32:
2533:, which were previously approachable only through perturbation theory, are in fact completely integrable. This discovery was quite dramatic, as it allowed exact solutions to be given. This, in turn, helped clarify the meaning of the perturbative series, as one could now compare the results of the series to the exact solutions.
2976:
Hamiltonian and the exact non-relativistic
Hamiltonian as the perturbation. The zero-order energy is the sum of orbital energies. The first-order energy is the Hartree–Fock energy and electron correlation is included at second-order or higher. Calculations to second, third or fourth order are
2514:
mean. The one-to-one correspondence between the diagrams, and specific integrals is what gives them their power. Although originally developed for quantum field theory, it turns out the diagrammatic technique is broadly applicable to many other perturbative series (although not always worthwhile).
2513:
by observing that many terms repeat in a regular fashion. These terms can be replaced by dots, lines, squiggles and similar marks, each standing for a term, a denominator, an integral, and so on; thus complex integrals can be written as simple diagrams, with absolutely no ambiguity as to what they
919:
in some "small" parameter, that quantifies the deviation from the exactly solvable problem. The leading term in this power series is the solution of the exactly solvable problem, while further terms describe the deviation in the solution, due to the deviation from the initial problem. Formally, we
2890:
Since the planets are very remote from each other, and since their mass is small as compared to the mass of the Sun, the gravitational forces between the planets can be neglected, and the planetary motion is considered, to a first approximation, as taking place along Kepler's orbits, which are
1421:
perturbation problem. In regular perturbation problems, the asymptotic solution smoothly approaches the exact solution. However, the perturbation series can also diverge, and the truncated series can still be a good approximation to the true solution if it is truncated at a point at which its
1207:
these higher-order terms in the series generally (but not always) become successively smaller. An approximate "perturbative solution" is obtained by truncating the series, often by keeping only the first two terms, expressing the final solution as a sum of the initial (exact) solution and the
2910:
were the first to advance the view that the so-called "constants" which describe the motion of a planet around the Sun gradually change: They are "perturbed", as it were, by the motion of other planets and vary as a function of time; hence the name "perturbation theory".
2950:
The standard exposition of perturbation theory is given in terms of the order to which the perturbation is carried out: first-order perturbation theory or second-order perturbation theory, and whether the perturbed states are degenerate, which requires
2868:, becoming as large or maybe larger than the zeroth order term. This situation signals a breakdown of perturbation theory: It stops working at this point, and cannot be expanded or summed any further. In formal terms, the perturbative series is an
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usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, usually by keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction.
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wave equations, but also since the quantum mechanical notation allows expressions to be written in fairly compact form, thus making them easier to comprehend. This resulted in an explosion of applications, ranging from the
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1605:
is created by adding successive corrections to the simplified problem. The corrections are obtained by forcing consistency between the unperturbed solution, and the equations describing the system in full. Write
2967:
use perturbation theory directly or are closely related methods. Implicit perturbation theory works with the complete
Hamiltonian from the very beginning and never specifies a perturbation operator as such.
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of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In perturbation theory, the solution is expressed as a
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Since astronomic data came to be known with much greater accuracy, it became necessary to consider how the motion of a planet around the Sun is affected by other planets. This was the origin of the
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observed (as perhaps had earlier mathematicians) that sometimes 2nd and higher order terms in the perturbative series have "small denominators": That is, they have the general form
2420:), statistical or quantum-mechanical systems of non-interacting particles (or in general, Hamiltonians or free energies containing only terms quadratic in all degrees of freedom).
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This unmanageability has forced perturbation theory to develop into a high art of managing and writing out these higher order terms. One of the fundamental breakthroughs in
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was an explanation of why this happened: The small divisors occur whenever perturbation theory is applied to a chaotic system. The one signals the presence of the other.
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led to incremental demands in the accuracy of solutions to Newton's gravitational equations, which led many eminent 18th and 19th century mathematicians, notably
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Perturbation theory has been used in a large number of different settings in physics and applied mathematics. Examples of the "collection of equations"
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This article is about perturbation theory as a general mathematical method. For perturbation theory applied specifically to quantum mechanics, see
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explained the gravitation between two astronomical bodies, but when a third body is added, the problem was, "How does each body pull on each?"
