Symbolic Operators:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Johannesburg
Witwatersrand Univ. Pr.
1954
|
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | 194 S. |
Internformat
MARC
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| 100 | 1 | |a DALTON, JOHN P. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Symbolic Operators |
| 264 | 1 | |a Johannesburg |b Witwatersrand Univ. Pr. |c 1954 | |
| 300 | |a 194 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Datensatz im Suchindex
| _version_ | 1804140534277603328 |
|---|---|
| adam_text | CONTENTS
Pabt I
ORDINARY OPERATORS
Chapter I. ELEMENTARY OPERATIONS
1.1. Operators and operands ...... 1
1.2. Inverse operators ....... 2
1.3. The operators D and (D—k) ..... 3
1.4. The operator Q . . . • • .4
1.5. DQ and QD . . . . ¦ . .6
1.6. A convergence theorem for Q . . ¦ .7
1.7. The inverse operator Q~l . . . . . .9
1.8. Examples . . . . . . . .11
Chapter II. THE OPERATORS to* AND fiA
2.1. The primary operator a n . . . ¦ • .14
2.2. The secondary operator Qx . . . . .16
2.2a. A-Differentiability of oA and tlx . . . . .17
2.3. The expansion of (1—ftQ^-1 . . . . .19
2.4. An exponential rule for fiA ..... 20
2.5. Term-by-term operation . . . . . .21
2.6. Some examples of w and (1 operations . . . .21
Chapter III. THE PRODUCT FORMULA
3.1. The standard polynomial . . . . . .24
3.2. The product theorem . . . . . .25
3.3. The product theorem in determinant form . . .27
3.4. Composite operators . . . . . .29
3.5. Heaviside s expansion theorem . . . . .30
Chapter IV. THE CONFLUENT PRODUCT FORMULA
4.1. A preliminary limit . . . . • .36
4.2. Confluent products . . . . . .36
4.3. Numerical example of a confluent product . . .39
Chapter V. ORDINARY DIFFERENTIAL EQUATIONS
WITH CONSTANT COEFFICIENTS
5.1. Initial conditions . . . . . . .41
5.2. The operational solution . . . . . .41
5.3. Evaluation of the complementary function . . .43
5.4. Effect of confluence upon the complementary function . . 45
5.5. Complementary function under simplified initial conditions . 47
xiv CONTENTS
Chapter VI. ORDINARY DIFFERENTIAL EQUATIONS:
EXAMPLES
6.1. The particular integral . . . . . .51
6.2. f(t) a constant . . . . . . .51
6.3. f(t) a polynomial . . . . . • .54
6.4. f(t) an exponential function . . . . .55
6.5. f(t) = ektg(t) where g(t) is a polynomial . . .56
6.6. Complex parameters and conjugate operators . . .57
6.7. Examples illustrating conjugate operators . . . .58
Chapter VII. TWO-POINT BOUNDARY CONDITIONS
7.1. Conditions other than initial . . . . .64
7.2. Green s function . . . . . . .65
7.3. The homogeneous boundary problem . . . .67
7.4. Expansion of O(x, £, A) . . . . .68
7.5. The non-homogeneous boundary problem . . . .69
Chapter VIII. SIMULTANEOUS DIFFERENTIAL EQUA¬
TIONS WITH CONSTANT COEFFICIENTS
8.1. First-order equations . . . . . .71
8.2. Equations of higher order . . . . . .73
pabt n
PARTIAL OPERATORS OF INTEGRAL ORDER
Chapter IX. PARTIAL OPERATORS
9.1. Derivatives and integrals . . . . . .75
9.2. Partial operators of integral order and of simple type . . 78
9.3. The operator wa, 8 = XDX . . . . . .81
9.4. A-derivatives of u t . . . . . .86
9.5. The product formula for a a . . . .87
9.6. The partial operator ojA, A = AjD|. . . . .88
9.7. Secondary partial operators of mixed type. . . .89
9.8. The mixed operator [1—AD^nj 1. . . . .90
Chapter X. FIRST-ORDER PARTIAL DIFFERENTIAL
EQUATION WITH CONSTANT COEFFICIENTS
10.1. Solution of the equation . . . . . .92
10.2. Phase propagation . . . . . .93
10.3. The Heaviside unit function . . . . .95
10.4. Heaviside s wave expansion . . . . .96
10.5. Operational parameters . . . . . .97
