Numerical CH27

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					                    Chapter 27


Boundary-Value and Eigenvalue Problems
                                                                                                               June 2011


         Ordinary differential equation is accompanied by auxiliary conditions.
  These conditions are used to evaluate the constants of integration that
  result during the solution of the equation. For an nth order equation,
  n conditions are required. If all the conditions are specified at the same
  value of the independent variable, then we are dealing with an
  initial-value problem (next figure a).




  In contrast, there is another application for which the conditions are not
  known at a single point, but rather, are known at different values of the
  independent variable. Because these values are often specified at the
  extreme points or boundaries of a system, they are customarily referred to
  as boundary-value problems (previous figure b).
                                                                              ‫من يبحث عن صددي بدع عيد ، يبقدى‬
                                                                                            ) ‫بع صدي ( مثل تركي‬
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  We will discuss two general approaches for obtaining a solution:
  the shooting method and the finite-difference approach.
  Additionally, we present techniques to approach a special type of
  boundary-value problem: the determination of eigenvalues.


    27.1 General Methods for Boundary-Value Problems


         The conservation of heat can be used to develop a heat balance for a
  long, thin rod. If the rod is not insulated along its length and the system is
  at a steady state, the equation that result is

                                                                                                                   (27.1)

  where h’ is a heat transfer coefficient (m-2) that parameterizes the rate of
  heat dissipation to the surrounding air and Ta is the temperature of the
  surrounding air (⁰C).


         To obtain a solution for the previous equation, there must be
  appropriate boundary conditions. A simple case is where the temperatures
  at the ends of the bar are held at fixed values. These can be expressed
  mathematically as
         T(0) = T1
         T(L) = T2
                                                                             ‫يمكندددددت أن تكدددددون أ دددددل مدددددن كدددددل‬
                                                                             ‫أصدددقا ت بأ ددد يددريقين: االجتهدداد أو‬
                                                                                                      ‫أن تصادق الحمقى‬
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  With these conditions, the previous equation can be solved analytically
  using calculus. For a 10-m rod with Ta = 20, T1 = 40, T2 = 200 and h’ = 0.01,
  the solution is
         T = 73.4523 e0.1x – 53.4523 e-0.1x + 20


  27.1.1 The Shooting Method

         The shooting method is based on converting the boundary-value
  problem into an equivalent initial-value problem. A trial-and-error
  approach is then implemented to solve the initial-value version.
  The approach can be illustrated by an example.


  EXAMPLE 27.1 The Shooting Method


  Problem Statement:
  Use the shooting method to solve Eq.(27.1) for a 10-m rod with h’=0.01
  m-2, Ta = 20, and the boundary conditions
         T(0) = 40          T(10) = 200


  Solution:
  The second-order equation can be expressed as two first-order ODEs:

                                                                                                                     ََْ ْ
                                                                             ‫ولَقَدددددددـد ُكرتردددددددت والرمدددددددا ر نَواهدددددددل‬
                                                                                     َ َ             َ ‫ل‬   َ
                                                                             ‫منلدددـي.......... وبَيدددـا الهندددـد تَقطردددـر‬
                                                                             ‫َ ْ ر َْ َ ْ ر‬                                           َ
                                                                                                                       ‫مـنْ دمـي‬
                                                                                                                           ََ َ

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  Continue:
         To solve these equations, we require an initial value for z.
  For the shooting method, we guess a value- say, z(0) = 10 and we can then
  obtain the solution by integration. For example, using a fourth-order RK
  method with a step size of 2, we obtain a value at the end of the interval
  of T(10) = 168.3797 (next figure a), which differs from the boundary
  condition of T(10) = 200. Therefore, we make another guess, z(0) = 20,
  and perform the computation again. This time, the result of
  T(10) = 285.8980 is obtained (next figure b).