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For physical problems involving interactions between particles, the terms of the perturbation series may be displayed (and manipulated) using
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only solve Newton's gravitational equations when the latter are limited to just two bodies interacting. The gradually increasing accuracy of
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1428:. If the perturbation series is divergent or not a power series (for example, if the asymptotic expansion must include non-integer powers
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developed quantum perturbation theory in 1927 to evaluate when a particle would be emitted in radioactive elements. This was later named
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MartĂnez-Carranza, J.; Soto-Eguibar, F.; Moya-Cessa, H. (2012). "Alternative analysis to perturbation theory in quantum mechanics".
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Examples of systems that can be solved with perturbations include systems with nonlinear contributions to the equations of motion,
2902:; thus, in studying the system Moon-Earth-Sun, the mass ratio between the Moon and the Earth was chosen as the "small parameter".
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These well-developed perturbation methods were adopted and adapted to solve new problems arising during the development of
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Martin C. Gutzwiller, "Moon-Earth-Sun: The oldest three-body problem", Rev. Mod. Phys. 70, 589 – Published 1 April 1998
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2525:, while the perturbed systems were not. This promptly lead to the study of "nearly integrable systems", of which the
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In practice, this process rapidly explodes into a profusion of terms, which become extremely hard to manage by hand.
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Michael A. Box, "Radiative perturbation theory: a review", Environmental
Modelling & Software 17 (2002) 95–106
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The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of
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Examples of the kinds of solutions that are found perturbatively include the solution of the equation of motion (
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2481:. Perturbation theory in quantum mechanics is fairly accessible, mainly because quantum mechanics is limited to
1522:. Many special techniques in perturbation theory have been developed to analyze singular perturbation problems.
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Perturbation theory is used in a wide range of fields, and reaches its most sophisticated and advanced forms in
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Advanced mathematical methods for scientists and engineers I : asymptotic methods and perturbation theory
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stand in for the problem to be solved. Quite often, these are differential equations, thus, the letter "D".
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Keeping the above example in mind, one follows a general recipe to obtain the perturbation series. The
1057:{\displaystyle A\equiv A_{0}+\varepsilon ^{1}A_{1}+\varepsilon ^{2}A_{2}+\varepsilon ^{3}A_{3}+\cdots }
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3240:"L. A. Romero, "Perturbation theory for polynomials", Lecture Notes, University of New Mexico (2013)"
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1381:{\displaystyle \;A=A_{0}+\varepsilon A_{1}+{\mathcal {O}}{\bigl (}\ \varepsilon ^{2}\ {\bigr )}~.}
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is the canonical example. At the same time, it was also discovered that many (rather special)
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1277:{\displaystyle A\approx A_{0}+\varepsilon A_{1}\qquad {\mathsf {for}}\qquad \varepsilon \to 0}
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Sergei
Winitzki, "Perturbation theory for anharmonic oscillations", Lecture notes, LMU (2006)
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1589:) and not quite correct when the gravitational interaction is stated using formulations from
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who successfully observed
Neptune through his telescope – a triumph of perturbation theory.
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903:. The field in general remains actively and heavily researched across multiple disciplines.
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In math and applied mathematics, methods for finding an approximate solution to a problem
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It is a good approximation, precisely because the parts that were ignored were of size
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which, in the general case, can be written in closed form as a sum over integrals over
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The process is generally mechanical, if laborious. One begins by writing the equations
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converges with a nonzero radius of convergence, the perturbation problem is called a
2640:{\displaystyle \ {\frac {\ \psi _{n}V\phi _{m}\ }{\ (\omega _{n}-\omega _{m})\ }}\ }
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Perturbation methods start with a simplified form of the original problem, which is
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would be the known solution to the exactly solvable initial problem, and the terms
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216:
3212:
Great
Physicists: The life and times of leading physicists from Galileo to Hawking
2408:
Examples of exactly solvable problems that can be used as starting points include
911:
Perturbation theory develops an expression for the desired solution in terms of a
3586:
2997:(sc) occurs in perturbation theory when matter trajectories intersect, forming a
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in the calculation of the motions of planets in the solar system. For instance,
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3001:. This limits the predictive power of physical simulations at small scales.
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are some complicated expressions pertinent to the problem to be solved, and
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between particles, terms of higher powers in the
Hamiltonian/free energy.
