors in cell C20 by settingC20 as "target cell" and searching for its minimal value by changing cells C10, C11 and C12.
1.0741E-135.68823sum=f(CD)^2+f(CX)^2+f(CZ)^2-2.9923E-070f(CZ) = CZ-KC3*CA*CX = 0-1.3358E-070f(CX) = CX*CY-KC2*CB*CC = 05.1760E-09-2.385f(CD) = CC*CD-KC1*CA*CB = 0
0.242901.5CB = CB0-CD-CY0.420691.5CA = CA0-CD-CZ0.153570CC = CD-CY0.551770CY = CX+CZ0.373980CZ0.177790CX0.705330CD
55KC3 = 52.632.63KC2 = 2.631.061.06KC1 = 1.061.51.5CB0 = 1.51.51.5CA0 = 1.5
SolutionInitial values
ODE ODE –– Initial Value Problems Initial Value Problems
TheThe Runge Runge--KuttaKutta Method Method
There are no tools in Excel to solve differential equations so the solution algorithm must bebuild into the solution worksheet. In this example a fixed step size, explicit, fourth-orderRunge-Kutta algorithm is used. The system of N first-order ODE for the functions
is written :
(1)
The fourth-order Runge-Kutta formula is written:
(2)
This formula advances a solution from xn to
Niyi ,,1, K=
),,,,()(
1 Nii yyxfdx
xdyK= Ni ,,1 K=
)22(61
),(
)2
,2
(
)2
,2
(
),(
43211
34
23
12
1
kkkkyy
kyhxhfk
kyhxhfk
kyhxhfk
yxhfk
nn
nn
nn
nn
nn
++++=
++=
++=
++=
=
+
hxx nn +≡+1
ODE ODE –– Initial Value Problems Instructions for Conversion (1) Initial Value Problems Instructions for Conversion (1)
Apply the Runge-Kutta algorithm to the system of first-order, ODE carry out the conversionfrom the POLYMATH file to the Excel spreadsheet in the following steps:
1.Copy the differential equation and the ordered explicit algebraic equations from thePOLYMATH solution report.
2. Paste the equations into an Excel worksheet; remove the text and the equation numbers.
3. Put the parameters: final value of the independent variable and integration step-size (h) inthe first cells of the worksheet. Rearrange the equations in the order: constant definitions,functions of the constants, independent variable, dependent variables, explicit functions ofthe variables and differential equations.
ODE ODE –– Initial Value Problems Initial Value Problems
Instructions for Conversion (2)Instructions for Conversion (2)
4. Copy the right hand side of the equations into the adjacent cell and replace the variable namesby variable addresses. Use absolute addressing for the constants and the functions of constantand relative addressing for the variables and functions of the variables. In the cell adjacent tothe variables put their initial values.
5. Copy the section starting with the independent variable up to the end of the equation set andpaste this section three times below, to obtain the values of k2, k3 and k4. Change the equationsas needed to reflect the change in the variable values, as shown in Equation (2).
6. In the next column write the equations to calculate the advanced values of the independent anddependent variables.
7. Copy and paste the columns (or rows) as many time as needed in order to reach the finalvalue of the independent variable.
ODE ODE –– Initial Value Problems Initial Value Problems
Ordered POLYMATH FileOrdered POLYMATH File
The use of this procedure is demonstrated in reference to Demo 9.
