CH-08 (523178)
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8Partial DifferentialEquations8.1 INTRODUCTIONDifferent engineering disciplines solve different types of problems in their respectivefields. For mechanical engineers, they may need to solve the temperature changewithin a solid when it is heated by the interior heat sources or due to a rise ordecrease of its boundary temperatures. For electrical engineers, they may need tofind the voltages at all circuit joints of a computer chip board. Temperature andvoltage are the variables in their respective fields. Hence, they are called fieldvariables. It is easy to understand that the value of the field variable is spacedependent and time-dependent. That is to say, that we are interested to know thespatial and temporal changes of the field variable. Let us denote the field variableas . and let the independent variables be xi which could be the time t, or, the spacecoordinates as x, y, and z.
In order not to overly complicate the discussion, weintroduce the general two-dimensional partial differential equation which governsthe field variable in the form of:A(x1, x 2 )∂2 φ∂2 φ∂2 φ+ B(x1, x 2 )+ C(x1, x 2 ) 22∂x1∂x1∂x 2∂x 2∂φ ∂φ = F x1, x 2 , φ,,∂x1 ∂x 2 (1)where the coefficient functions A, B, and C in the general cases are dependent onthe variables x1 and x2, and the right-hand-side function F, called forcing functionmay depend not only on the independent variables x1 and x2 but may also dependon the first derivatives of . There are innumerable of feasible solutions for Equation1. However, when the initial and/or boundary conditions are specified, only particularsolution(s) would then be found appropriate.In this chapter, we will discuss three simple cases when A, B, and C are allconstants.
The first case is a two-dimensional, steady-state heat conduction probleminvolving temperature as the field variable and only the spatial distribution of thetemperature needs to be determined, Equation 1 is reduced to a parabolic partialdifferential equation named after Poisson and Laplace when the forcing function Fis not equal to, or, equal to zero, respectively. This is a case when the coefficientfunctions in Equation 1 are related by the condition B2–4AC<0.The second case is a one-dimensional, transient heat conduction problem.
Again,the field variable is the temperature which is changing along the longitudinal x-axis© 2001 by CRC Press LLCof a straight rod and also in time. That is, x1 becomes x and x2 become the time t.Equation 1 is reduced to an elliptical partial differential equation. This is a casewhen B2–4AC = 0.The third case is the study of the vibration of a tightened string. The field variableis the lateral deflection of this string whose shape is changing in time. Equation 1is reduced to a hyperbolic partial differential equation. If x is the longitudinal axisof the string, then same as in the second case, the two independent variables are xand t. This is a case when B2–4AC>0.The reason that these problems are called parabolic, elliptical, and hyperbolicis because their characteristic curves have such geometric features.
Readers interested in exploring these features should refer to a textbook on partial differentialequations.Details will be presented regarding how the forward, backward, and centraldifferences discussed in Chapter 4 are to be applied for approximating the first andsecond derivative terms appearing in Equation 1. Repetitive algorithms can bedevised to facilitate programming for straight-forward computation of the spatialand temporal changes of the field variable. Numerical examples are provided toillustrate how these changes can be determined by use of either QuickBASIC,FORTRAN, MATLAB, or, Mathematica programs.Although explanation of the procedure for numerical solution of these threetypes of problems is given only for the simple one- and two-dimensional cases, butits extension to the higher dimension case is straight forward.
For example, one mayattempt to solve the transient heat conduction problem of a thin plate by having twospace variables instead of one space variable for a long rod. The steady-state heatconduction problem of a thin plate can be extended for the case of a three-dimensional solid, and the string vibration problem can be extended to a two-dimensionalmembrane problem.8.2 PROGRAM PARABPDE — NUMERICAL SOLUTIONOF PARABOLIC PARTIAL DIFFERENTIAL EQUATIONSThe program ParabPDE is designed for numerically solving engineering problemsgoverned by parabolic partial differential equation in the form of:∂φ∂2φ=a 2∂t∂x(1)and is a function of t and x and satisfies a certain set of supplementary conditions.Equation 1 is called a parabolic partial differential equation.
For example, couldbe the temperature, T, of a longitudinal rod shown in Figure 1 and the parameter ain Equation 1 could be equal to k/c where k, c, and are the thermal conductivity,specific heat, and specific weight of the rod, respectively. To make the problem morespecific, the rod may have an initial temperature of 0°F throughout and it is completely insulated around its lateral surface and also at its right end.
If its left end isto be maintained at 100°F beginning at the time t = 0, then it is of interest to know© 2001 by CRC Press LLCFIGURE 1. could be the temperature, T, of a longitudinal rod.how the temperatures along the entire length of the rod will be changing as the timeprogresses. This is therefore a transient heat conduction problem.