  Now, because the original ODE is linear, the values
         z(0) = 10                   T(10) = 168.3797
  and
         z(0) = 20                   T(10) = 285.8980
  are linearly related. As such, they can be used to compute the value of z(0)
  that yields T(10) = 200. A linear interpolation formula can be employed for
  this purpose:

                                                              (200 – 168.3797) = 12.6907


  This value can then be used to determine the correct solution, as shown in
  the next figure (c).
                                                                                  ،‫مدددن كدددعس بلدددي : عجبدددب البدددن دس‬
                                                                                  ‫يح هللا ويعصاه، ويبغ ني ويطيعني‬

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  Nonlinear Two-Point Problems                                               ‫قد يرى الناس الجر الذي ي رأست‬
                                                                              ‫لكنهم ال يشعرون باأللم الذي تعانيه‬

         For nonlinear boundary-value problems, linear interpolation or
  extrapolation through two solution points will not necessarily result in an
  accurate estimate of the required boundary condition to attain an exact
  solution. An alternative is to perform three applications of the shooting
  method and use a quadratic interpolating polynomial to estimate the
  proper boundary condition.
         However, it is unlikely that such approach would yield the exact
  answer, and additional iterations would be necessary to obtain the
  solution.
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         Another approach for a nonlinear problem involves recasting it as a
  roots problem. Recall that the general form of a root problem is to find the
  value of x that makes the function f(x) = 0. Now, let us use Example 27.1
  to understand how the shooting method can be recast in this form.


         First, recognize that the solution of the pair of differential equations
  is also a “function” in the sense that we guess a condition at the left-hand
  end of the bar, z0, and the integration yields a prediction of the
  temperature at the right-hand end, T10.
  Thus, we can think of the integration as
         T10 = f(z0)                                                           ‫قد يرى الناس الجر الذي دي رأسدت‬
                                                                                 ‫لكنهم ال يشعرون باأللم الذي تعانيه‬

         That is, it represents a process whereby a guess of z 0 yields a
  prediction of T10. Viewed in this way, we can see that what we desire is the
  value of z0 that yields a specific value of T10. If, as in the example,
  we desire T10 = 200, the problem can be posed as
         200 = f(z0)

         By bringing the goal of 200 over to the right-hand side of the
  equation, we generate a new function, g(z0), that represents the
  difference between what we have, f(z0), and what we want, 200.
         g(z0) = f(z0) – 200
  if we drive this new function to zero, we will obtain the solution.
  The next example illustrates the approach.
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  EXAMPLE 27.2 The Shooting Method for Nonlinear Problems
  Problem Statement:
  Suppose that the following nonlinear ODE is used to simulate the
  temperature of the heated bar:



  where h’’ = 5 × 10-8. The reaming problem conditions are as specified in
  Example 27.1
  Solution:
  The second-order equation can be expressed as two first-order ODEs:
                                                                        ‫الندداس ثعثددة أنددوا : نددو كالغددذات تحتددا‬
                                                                      ‫ليددده دا مدددا، وندددو كالددددوات تحتدددا ليددده‬
                                                                                 ‫أ يانا، والثالث كالدات ي رك قط‬

  Now, these equations can be integrated using any of the methods
  described in the previous chapter. We used the constant step-size version
  of the fourth-order RK and we implemented this approach as an Excel
  macro function written in Visual BASIC. The function integrated the
  equations based on an initial guess for z(0) and returned the temperature
  at x = 10. The difference between this value and the goal of 200 was then
  placed in a spreadsheet cell. The Excel Solver was then invoked to adjust
  the value of z(0) until the difference was driven to zero.

  The result is shown in the next figure along with the original linear case.
  As might be expected, the nonlinear case is curved more than the linear
  model. This is due to the power of four term in heat transfer relationship.
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  27.1.2 Finite-Difference Methods


         The most common alternatives to the shooting method are
  finite-difference            approaches.               In     these         techniques,              finite    divided
  differences are substituted for the derivatives in the original equation.
  Thus, a linear differential equation is transformed into a set of
  simultaneous algebraic equations that can be solved using other methods.