Differential equations as entered by the user [1] d(T1)/d(t) = (W*Cp*(T0-T1)+UA*(Tsteam-T1))/(M*Cp) [2] d(T2)/d(t) = (W*Cp*(T1-T2)+UA*(Tsteam-T2))/(M*Cp) [3] d(T3)/d(t) = (W*Cp*(T2-T3)+UA*(Tsteam-T3))/(M*Cp)Explicit equations as entered by the user [1] W = 100 [2] Cp = 2.0 [3] T0 = 20 [4] UA = 10. [5] Tsteam = 250 [6] M = 1000
ODE ODE –– Initial Value Problems Excel Formulas (1) Initial Value Problems Excel Formulas (1)
=$C$6*($C$7*$C$8*(C14-C15)+$C$10*($C$11-C15))/($C$12*$C$8)
k12=h*d(T2)/d(t)=h*(W*Cp*(T1-T2)+UA*(Tsteam-T2))/(M*Cp) 18
=$C$6*($C$7*$C$8*($C$9-C14)+$C$10*($C$11-C14))/($C$12*$C$8)
k11=h*d(T1)/d(t)=h*(W*Cp*(T0-T1)+UA*(Tsteam-T1))/(M*Cp)Differential equations17
20T3 1620T2 1520T1Dependent variables140tIndependent variable13=1000M=1000 12=250Tsteam=250 11=10UA=10. 10=20T0=20 9=2Cp=2.0 8=100W=100Constants7=($C$5-$C$13)/200hIntegration step size6200tf=200Final value (ind. Var.)5Equations/valuesDefinitions 4
CBA
ODE ODE –– Initial Value Problems Initial Value Problems
Excel Formulas (2)Excel Formulas (2)
=$C$6*($C$7*$C$8*(C27-C28)+$C$10*($C$11-C28))/($C$12*$C$8)
k3331
=$C$6*($C$7*$C$8*(C26-C27)+$C$10*($C$11-C27))/($C$12*$C$8)
k3230
=$C$6*($C$7*$C$8*($C$9-C26)+$C$10*($C$11-C26))/($C$12*$C$8)
k3129
=C16+$C$6*C25/2T3+k23/228
=C15+$C$6*C24/2T2+k22/227
=C14+$C$6*C23/2T1+k21/226
=$C$6*($C$7*$C$8*(C21-C22)+$C$10*($C$11C22))/($C$12*$C$8)
k2325
=$C$6*($C$7*$C$8*(C20-C21)+$C$10*($C$11-C21))/($C$12*$C$8)
k2224
=$C$6*($C$7*$C$8*($C$9-C20)+$C$10*($C$11-C20))/($C$12*$C$8)
k2123
=C16+$C$6*C19/2T3+k13/222
=C15+$C$6*C18/2T2+k12/221
=C14+$C$6*C17/2T1+k11/220
CBA
ODE ODE –– Initial Value Problems Initial Value Problems
Excel Formulas (3)Excel Formulas (3)
In column D the solution is advanced from xn to hxx nn +≡+1
=$C$6*($C$7*$C$8*(C33-C34)+$C$10*($C$11-C34))/($C$12*$C$8)
k4337
=$C$6*($C$7*$C$8*(C32-C33)+$C$10*($C$11-C33))/($C$12*$C$8)
k4236
=$C$6*($C$7*$C$8*($C$9-C32)+$C$10*($C$11-C32))/($C$12*$C$8)
k4135
=C16+$C$6*C31T3+k3334
=C15+$C$6*C30T2+k3233
=C14+$C$6*C29T1+k3132
CBA
=C16+(1/6)*(C19+2*C25+2*C31+C37)20T3 16
=C15+(1/6)*(C18+2*C24+2*C30+C36)20T2 15
=C14+(1/6)*(C17+2*C23+2*C29+C35)20T1Dependent variables14
=C13+$C$60tIndependent variable13
DCBA
ODE ODE –– Initial Value Problems Initial Value Problems
Results for Results for t=1 min and t=80 mint=1 min and t=80 min (1) (1)
1.0603E-021.144091.15k13
2.3274E-031.138911.15k12
2.5860E-041.035371.15k11Differential equations
51.1930321.1470820T3
41.3587121.1453220T2
30.9499221.0916820T1Dependent variables
8010tIndependent variable
1000M=1000
250Tsteam=250
10UA=10.
20T0=20
2Cp=2.0
100W=100Constants
1hIntegration step size
200tf=200Final value (ind. var.)