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The
Unreasonable Effectiveness of Mathematics in the Natural Sciences
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2874:: A useful approximation for a few terms, but at some point becomes
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developed, it became clear that unperturbed systems were in general
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was to deal with the otherwise unsolvable mathematical problems of
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2923:
2540:
coming from chaos theory helped shed light on what was termed the
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Perturbation theory was investigated by the classical scholars –
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1720:
which can be solved exactly, and some additional remaining part
1687:
so that they split into two parts: some collection of equations
1578:
1570:
1291:
to indicate the order of the error in the approximate solution:
4388:
European
Community on Computational Methods in Applied Sciences
3605:
2466:, to extend and generalize the methods of perturbation theory.
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25:
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accurate if even more terms are added. The breakthrough from
2501:
Despite the simpler notation, perturbation theory applied to
4383:
International
Council for Industrial and Applied Mathematics
3322:"The quantum theory of emission and absorption of radiation"
1337:
1632:
for this collection of equations; that is, let the symbol
1546:
because of the competing gravitation of the Earth and the
2259:
The process can then be repeated, to obtain corrections
2818:. The small divisor problem arises when the difference
2302:
is reported to have said, regarding the problem of the
4002:
Numerical methods for ordinary differential equations
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Société de Mathématiques Appliquées et Industrielles
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Japan Society for Industrial and Applied Mathematics
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Numerical methods for partial differential equations
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Alternative approach to quantum perturbation theory
3416:
King, Matcha (1976). "Theory of the Chemical Bond".
3382:
Encyclopedia of Mathematics (encyclopediaofmath.org)
2187:{\displaystyle \ A\approx A_{0}+\varepsilon A_{1}\ }
1991:{\displaystyle \ A\approx A_{0}+\varepsilon A_{1}\ }
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2473:in 20th century atomic and subatomic physics.
2322:perturbation series to be represented by a sketch.
1542:, which moves noticeably differently from a simple
1472:{\displaystyle \ \varepsilon ^{\left(1/2\right)}\ }
56:. Unsourced material may be challenged and removed.
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3449:Monthly Notices of the Royal Astronomical Society
2864:is small, causing the perturbative correction to
2517:In the second half of the 20th century, as
920:have for the approximation to the full solution
2895:, the two bodies being the planet and the Sun.
2442:Perturbation theory was first devised to solve
1857:) is known, and one seeks the general solution
4366:Society for Industrial and Applied Mathematics
3617:
3510:"Introduction to regular perturbation theory"
3443:Rampf, Cornelius; Hahn, Oliver (2021-02-01).
2977:very common and the code is included in most
1934:{\displaystyle \ D=D_{0}+\varepsilon D_{1}~.}
1530:The earliest use of what would now be called
1367:
1344:
1158:{\displaystyle \ A_{1},A_{2},A_{3},\ldots \ }
949:a series in the small parameter (here called
800:
8:
4184:Supersymmetric theory of stochastic dynamics
3296:(2nd ed.). Prentice Hall. p. 443.
3103:. Steven A. Orszag. New York, NY: Springer.
2981:. A related but more accurate method is the
1515:) then the perturbation problem is called a
2886:Beginnings in the study of planetary motion
2857:{\displaystyle \ \omega _{n}-\omega _{m}\ }
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835:to a problem, by starting from the exact
116:Learn how and when to remove this message
3529:"Perturbation method of multiple scales"
3418:Journal of the American Chemical Society
2970:Møller–Plesset perturbation theory
2412:, including linear equations of motion (
2405:energy of a quantum mechanical problem.
1422:elements are minimum. This is called an
3292:Bransden, B.H.; Joachain, C.J. (1999).
3078:
3062:Perturbation theory (quantum mechanics)
1573:) but not quite correct when there are
1208:"first-order" perturbative correction
897:Perturbation theory (quantum mechanics)
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21:Perturbation theory (quantum mechanics)
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2314:for controlling the expansion are the
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3178:. Ohio: Aphelion Press. p. 107.
2448:Newton's law of universal gravitation
1784:{\displaystyle \ \varepsilon \ll 1~.}
1749:{\displaystyle \ \varepsilon D_{1}\ }
1508:{\displaystyle \ \varepsilon ^{-2}\ }
7:
3145:(2nd ed.). New York: Springer.
3142:Introduction to perturbation methods
3134:
3132:
3130:
3128:
3092:
3090:
3088:
3086:
3084:
3082:
2979:ab initio quantum chemistry programs
2252:{\displaystyle \ \varepsilon ^{2}~.}
899:describes the use of this method in
255:List of named differential equations
54:adding citations to reliable sources
3527:Chow, Carson C. (23 October 2007).
