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Solutions for Chapter 9.6: Greens Functions for Time-Independent Problems

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition | ISBN: 9780321797056 | Authors: Richard Haberman

Full solutions for Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition

ISBN: 9780321797056

Applied Partial Differential Equations with Fourier Series and Boundary Value Problems | 5th Edition | ISBN: 9780321797056 | Authors: Richard Haberman

Solutions for Chapter 9.6: Greens Functions for Time-Independent Problems

This expansive textbook survival guide covers the following chapters and their solutions. Chapter 9.6: Greens Functions for Time-Independent Problems includes 9 full step-by-step solutions. Since 9 problems in chapter 9.6: Greens Functions for Time-Independent Problems have been answered, more than 7838 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, edition: 5. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems was written by and is associated to the ISBN: 9780321797056.

Key Math Terms and definitions covered in this textbook
  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Companion matrix.

    Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

  • Dimension of vector space

    dim(V) = number of vectors in any basis for V.

  • Ellipse (or ellipsoid) x T Ax = 1.

    A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • Linear combination cv + d w or L C jV j.

    Vector addition and scalar multiplication.

  • Lucas numbers

    Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

  • Matrix multiplication AB.

    The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Particular solution x p.

    Any solution to Ax = b; often x p has free variables = o.

  • Pivot columns of A.

    Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Row space C (AT) = all combinations of rows of A.

    Column vectors by convention.

  • Semidefinite matrix A.

    (Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • Singular matrix A.

    A square matrix that has no inverse: det(A) = o.

  • Spectrum of A = the set of eigenvalues {A I, ... , An}.

    Spectral radius = max of IAi I.

  • Vector addition.

    v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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