Equations/valuesDefinitions
ODE ODE –– Initial Value Problems Initial Value Problems
Results for t=1 min and t=80 min (2)Results for t=1 min and t=80 min (2)
9.7559E-031.137171.14398k432.1185E-031.118801.13913k422.3280E-040.932071.03526k4151.2032122.2878121.14713T3+k3341.3609322.2739121.14426T2+k3230.9501622.0755521.09279T1+k31
1.0180E-021.140731.14713k332.2232E-031.128591.14426k322.4574E-040.983871.09279k3151.1981121.7175720.57356T3+k23/241.3598221.7107620.57356T2+k22/230.9500421.5821920.54481T1+k21/2
1.0162E-021.140971.147125k232.2181E-031.130891.147125k222.4502E-040.981021.089625k2151.1983321.7191320.575T3+k13/241.3598721.7147720.575T2+k12/230.9500521.6093720.575T1+k11/2
ODE ODE –– Initial Value Problems Initial Value Problems
Plot of the resultsPlot of the results
H eat E xch an g e in a S eries o f T an ks
20
25
30
35
40
45
50
55
0 50 100
Tim e (m in)
Temperature (deg. C
T 1 (t)
T2(t)
T3(t)
ODE ODE –– Boundary Value Problems Boundary Value Problems
Solution MethodSolution Method
There are no tools in Excel to solve differential equations so the solution algorithm must bebuild into the solution worksheet. In this example a fixed step size, explicit, Euler algorithmis used. After setting up the worksheet for integrating the differential equations the "GoalSeek" (for the case of one boundary value) or the "Solver" (for the case of several boundaryvalues) is used for converging to the proper initial values.The formula for the Euler method is
(3)
This formula advances a solution from xn to
Steps of the Solution.
1. Copy the differential equation and the ordered explicit algebraic equations from thePOLYMATH solution report.
2. Paste the equations into an Excel worksheet; remove the text and the equation numbers.
),(1 nnnn yxhfyy +=+
hxx nn +≡+1
ODE ODE –– Boundary Value Problems Boundary Value Problems
Steps of the SolutionSteps of the Solution
3. Put the parameters: final value of the independent variable and integration step-size (h)in the first cells of the worksheet. Rearrange the equations in the order: constant definitions,functions of the constants, independent variable, dependent variables, explicit functionsof the variables and differential equations.
4. Copy the right hand side of the equations into the adjacent cell and replace the variablenames by variable addresses. In the cell adjacent to the variables put their initial values. If theinitial value is not known put initial estimates, instead.
5. In the next column write the equations to calculate the advanced values of the variablesusing Equation 3.
6. Copy and paste the columns as many times as needed in order to reach the final value ofthe independent variable.
7. Use the "Goal Seek" (for the case of one boundary value) or the "Solver" (for the case ofseveral boundary values) to converge to the desired final value of the variables whilechanging their initial values.
ODE ODE –– Boundary Value Problems Boundary Value Problems
POLYMATH File and Excel FormulasPOLYMATH File and Excel Formulas
The use of this procedure is demonstrated in reference to Demo 8.
Differential equations as entered by the user [1] d(CA)/d(z) = y [2] d(y)/d(z) = k*CA/DAB Explicit equations as entered by the user [1] k = 0.001 [2] DAB = 1.2E-9
=$C$8*D11/$C$9=$C$8*C11/$C$9f2=d(y)/d(z) = k*CA/DAB 14
=D12=C12f1=d(CA)/d(z) = yDifferential equations13
=C12+$C$7*C14-150y 12
=C11+$C$7*C130.2CADependent variables11
=C10+$C$70zIndependent variable10
0.0000000012DAB = 1.2E-9 9
0.001 k = 0.001Constants8
=(C6-C10)/100hIntegration step-size7
0.001zf=0.001Final value ind.var.6
5
Equations/valuesDefinitions 4
DCBA
ODE ODE –– Boundary Value Problems Boundary Value ProblemsResults at z = 0, 0.00001 and 0.001 mResults at z = 0, 0.00001 and 0.001 m