  The finite divided-difference approximation for the second derivative is



  This approximation can be substituted into Eq.(27.1) to give



  Collecting terms gives
                                                                                          ‫أعددددر أن معبدددد وسدددديارتت‬
                                                                                          ‫وموبايلددت قيمتهدددا عاليدددة، لكدددن‬
                                                                                                 ‫اول أن تكون أنب األغلى‬
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  This equation applies for each of the interior nodes of the rod.
  The first and last interior nodes,                              and            , respectively, are specified
  by the boundary conditions. Therefore, the resulting set of linear algebraic
  equations will be tridiagonal.


  EXAMPLE 27.3 Finite Difference Approximation of Boundary-Value
  Problems


  Problem Statement:
  Use the finite-difference approach to solve the same problem as in
  Example 27.1


  Solution:
  Employing the parameters in Example 27.1, we can write the above
  equation for the rod.
  Using four interior nodes with a segment length of                                                 m results in the
  following equations:


                                                                             ‫ال يمكنت أن تصنع ما هدو أكثدر ماقدة‬
                                                                             ‫من أن تجل على جان الطري تى‬
                                                                                   ‫يأتيت أ دهم ويحاول مساعدتت‬
  which can be solved for




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  The next table provides a comparison between the analytical solution and
  the numerical solutions obtained in Examples 27.1 and 27.3




  For both numerical methods, these errors can be omitted by decreasing
  their respective step sizes. Although both techniques perform well for the
  present case, the finite-difference approach is preferred because of the
  ease with which it can be extended to more complex cases.


    27.2 Eigenvalue Problems


         Eigenvalue, or characteristic-value, problems are a special class of
  boundary-value problems that are common in engineering problem
  contexts involving vibrations, elasticity, and other oscillating systems.


  27.2.1 Mathematical Background


         We dealt with methods for solving sets of linear algebraic equations
  of the general form                                                          ‫صــداقـددددـات ال تـنـتـهددددـي (( الــمددددـرأ‬
                                                                              .. ‫بــمـر تـهددددـا .. الـقددددـارك بـكـتـابددددـة‬
                                                                                   ))‫الـمـنـا ـ بـحـذات مـن يـنـا ـقـة‬
                      ‫ أو بالبريد االلكتروني‬SMS ‫ديناران هدية عنـد التنبيه على كـل خطـأ بمذكرات الموقع برسالة‬
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  If the equations comprising such a system are linearly independent
  (that is, have a nonzero determinant), they will have a unique solution.


         In contrast, a homogenous linear algebraic system has the general
  form
         [A]{X} = 0


         Although nontrivial solutions (that is, solutions other than all x’s = 0)
  of such systems are possible, they are generally not unique.
  Rather, the simultaneous equations establish relationships among the x’s
  that can be satisfied by various combinations of values.

  Eigenvalue problems associated with engineering are typically of the
  general form




                            .                                       .
                            .                                       .
                            .                                       .


  where        is an unknown parameter called the eigenvalue, or characteristic
  value. A solution {X} for such a system is referred to as an eigenvector.
                                                                         ‫رو وا ددد لددب ددي جسدددين، تلددت هددي‬
                                                                             ‫الصداقة ن سألتني عنها...... أرسطو‬
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  The above set of equations may also be expressed concisely as




         The solution of the previous equation depends on determining                                                      .
  One way to accomplish this is based on the fact that the determinant of
  the matrix                               must be equal to zero for non-trivial solutions to
  be possible. Expanding the determinant yields a polynomial in                                                            .
  The roots of this polynomial are the solutions for the eigenvalues.


  27.2.2 Physical Background

         The mass-spring system in the next figure (a) is a simple context to
  illustrate how eigenvalues occur in physical problem settings.




                                                                                ‫ال تددأمن مددن كددذن لددت أن يكددذن عليددت‬
                                                                                ‫وال مددن اغتددان عندددك أن يغتابددت عنددد‬
                                                                                                                  ‫غيرك‬
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         Assume that each mass has no external or damping forces acting on
  it. In addition, assume that each spring has the same natural length l and
  the same spring constant k.