10.6. Examples of first-order equations . . . .99
10.7. Note on impulsive functions ..... 100
CONTENTS xv
Chapter XL SECOND-ORDER EQUATIONS WITH CON¬
STANT COEFFICIENTS
11.1. The factorizable equation . . . . .105
11.2. Hyperbolic equation—homogeneous .... 106
11.3. Hyperbolic equation ...... 108
11.4. Riemann s method for more general boundary conditions . 110
11.5. Hyperbolic equation—second canonical form . . .113
Part III
PARTIAL OPERATORS OF FRACTIONAL ORDER
Chapter XII. FRACTIONAL INTEGRALS AND
DERIVATIVES
12.1. A preliminary integral . . . . . .116
12.2. Fractional integration . . . . .117
12.3. Term-by-term operation by Q . . . . .119
12.4. A convergence theorem for Q . . . . 120
12.5. The inverse of Q . . . . . . .121
12.6. Fractional differentiation . . . . .122
12.7. Term-by -term operation by D . . . . .124
12.8. Fractional operators of order v = £ . . . .125
12.9. Some operational forms of error and Bessel functions . .125
Chapter XIII. THE ERROR FUNCTION
13.1. Error function of a single variable . . . .129
13.2. Integrals and derivatives involving erf -J(X/t) . . . 132
13.3. The function er£{xl2y/(kt)}. . . . . .134
13.4. Higher derivatives of etf{x/2j(kt)} . . . .140
13.5. Further operations on erf{x/2V(/t )} - ¦ ¦ .142
ChapterXIV. SECOND-ORDER EQUATIONS OF PARABOLIC
TYPE
14.1. Some auxiliary theorems ...... 146
14.2. Diffusion equation under one-point boundary conditions . . 155
14.3. The diffusion operator er^, q = k~*D . . . .158
14.4. q as an operational parameter ..... 163
14.5. Examples illustrating the use of e~xq . . . 166
14.6. Solution of the diffusion equation subject to prescribed values
along the positive semi-axes . . . . .168
xvi CONTENTS
ChaptebXV. PARTIAL DIFFERENTIAL EQUATIONS UNDER
TWO-POINT BOUNDARY CONDITIONS
15.1. Hyperbolic equation . . . . . . 170
15.2. Elliptic equation ....... 173
15.3. Parabolic equation ...... 175
15.4. The equation of telegraphy ..... 177
15.5. Heaviside s wave expansions ..... 181
APPENDIX
Asymptotic approximation . . . . . .186
INDEX ........ .193
|
| any_adam_object | 1 |
| author | DALTON, JOHN P. |
| author_facet | DALTON, JOHN P. |
| author_role | aut |
| author_sort | DALTON, JOHN P. |
| author_variant | j p d jp jpd |
| building | Verbundindex |
| bvnumber | BV024530656 |
| classification_rvk | SK 620 |
| ctrlnum | (OCoLC)833156403 (DE-599)BVBBV024530656 |
| dewey-full | 517.7 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 517 - [Unassigned] |
| dewey-raw | 517.7 |
| dewey-search | 517.7 |
| dewey-sort | 3517.7 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV024530656 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:01:30Z |
| institution | BVB |
| language | English |
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| physical | 194 S. |
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| publishDate | 1954 |
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| publisher | Witwatersrand Univ. Pr. |
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| spelling | DALTON, JOHN P. Verfasser aut Symbolic Operators Johannesburg Witwatersrand Univ. Pr. 1954 194 S. txt rdacontent n rdamedia nc rdacarrier HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018504610&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | DALTON, JOHN P. Symbolic Operators |
| title | Symbolic Operators |
| title_auth | Symbolic Operators |
| title_exact_search | Symbolic Operators |
| title_full | Symbolic Operators |
| title_fullStr | Symbolic Operators |
| title_full_unstemmed | Symbolic Operators |
| title_short | Symbolic Operators |
| title_sort | symbolic operators |
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