1. Initial estimate: y = -150
2. After using of the "Goal Seek" tool to set the value of y(0.001) at zero while changing y(0).
9.7563E+04165416.6671.6667E+05f2=d(y)/d(z) =k*CA/DAB
-26.052-148.333-150f1=d(CA)/d(z) = yDifferential equations
-26.052-148.333-150y
0.117080.19850.2CADependent variables
0.0010.000010zIndependent variable
1.1475E+05-130.2471.6667E+05f2=d(y)/d(z) =k*CA/DAB
-7.53E-140.13770-131.913f1=d(CA)/d(z) = yDifferential equations-7.53E-140.19868-131.913y 0.137700.0010.2CADependent variables0.0010.000010zIndependent variable
DAE DAE –– Initial Value Problems Initial Value Problems
Solution MethodSolution Method
There are no tools in Excel to solve differential equations so the solution algorithm must be buildinto the solution worksheet. In this example a fixed step size, implicit, Euler algorithm is used.Using this method the differential equations are converted into nonlinear algebraic equations.Thus, in each integration step a system of nonlinear algebraic equations is solved using the"Solver" tool. The formula for the implicit Euler method is
(4)
This formula advances a solution from xn-1 to for n>1.
0)},(),({2 111 =
++−= −−− nnnnnnn yxfyxfhyyF
hxx nn +≡ −1
DAE DAE –– Initial Value Problems Initial Value Problems
Steps of the Solution (1)Steps of the Solution (1)
1. Copy the differential equations and the ordered explicit algebraic equations from thePOLYMATH solution report.
2. Paste the equations into an Excel worksheet; remove the text and the equation numbers.
3. Put the parameters: final value of the independent variable and integration step-size (h) inthe first cells of the worksheet. Rearrange the equations in the order: constant definitions,functions of the constants, independent variable, dependent variables, explicit functions ofthe variables, differential equations and implicit algebraic equations.
4. Add an equation with the sum of squares of the implicit functions (the algebraic equationsand the implicit Euler method representation of the differential equations).
DAE DAE –– Initial Value Problems Initial Value Problems
Steps of the Solution (2)Steps of the Solution (2)
5. Copy the right hand side of the equations into the adjacent cells and replace the variablenames by variable addresses. Use absolute addressing for the constants and the functions ofconstant and relative addressing for the variables and functions of the variables. In the celladjacent to the variables put their initial values. In the cell containing the sum of squares ofthe function values include only the functions associated with the implicit algebraic equations.
6. Use the "Solver" (or "Goal Seek" tools) to find the initial values of the unknownsassociated with the implicit algebraic equations.
7. In the next column write the equations to calculate the advanced values of the independentand dependent variables.
8. From this point on the columns can be copied and pasted, as many time as needed to reachthe final value of the independent variable. The "Solver" tool must be applied on thecolumns sequentially, to solve the system of nonlinear algebraic equations for each step
DAE DAE –– Initial Value Problems Initial Value Problems
POLYMATH FilePOLYMATH File
The use of this procedure is demonstrated in reference to Demo 11 The differential equationsand the ordered explicit algebraic equations as copied from the POLYMATH solution reportare the following.
Differential equations as entered by the user [1] d(L)/d(x2) = L/(k2*x2-x2) [2] d(T)/d(x2) = Kc*err Explicit equations as entered by the user [1] Kc = 0.5e6 [2] k2 = 10^(6.95464-1344.8/(T+219.482))/(760*1.2) [3] x1 = 1-x2 [4] k1 = 10^(6.90565-1211.033/(T+220.79))/(760*1.2) [5] err = (1-k1*x1-k2*x2)
DAE DAE –– Initial Value Problems Excel formulas Initial Value Problems Excel formulas
In the next column (column D) the definition of the independent variable is changed to:=C8+$C$7 and the definition of the sum of squares of errors ischanged to: =(D9-(C9+($C$7/2)*(C14+D14)))^2+D15^2 .