         Finally, assume that the displacement of each spring is measured
  relative to its own local coordinate system (previous figure a).


         Under these assumptions, Newton’s second law can be employed to
  develop a force balance for each mass



  and




  where         is the displacement of mass i away from its equilibrium position
  (previous figure b).


  These equations can be expressed as




                                                                            ‫تستغرق مناقشة المسا ل التا هة وقتدا‬
                                                                            ‫يددويع ألن بع ددنا يعددر عنهددا أكثددر‬
                                                                                      ‫مما يعر عن المسا ل الهامة‬
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  From vibration theory, it is known that solutions to the previous equations
  can take the form


  where          = the amplitude of the vibration of mass and                                        = the frequency
  of the vibration, which is equal to




  where          is the period. It follows that




  The previous two equations can be transformed to




  at this point, the solution has been reduced to an eigenvalue problem.




                                                                                ‫مددن الحكمددة أن تعتددذر لرجددل ُا كنددب‬
                                                                                ‫مخطئددا ....... وأن تعتددذر المددرأ تددى‬
                                                                                                  ‫ولو كنب على صوان‬
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  EXAMPLE 27.4 Eigenvalues and Eigenvectors for a Mass-Spring System


  Problem Statement:
  Evaluate the eigenvalues and the eigenvectors of the previous equations
  for the case where                                         kg and                      N/m.

  Solution:
  Substituting the parameter values into the equations yields




  The determinant of this system is


  which can be solved by the quadratic formula for                                                             and 5 s-2.
  Therefore, the frequencies for the vibrations of the masses are
                           and                      , respectively. These values can be used to
  determine the periods for the vibrations. For the first mode, T p = 1.62 s,
  and for the second, Tp = 2.81 s.


         As stated before, a unique set of value cannot be obtained for the
  unknowns. However, their ratios can be specified by substituting the
  eigenvalues back into the equations. For example, for the first mode
                                              . For the second mode                                                     .
                                                                                   ‫الطفدددل يلهدددو بالحيدددا صدددغيرا دون أن‬
                                                                                    ..‫يعلم أن الحيا سو تلع به كبيرا‬
                      ‫ أو بالبريد االلكتروني‬SMS ‫ديناران هدية عنـد التنبيه على كـل خطـأ بمذكرات الموقع برسالة‬
               PhysicsI/II, Circuits, English 123, Numerical, Dynamics, Strength, Statics:‫مواد عامة‬
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                                                                                                               June 2011


  Continue:
         This example provides valuable information regarding the behavior of
  the system in the previous figure. Aside from its period, we know that if
  the system is vibrating in the first mode, the amplitude of the second mass
  will be equal but of opposite sign to the amplitude of the first.
  As in the next figure (a), the masses vibrate apart and then together
  indefinitely.
         In the second mode, the two masses have equal amplitudes at all
  times. Thus, as in the figure (b), they vibrate back and forth in unison.
  It should be noted that the configuration of the amplitudes provides
  guidance on how to set their initial values to attain pure motion in either
  of the two modes. Any other configurations will lead to superposition of
  the modes.




                      ‫ أو بالبريد االلكتروني‬SMS ‫ديناران هدية عنـد التنبيه على كـل خطـأ بمذكرات الموقع برسالة‬
               PhysicsI/II, Circuits, English 123, Numerical, Dynamics, Strength, Statics:‫مواد عامة‬
    C++, Java, MATLAB, Data Structures, Algorithms, Discrete Math, Digital Logic, Concepts :‫مواد كمبيوتر‬
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                                                                                                               June 2011


  Problem 27.4
  Use the shooting method to solve



  with the boundary conditions y(0) = 5 and y(20) = 8




  Solution
  The second-order ODE can be expressed as the following pair of first-order
  ODEs,