One would liketo know how long would it take to have the entire rod reaching a uniform temperatureof 100°F.If the rod is made of a single material, k/c would then be equal to a constant.Analytical solution can be found for this simple case.1 For the general case that therod may be composed of a number of different materials and the physical propertiesk, c, and would not only depend on the spatial variable x but may also depend onthe temporal variable t. The more complicated the variation of these properties in xand t, the more likely no analytical solution is possible and the problem can onlybe solved numerically. The finite-difference approximation of Equation 1 can beachieved by applying the forward difference for the first derivative with respectiveto t and central difference for the second derivative with respect to x as follows (fort at ti and x at xj):∂T Ti +1, j − Ti, j=˙∂t∆tand∂2 T Ti, j−1 − 2 Ti, j + Ti, j+1=˙∂x 2( ∆x)2If k/c is changing in time and also changing from one location to another, wecould designate it as ai,j.
As a consequence, Equation 1 can then be written as:Ti +1, j − Ti, j∆t= a i, jTi, j−1 − 2 Ti, j + Ti, j+1( ∆x)2(2)Since the initial temperature distribution T is known, the above expressionsuggests that for a numerical solution we may select an appropriate increment in t,t, and the temperature be determined at a finite number of stations, N.
It is advisableto have these stations be equally spaced so that the increment x is equal to L/(N–1)where L is the length of the rod, and the instants are to be designated as t1 = 0, t2 =t,…, ti = (i–1)t, and the stations as x1 = 0, x2 = x,…, xj = (j–1)∆x,…, and xN =(N–1)x = L. The task at hand is then to find T(ti,xj) for i = 1,2,… and j = 1,2,…,N.It can be noticed from Equation 2 that the there is only one temperature at ti + 1 andcan be expressed in terms of those at the preceding instant ti as:© 2001 by CRC Press LLCTi +1, j = Ti, j +a i, j∆t( ∆x)2(Ti , j−1− 2 Ti, j + Ti, j+1)(3)Equation 3 is to be used for j = 2 through j = N–1.
For the last station, j = N,which is insulated,the temperatures on both side of this station can be assumed tobe equal (the station N + 1 is a fictitious one!). The modified equation for thisparticular station is:Ti +1,N = Ti,N +2a i,N ∆t( ∆x)2(Ti ,N −1− Ti,N)(4)For generating the temperatures of the rod at N stations for any specified timeincrement t until the temperatures are almost all equal to 100°F throughout, theprogram ParabPDE has been applied. It is listed below along with a typical printoutof the results.FORTRAN VERSION© 2001 by CRC Press LLCSample Output© 2001 by CRC Press LLCQUICKBASIC VERSIONMATLAB APPLICATIONSA MATLAB version of ParabPDE can be created easily by converting theQuickBASIC program.
The m file may be arranged as follows:© 2001 by CRC Press LLCFor solving the sample transient temperature problem, this m file can be calledand interactive MATLAB instructions can be entered through keyboard to obtainthe temperature distribution of the rod at various times:© 2001 by CRC Press LLCFIGURE 2. A composite graph with axes labels and markings of the curves by making useof the MATLAB commands xlabel, ylabel, and text.Notice that results of temperature distributions which have been terminated usingfive differentials of 1, 0.1, 0.01, 0.001, and 0.0001 Fahrenheit are saved in Tsaveand then later plotted.
Since T is a row matrix, the transpose of T, T, is stored inan appropriate column of Tsave. If the temperatures were rounded, the rod reachesa uniform temperature distribution of 100°F when the required temperature differential is selected to be 0.001°F. If the fifth curve for the temperature differentialequal to 0.0001 is plotted, it will be too close to the forth curve and also the plotfunction provides only four line types (solid, broken, dot, and center lines), the fifthset of results is therefore not saved in Tsave.
Figure 2 shows a composite graph withaxes labels and markings of the curves by making use of the MATLAB commandsxlabel, ylabel, and text.MATHEMATICA APPLICATIONSThe heat conduction problem of an insulated rod previously discussed in theversions for FORTRAN, QuickBASIC, and MATLAB can be solved by applicationof Mathematica as follows:© 2001 by CRC Press LLCIn[1]: = (n = 11; rk = 0.037; c = 0.212; rho = 168; dt = 1; dx = 0.1;c1 = rk/c/rho*dt/dx^2; nm1 = n–1;)In[2]: = (t = Table[0,{n}]; tn = Table[0,{n}]; t[[1]] = 100; tn[[1]] = 100; tdf = 1;tm = 0; flag1 = 0;In[3]: = (While[flag1 = = 0, Do[tn[[j]] = t[[j]] + c1*(t[[j–1]]–2*t[[j]] + t[[j + 1]]),{j,2,nm1}];tn[[n]] = t[[n]] + 2*c1*(t[[nm1]]-t[[n]]);tm = tm + dt;flag1 = 1;Do[If[Abs[tn[[I]]-t[[I]]]>tdf, flag1 = 0; Break,Continue],{i,n}];Do[t[[I]] = tn[[I]],{i,n}]]; Print[“t = “,tm]; Print[N[tn,3]])Out[3] = t = 24{100., 65.7, 37.5, 18.5, 7.83, 2.85, 0.892, 0.241, 0.0561, 0.0116, 0.00394}The temperature distribution of the heated rod after 24 seconds is same asobtained by the FORTRAN, QuickBASIC, and MATLAB versions when everycomponent of two consecutive temperature distribution, kept as {t} and {tn}, differno more than the allowed set value of tdf = 1 degree in In[2].
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