[Ln-(Ln-1+h/2(f1n+f1n-1))]^2+f2n+1^2Sum of squares oferrors16
=(1-C12*C11-C13*C8)f2=f(T)=(1-k1*x1-k2*x2)=0 15= C9/(C13*C8-C8)f1=d(L)/d(x2) = L/(k2*x2-x2)Differential equations14
= 10^(6.95464-1344.8/(C10+219.482))/(760*1.2)
k2 = 10^(6.95464-1344.8/(T+219.482))/(760*1.2) 13
= 10^(6.90565-1211.033/(C10+220.79))/(760*1.2)
k1 = 10^(6.90565-1211.033/(T+220.79))/(760*1.2) 12
= 1-C8x1 = 1-x2Explicit equations1195T= 10100L=Dependent variables90.4x2Independent variable8=($C$6-$C$8)/20hIntegration step-size70.8x2(f)=Final value ind.var.6
Equations/valuesDefinitions 5CBA
DAE DAE –– Initial Value Problems Initial Value Problems
Results for x2 = 0.4 and 0.42Results for x2 = 0.4 and 0.42
Results obtained by applying "Goal Seek" to set cell C15 at zero while changing the initialtemperature (cell C10) and subsequently applying the "Solver" tool to minimize the value incell D16 while changing the contents of cells D9 and D10.
4.4521E-08[Ln-(Ln-1+h/2(f1n+f1n-1))]^2+f2n+1^2Sum of squares of errors
-2.1100E-046.8122E-05f2=f(T)=(1-k1*x1-k2*x2)=0
-467.601-534.754f1=d(L)/d(x2) = L/(k2*x2-x2)Differential equations
0.541850.53250k2 = 10^(6.95464-1344.8/(T+219.482))/(760*1.2)
1.33211.3116k1 = 10^(6.90565-1211.033/(T+220.79))/(760*1.2)
0.580.6x1 = 1-x2Explicit equations96.14295.583T= 89.976100L=Dependent variables0.420.4x2Independent variable
0.02hIntegration step-size 0.8x2(f)=Final value ind.var.
DAE DAE –– Initial Value Problems Initial Value Problems
Results for x2 = 0.8Results for x2 = 0.8
Column D is copied and pasted as many time as necessary to reach the final value of x2 (= 0.8).The "Solver" tool is applied sequentially, for every column to minimize the value in row 16.
5.1713E-07[Ln-(Ln-1+h/2(f1n+f1n-1))]^2+f2n+1^2Sum of squares of errors
-6.4625E-04f2=f(T)=(1-k1*x1-k2*x2)=0
3.1544E-04f1=d(L)/d(x2) = L/(k2*x2-x2)Differential equations
0.78634k2 =
1.8579k1 =
0.2x1 = 1-x2Explicit equations
108.595T=
14.006L=Dependent variables
0.8x2Independent variable
Partial Differential Equations Excel Formulas for Demo 12 (1)Partial Differential Equations Excel Formulas for Demo 12 (1)
The system of PDEs is converted into a system of first order ODEs using the method of lines.Explicit Euler's method is used for solution.