3445:"Shell-crossing in a ΛCDM Universe"
2965:ab initio quantum chemistry methods
2928:The discovery of the planet Neptune
2024:{\displaystyle \ \varepsilon D_{1}}
328:Dependent and independent variables
3633:Industrial and applied mathematics
3327:Proceedings of the Royal Society A
2401:, average magnetization), and the
2031:. This results in an equation for
14:
3863:Stochastic differential equations
3362:See equations (24) and (32).
831:comprises methods for finding an
4179:Supersymmetric quantum mechanics
3522:from the original on 2004-09-20.
3017:Cosmological perturbation theory
2972:uses the difference between the
2891:defined by the equations of the
1410:{\displaystyle \ \varepsilon \ }
1200:{\displaystyle \ \varepsilon \ }
463:Carathéodory's existence theorem
30:
4452:Ordinary differential equations
4061:Stochastic variational calculus
3853:Ordinary differential equations
3557:The European Physical Journal D
2803:{\displaystyle \ \omega _{m}\ }
2770:{\displaystyle \ \omega _{n}\ }
2505:still easily gets out of hand.
1264:
1250:
41:needs additional citations for
3858:Partial differential equations
3731:Arbitrary-precision arithmetic
2676:{\displaystyle \ \psi _{n}\ ,}
2625:
2599:
2536:The improved understanding of
2444:otherwise intractable problems
1268:
550: / Integral solutions
1:
3746:Interactive geometry software
2938:. He sent the coordinates to
2737:{\displaystyle \ \phi _{m}\ }
2523:completely integrable systems
2308:"It causeth my head to ache."
3210:Cropper, William H. (2004).
3037:Homotopy perturbation method
3027:Dynamic nuclear polarisation
880:{\displaystyle \varepsilon }
858:{\displaystyle \varepsilon }
594:Exponential response formula
340:Coupled / Decoupled
3798:Computational number theory
3761:Numerical-analysis software
2548:. In the 19th century
2397:of some physical quantity (
2194:is a good approximation to
2103:Thus, one has obtained the
4483:
3587:10.1140/epjd/e2011-20654-5
3174:William E. Wiesel (2010).
3052:Method of dominant balance
2452:Kepler's orbital equations
2374:, radiative transfer, and
2288:{\displaystyle \ A_{2}\ ,}
2060:{\displaystyle \ A_{1}\ ,}
18:
4396:
4204:Algebra of physical space
3671:Automated theorem proving
3546:10.4249/scholarpedia.1617
3022:Deformation (mathematics)
2542:small denominator problem
2509:developed the celebrated
2456:astronomical observations
2368:thermodynamic free energy
2132:{\displaystyle \ A_{1}\ }
2096:{\displaystyle \ A_{0}~.}
1850:{\displaystyle \ D_{0}\ }
1817:{\displaystyle \ A_{0}\ }
1713:{\displaystyle \ D_{0}\ }
1557:to be solved exactly. In
1093:{\displaystyle \ A_{0}\ }
728:Józef Maria Hoene-Wroński
674:Undetermined coefficients
583:Method of characteristics
468:Cauchy–Kowalevski theorem
3997:Numerical linear algebra
3139:Holmes, Mark H. (2013).
3097:Bender, Carl M. (1999).