  These can be solved for two guesses for the initial condition of z.
  For our cases we used –1 and                                . We solved the ODEs with the Heun
  method without iteration using a step size of 0.125. The results are


        z(0)                 1                              0.5                      ‫أ ددل يبي د ددي العددالم هددو البيطددري‬
                                                                                       ‫هو ال يستطيع أن يسدأل مري ده عدن‬
       y(20) 11,837.64486                          22,712.34615                                ‫شكواه بل يكتشف ُلت بنفسه‬


  Clearly, the solution is quite sensitive to the initial conditions.
  These values can then be used to derive the correct initial condition,




                      ‫ أو بالبريد االلكتروني‬SMS ‫ديناران هدية عنـد التنبيه على كـل خطـأ بمذكرات الموقع برسالة‬
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    C++, Java, MATLAB, Data Structures, Algorithms, Discrete Math, Digital Logic, Concepts :‫مواد كمبيوتر‬
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                                                                                                                June 2011


  Continue:
  The resulting fit is displayed below:


         x           y
       0      5
       2      4.151601
       4      4.461229
       6      5.456047
       8      6.852243
       10 8.471474
       12 10.17813
       14 11.80277
       16 12.97942
       18 12.69896
       20 8




                                                                               ‫أجمدددل األنهدددار لدددم نرهدددا بعدددد.. أجمدددل‬
                                                                               ‫الكت لم نقرأها بعد.. أجمل أياس ياتنا‬
                                                                                         ‫لم تأت بعد........ ناظم كمب‬     ٍ
                      ‫ أو بالبريد االلكتروني‬SMS ‫ديناران هدية عنـد التنبيه على كـل خطـأ بمذكرات الموقع برسالة‬
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    C++, Java, MATLAB, Data Structures, Algorithms, Discrete Math, Digital Logic, Concepts :‫مواد كمبيوتر‬
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                                                                                                               June 2011


  Problem 27.3
  Use the finite-difference approach with                                         to solve




  Solution


  A centered finite difference can be substituted for the second derivative to
  give,



  or for h = 1,
                                   –


  The first node would be
                  –

  and the last node would be




  The tridiagonal system can be solved with the Thomas algorithm or
  Gauss-Seidel for (the analytical solution is also included)
                                                                               ‫الخبددر هددي المشددط الددذي تقدمدده الطبيعددة‬
                                                                                       ‫لإلنسان عندما يتساقط شعر رأسه‬
                      ‫ أو بالبريد االلكتروني‬SMS ‫ديناران هدية عنـد التنبيه على كـل خطـأ بمذكرات الموقع برسالة‬
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    C++, Java, MATLAB, Data Structures, Algorithms, Discrete Math, Digital Logic, Concepts :‫مواد كمبيوتر‬
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                                                                                                               June 2011


  Continue:
          x           T              Analytical
      0       240                  240
      1       165.7573             165.3290
      2       116.3782             115.7689
      3       84.4558              83.7924
      4       65.2018              64.5425
      5       55.7281              55.0957
      6       54.6136              54.0171
      7       61.6911              61.1428
      8       78.0223              77.5552
      9       106.0569             105.7469
      10 150                       150




                                                                                ‫ُا انتقمددب لنفسددت مددن المسدديت ليددت‬
                                                                                ‫سدداويب نفسددت بدده.. و ُا صددفحب عندده‬
                                                                                          )‫استعبدته. ( رنسي بيكون‬
                      ‫ أو بالبريد االلكتروني‬SMS ‫ديناران هدية عنـد التنبيه على كـل خطـأ بمذكرات الموقع برسالة‬
               PhysicsI/II, Circuits, English 123, Numerical, Dynamics, Strength, Statics:‫مواد عامة‬
    C++, Java, MATLAB, Data Structures, Algorithms, Discrete Math, Digital Logic, Concepts :‫مواد كمبيوتر‬
Mechanical Design I/II, Structural Analysis I/II, Concrete I/II, Soil, Fluid Mechanics, System Dynamics :‫مواد تصميم‬
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