=(4*D$22-D$21)/3=(4*C$22-C$21)/3T11 = (4*T10-T9)/3 23=C22+($F$29-$F$28)*C32100T10 22=C21+($F$29-$F$28)*C31100T9 21=C20+($F$29-$F$28)*C30100T8 20=C19+($F$29-$F$28)*C29100T7 19=C18+($F$29-$F$28)*C28100T6 18=C17+($F$29-$F$28)*C27100T5 17=C16+($F$29-$F$28)*C26100T4 16=C15+($F$29-$F$28)*C25100T3 15=C14+($F$29-$F$28)*C24100T2Variables14=C13+$C$80tIndependent Variable13 =C10/C11^2alpha/deltax^2 12 0.1deltax = .10 11 0.00002alpha = 2.e-5 10 0T1 = 0Constants9 =(C7-C13)/200hIntegration Step-size8 6000tf=6000Final value ind.var.7 Equations/valuesDefinitions 6
DCBA
Partial Differential EquationsPartial Differential Equations
Excel Formulas for Demo 12 (2)Excel Formulas for Demo 12 (2)
=$C$12*($P28-2*$O28+$N28)f9=d(T10)/d(t) = alpha/deltax^2*(T11-2*T10+T9) 32
=$C$12*($O28-2*$N28+$M28)f8=d(T9)/d(t) = alpha/deltax^2*(T10-2*T9+T8) 31
=$C$12*($N28-2*$M28+$L28)f7=d(T8)/d(t) = alpha/deltax^2*(T9-2*T8+T7) 30
=$C$12*($M28-2*$L28+$K28)f6=d(T7)/d(t) = alpha/deltax^2*(T8-2*T7+T6) 29
=$C$12*($L28-2*$K28+$J28)f5=d(T6)/d(t) = alpha/deltax^2*(T7-2*T6+T5) 28
=$C$12*($K28-2*$J28+$I28)f4=d(T5)/d(t) = alpha/deltax^2*(T6-2*T5+T4) 27
=$C$12*($J28-2*$I28+$H28)f3=d(T4)/d(t) = alpha/deltax^2*(T5-2*T4+T3) 26
=$C$12*($I28-2*$H28+$G28)f2=d(T3)/d(t) = alpha/deltax^2*(T4-2*T3+T2) 25
=$C$12*($H28-2*$G28+$C$9)f1=d(T2)/d(t) = alpha/deltax^2*(T3-2*T2+T1)
Differentialequations24
CBA
Column D for this section is obtained by copying and pasting the same section in column C.To obtain the complete solution column D is copied and pasted as many times as needed forreaching the final time.
Partial Differential EquationsPartial Differential Equations
Results for Results for t=0, 30 and 6000 mint=0, 30 and 6000 min
91.6912100100T11 = (4*T10-T9)/390.8270100100T1088.2343100100T983.8125100100T877.4300100100T768.9780100100T658.4307100100T545.8980100100T431.6589100100T316.163294100T2
6000300t 2.00E-03alpha/deltax^2 0.1deltax = .10 2.00E-05alpha = 2.e-5 0T1 = 0 30h 6000tf=6000 Equations/valuesDefinitions
Partial Differential EquationsPartial Differential Equations
Plot of Some Results for Demo 12Plot of Some Results for Demo 12
T em perature P rofiles for a O ne-D im entional S lab
0
20
40
60
80
100
120
0 2000 4000 6000
T im e (s)
Temperature (C)
T2
T3
T4
T5
Multiple Linear RegressionMultiple Linear Regression
Copying the Data from POLYMATHCopying the Data from POLYMATHIn this demonstration Riedel's equation is fitted to the data of Demo 6.
Multiple Linear RegressionMultiple Linear Regression
Pasting the Data into Excel and Adding TitlesPasting the Data into Excel and Adding Titles
2.8808141.25E+052.54808220.0028308611
2.602061.11E+052.52342130.0029962510
2.3010399445.622.49879280.003171089
289550.562.47603420.003341698
1.77815183261.12.46022110.00346567
1.6020678820.562.44831980.003561896
1.3010373197.32.43224750.003696175
168460.722.41772070.00382194
0.6989764287.62.40406360.0039443
055908.62.37373930.004229222
logPT2logTTrec1
DCBA
Multiple Linear RegressionMultiple Linear Regression
Using the LINEST FunctionUsing the LINEST Function
The LINEST function puts the full set of results in an area that includes 5 rows and numberof columns as the number of the parameters.
For this problem mark an area of 5 rows and 4 columns. Type in LINEST(D2:D11, A2:C11,TRUE,TRUE) and press CONTROL+SHIFT+ENTER to enter this formula into all themarked cells.
