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1: 3.8 Nonlinear Equations

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Regula Falsi

… ►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: …

2: Mark J. Ablowitz

… ►for appropriate data they can be linearized by the Inverse Scattering Transform (IST) and they possess solitons as special solutions. …

3: 31.16 Mathematical Applications

… ► thesis “Inversion problem for a second-order linear differential equation with four singular points”. …

4: 1.2 Elementary Algebra

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The Inverse

►If det($𝐀$) $\ne$ $0$, $𝐀$ has a unique inverse, ${𝐀}^{-1}$, such that … ►

$n$ Linear Equations in $n$ Unknowns

… ►If $det(𝐀)\ne 0$ the system of $n$ linear equations in $n$ unknowns, … ►and for the corresponding eigenvectors one has to solve the linear system …

5: Bibliography

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Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-38730-2, Document, Link, MathReview (Walter Oevel), ZentralBlatt

Referenced by:

§22.19(ii), §22.19(iii), §32.11(ii), §32.5, §9.16, Profile Mark J. Ablowitz, Profile Peter A. Clarkson.

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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External Links:

Document, MathReview (Mark Adler)

Referenced by:

§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:

ISBN 0-89871-174-6, MathReview (Benno Fuchssteiner), ZentralBlatt

Referenced by:

§21.9, §21.9, §32.5, §9.16, Profile Mark J. Ablowitz.

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Linear algebra done right. Third edition, Undergraduate Texts in Mathematics, Springer, Cham.

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External Links:

ISBN 978-3-319-11079-0; 978-3-319-11080-6, Document, Link, MathReview Entry

Referenced by:

§1.2(v), §1.2(vi), §1.3(iv).

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6: 1.1 Special Notation

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$x,y$	real variables.
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${𝐀}^{-1}$	inverse of the square matrix $𝐀$
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$ℒ$	linear operator defined on a manifold $ℳ$
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7: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

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On the calculation of Stokes multipliers for linear differential equations of the second order. Methods Appl. Anal. 2 (3), pp. 348–367.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.7(ii).

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On the asymptotic and numerical solution of linear ordinary differential equations. SIAM Rev. 40 (3), pp. 463–495.

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External Links:

ISSN 1095-7200, Document, MathReview (W. Balser), ZentralBlatt

Referenced by:

§16.25, §2.7(ii), §3.7(iii).

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Hyperasymptotic solutions of second-order linear differential equations. II. Methods Appl. Anal. 2 (2), pp. 198–211.

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External Links:

ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §2.11(v), §9.7(v).

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

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8: Errata

… ►This especially included updated information on matrix analysis, measure theory, spectral analysis, and a new section on linear second order differential operators and eigenfunction expansions. … ►The specific updates to Chapter 1 include the addition of an entirely new subsection §1.18 entitled “Linear Second Order Differential Operators and Eigenfunction Expansions” which is a survey of the formal spectral analysis of second order differential operators. The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. … ►

The wording was changed to make the integration variable more apparent.

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These equations have been rewritten to improve the numerical computation of $\arctan x$.

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9: Bibliography B

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Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. The Macmillan Co., New York.

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External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§11.4(i), §11.4(v), §11.5(ii), §11.5(i), §11.5(ii), §11.5(iii), §11.7(ii), §11.7(iii), §11.7(v), §11.9(i), §11.9(iv), §11.9(iv), Ch.11, §14.29.

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

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Discrete ordinate solution of Fokker-Planck equations with non-linear coefficients. Phys. Rev. A 31 (3), pp. 1855–1868.

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Referenced by:

§18.39(iii).

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Error estimates for the solution of linear systems. SIAM J. Sci. Comput. 21 (2), pp. 764–781.

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External Links:

ISSN 1095-7197, Document, MathReview (Dimitrios Noutsos), ZentralBlatt

Referenced by:

§3.2(iii).

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The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods. John Wiley & Sons Ltd., Chichester.

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External Links:

ISBN 0-471-91046-5, MathReview (Peter Alfeld), ZentralBlatt

Referenced by:

§3.7(v).

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10: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(iv) Matrices as Linear Operators

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Linear Operators in Finite Dimensional Vector Spaces

►Square matices can be seen as linear operators because $𝐀(\alpha 𝐚+\beta 𝐛)=\alpha 𝐀𝐚+\beta 𝐀𝐛$ for all $\alpha ,\beta \in ℂ$ and $𝐚,𝐛\in {𝐄}_n$, the space of all $n$-dimensional vectors. … ►Let the columns of matrix $𝐒$ be these eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$, then ${𝐒}^{-1}={𝐒}^{\mathrm{H}}$, and the similarity transformation (1.2.73) is now of the form ${𝐒}^{\mathrm{H}}𝐀𝐒={\lambda }_i{\delta }_{i,j}$. …