Note that the range D2:D11 is the range where the dependent variable values are stored, therange A2:C11 is the range where the independent variable values are stored, the first logicalvariable TRUE (or the number 1) indicates that the parameter a0 cannot be assumed to be zeroand the second logical variable TRUE indicates that a matrix of regression statistics shouldalso be returned.
It is permitted to mark a one-, two-, three-, four-, or five-row array depending on the amount ofinformation desired.
The results obtained do not include any labeling and labeling should be added manually.
Multiple Linear RegressionMultiple Linear Regression
Results (1)Results (1)
For obtaining the results reported by POLYMATH the first three rows are significant. Thefirst row (coeff.s) contains the values of the parameters. The second row (std. dev. S.) containsthe standard deviation of the parameters. These values can be multiplied by the appropriatevalue from the t distribution to obtain the 95% confidence intervals. The square of thestandard error in y (SE y) is the variance as reported by POLYMATH.
#N/A#N/A0.0017777.1446SS(reg),SS(resid)
#N/A#N/A68042.39F, df
#N/A#N/A0.0172080.99975R2, SE (y)
63.9211984.9623.877062.0439E-05std.dev.s
216.721-9318.66-75.74824.4446E-05coeff.s
a0a1a2a3
Multiple Linear RegressionMultiple Linear Regression
Variance and Confidence intervals and ResidualsVariance and Confidence intervals and Residuals
Removing the extra rows from the results table and adding the calculations of the confidenceintervals and the variance yields the following table (only the first two columns out of the fourare shown).
Note that the t value for 95% confidence intervals with 6 degrees if freedom is: t = 2.4469.
=C16^2Variance18
=C15*2.4469=B15*2.446995% conf. int.17
=LINEST(D2:D11,A2:C11,1,1)=LINEST(D2:D11,A2:C11,1,1)R2, SE (y)16
=LINEST(D2:D11,A2:C11,1,1)=LINEST(D2:D11,A2:C11,1,1)std.dev.s15
=LINEST(D2:D11,A2:C11,1,1)=LINEST(D2:D11,A2:C11,1,1)coeff.s14
a2a3 13
CBA
=D2-E2=$E$14+$D$14*A2+$C$14*B2+$B$14*C22
residuallogP(calc)1
FE
Polynomial RegressionPolynomial Regression
Options and InstructionsOptions and Instructions
The LINEST function and "Regression" tool from the "Analysis ToolPak" can be used forcarrying out linear regression. The LINEST function has the advantages over the "Regression"tool that the calculation results are automatically updated when the data is modified and theresults are easier to rearrange for documentation purposes. The "Regression" tool provides morestatistical data and the output is clearly labeled.
The use of the LINEST function for carrying out polynomial regression will be demonstratedhere in reference to Problem 2.3a in the book of Cutlip and Shacham. To prepare the data filearrange the columns of data so that the column of the dependent variable and the column of theindependent variable are next to each other and put the column of the independent variable asthe last one.
Copy these columns of the data from the POLYMATH data table and paste them into an Excelworksheet. Define additional columns that contain increasing powers of the independentvariable, up to the 5th degree.
Polynomial RegressionPolynomial Regression
Excel Formulas and Numerical ValuesExcel Formulas and Numerical Values
Numerical values
=B3^5=B3^4=B3^3=B3^210041.33
=B2^5=B2^4=B2^3=B2^25034.062
TK5TK4TK3TK2TKCp1
FEDCBA
…
2.356E+127.902E+092.650E+078.889E+04298.1573.67
7.594E+155.063E+123.375E+092.250E+061500205.8920
1.521E+125.568E+092.038E+077.462E+04273.1668.746
3.200E+111.600E+098.000E+064.000E+0420056.075
7.594E+105.063E+083.375E+062.250E+0415048.794
1.000E+101.000E+081.000E+061.000E+0410041.33
3.125E+086.250E+061.250E+052.500E+035034.062
TK5TK4TK3TK2TKCp1
FEDCBA
Polynomial RegressionPolynomial Regression
Using the LINEST Function for a 2Using the LINEST Function for a 2ndnd Degree Polynomial Degree Polynomial
To solve for a second order polynomial (with three parameters) mark an area of 3 rows and 3columns.
Type in LINEST(A2:A20, B2:C20, TRUE,TRUE) and press CONTROL+SHIFT+ENTERto enter this formula into all the marked cells.
Note that the range A2:A20 is the range where the dependent variable values are stored, therange B2:C20 is the range where the independent variable values are stored, the first logicalvariable TRUE (or the number 1) indicates that the parameter a0 cannot be assumed to be zeroand the second logical variable TRUE indicates that a matrix of regression statistics should alsobe returned.
The results obtained do not include any labeling and labeling is added manually.
Polynomial RegressionPolynomial Regression
Results for a 2Results for a 2ndnd Degree Polynomial Degree Polynomial
#N/A2.6106265970.998331R2, SE (y)26
1.609708680.0054367943.53E-06std.dev.s25
17.74273280.21778651-6.16E-05coeff.s24
a0a1a2 23
DCBA
=C26^2Variance28
=2.1199*C25=2.1199*B2595% conf. int.27
=LINEST(A2:A20,B2:C20,1,1)=LINEST(A2:A20,B2:C20,1,1)R2, SE (y)26
=LINEST(A2:A20,B2:C20,1,1)=LINEST(A2:A20,B2:C20,1,1)std.dev.s25
=LINEST(A2:A20,B2:C20,1,1)=LINEST(A2:A20,B2:C20,1,1)coeff.s24
a1a2 23
CBA
Calculation of the confidence intervals and the variance (only the first two columns out of thefour are shown).
Multiple Nonlinear RegressionMultiple Nonlinear Regression
InstructionsInstructions
To carry out multiple nonlinear regression an objective function containing the sum ofsquares of the errors is prepared and this objective function is be minimized by means ofthe "Solver" tool by changing the regression model parameters.
Demo 6c is used as an Example.
In this particular example the Antoine equation is fitted to vapor pressure (Vp) versustemperature (T °C) data. Thus, the objective function to be minimized is the following.
(5)
After copying the independent and dependent variable data from the POLYMATH file andpasting them into an Excel worksheet the objective function can be calculated in threesuccessive columns.
2
1
2
)(^10∑
=
++−=
N
j jj TC
BAVps
Multiple Nonlinear RegressionMultiple Nonlinear Regression
Excel FormulasExcel Formulas
In this table the initial estimates for the parameters A, B and C are also shown.
=D9^2=B9-C9=10^($B$1+$B$2/(A9+$B$3))5-19.69
=D8^2=B8-C8=10^($B$1+$B$2/(A8+$B$3))1-36.78
of Residuals(Pv)-(Pv)calc10^(A+B/(TC+C)) 7
Sum of Sqrs.Residual(Pv)calcPv (mmHg)TC6
5
=E18/(10-3)Variance4
273C3
-2035.33B2
8.752A1
EDCBA
Multiple Nonlinear RegressionMultiple Nonlinear Regression
Numerical Values at the Initial EstimateNumerical Values at the Initial Estimate
47701.1174Sum 1845102.0196-212.3724972.372476080.117
131.242211.456188.543910026.114…
110.046710.490349.50976015.41371.99968.485331.5147407.61210.34203.215916.784120-2.6110.48300.69509.305010-11.5100.0611-0.24715.24715-19.690.1415-0.37621.37621-36.78
of Residuals(Pv)-(Pv)calc10^(A+B/(TC+C)) 7Sum of SquaresResiduallog(Pv)calcPv (mmHg)TC6
5 6814.45Variance4EDCBA
Multiple Nonlinear RegressionMultiple Nonlinear Regression
Numerical Values at the SolutionNumerical Values at the Solution
The sum of squares of errors is stored in cell C18. The "Solver" tool is used to minimizethis value while changing the values of the parameters A, B and C (in cells B1, B2 and B3).
2.2819Variance4
202.14C3
-1054.98B2
6.6185A1
BA
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