453:Picard–Lindelöf theorem
447:Existence and uniqueness
4462:Computational chemistry
3736:Finite element analysis
3686:Constraint satisfaction
3216:Oxford University Press
3042:Interval finite element
3032:Eigenvalue perturbation
1943:Next the approximation
1391:If the power series in
953:), like the following:
679:Variation of parameters
669:Separation of variables
458:Peano existence theorem
4291:Mathematical economics
3965:Multivariable calculus
3848:Differential equations
3691:Constraint programming
3681:Computational geometry
3480:10.1093/mnrasl/slaa198
3349:10.1098/rspa.1927.0039
3057:Order of approximation
2858:
2804:
2771:
2738:
2705:
2704:{\displaystyle \ V\ ,}
2677:
2641:
2356:differential equations
2344:
2289:
2253:
2217:
2188:
2133:
2105:first-order correction
2097:
2061:
2025:
1992:
1935:
1877:
1851:
1818:
1785:
1750:
1714:
1681:
1652:
1626:
1603:perturbative expansion
1597:Perturbative expansion
1585:, and the rest of the
1509:
1473:
1411:
1382:
1278:
1201:
1159:
1094:
1058:
943:
942:{\displaystyle \ A\ ,}
881:
859:
748:Carl David Tolmé Runge
291:Differential-algebraic
132:Differential equations
4244:Supersymmetry algebra
4229:Representation theory
4224:Renormalization group
3870:Differential geometry
3751:Optimization software
3723:Mathematical software
3506:van den Eijnden, Eric
3378:"Perturbation theory"
2953:singular perturbation
2859:
2805:
2772:
2739:
2706:
2678:
2642:
2546:small divisor problem
2376:Hamiltonian operators
2372:statistical mechanics
2345:
2290:
2254:
2218:
2216:{\displaystyle \ A~.}
2189:
2134:
2098:
2062:
2026:
1993:
1936:
1878:
1876:{\displaystyle \ A\ }
1852:
1819:
1786:
1751:
1715:
1682:
1680:{\displaystyle \ D\ }
1653:
1651:{\displaystyle \ D\ }
1627:
1625:{\displaystyle \ D\ }
1575:three or more objects
1510:
1474:
1412:
1383:
1279:
1202:
1160:
1095:
1059:
944:
882:
860:
844:in a small parameter
738:Augustin-Louis Cauchy
723:Joseph-Louis Lagrange
555:Numerical integration
537:Exponential stability
400:Relation to processes
65:"Perturbation theory"
4457:Mathematical physics
4296:Mathematical finance
4281:Social choice theory
4196:Algebraic structures
4145:in quantum mechanics
4081:Analytical mechanics
4047:Stochastic processes
4019:Variational calculus
3831:Approximation theory
3756:Statistical software
3176:Modern Astrodynamics
3067:Structural stability
2822:
2781:
2748:
2715:
2686:
2651:
2556:
2503:quantum field theory
2418:linear wave equation
2393:of a particle), the
2334:
2263:
2227:
2198:
2143:
2110:
2071:
2035:
2002:
1947:
1887:
1861:
1828:
1795:
1760:
1724:
1691:
1665:
1636:
1610:
1561:, this is usually a
1526:Prototypical example
1520:perturbation problem
1483:
1432:
1395:
1296:
1215:
1185:
1104:
1071:
960:
924:
893:quantum field theory
871:
849:
833:approximate solution
560:Dirac delta function
296:Integro-differential
50:improve this article
4467:Asymptotic analysis
4447:Functional analysis
4442:Concepts in physics
4437:Perturbation theory
4271:Operations research
4140:Perturbation theory
3938:Multilinear algebra
3909:Functional analysis
3766:Numerical libraries
3698:Computational logic
3579:2012EPJD...66...22M
3471:2021MNRAS.501L..71R
3430:10.1021/ja00428a004
3340:1927RSPSA.114..243D
2946:Perturbation orders
2492:hyperfine splitting
2479:Fermi's golden rule
2414:harmonic oscillator
2395:statistical average
2360:equations of motion
2352:algebraic equations
1559:celestial mechanics
1536:celestial mechanics
1532:perturbation theory
1479:or negative powers
917:perturbation series
913:formal power series
829:perturbation theory
825:applied mathematics
656:Perturbation theory
651:Integral transforms
542:Rate of convergence
408:(discrete analogue)
245:Population dynamics
212:Continuum mechanics
203:Applied mathematics
4408:Mathematics portal
4305:Other applications
4029:Probability theory
4012:Validated numerics
3992:Numerical analysis
3885:Geometric analysis
3875:Differential forms
3708:Information theory
3397:Unknown parameter
3047:Lyapunov stability
2974:Hartree–Fock
2900:three-body problem
2854:
2800:
2767:
2734:
2701:
2673:
2637:
2531:non-linear systems
2340:
2320:quantum mechanical
2285:
2249:
2213:
2184:
2129:
2093:
2057:
2021:
1988:
1931:
1873:
1847:
1814:
1781:
1746:
1710:
1677:
1648:
1622:
1591:general relativity
1538:: for example the
1505:
1469:
1407:
1378:
1274:
1197:
1179:higher-order terms
1155:
1090:
1054:
939:
877:
855:
646:Integrating factor
487:Initial conditions
422:Stochastic partial
4424:
4423:
4258:Decision sciences
4252:
4251:
4234:Spacetime algebra
3926:Harmonic analysis
3892:Dynamical systems
3836:Clifford analysis
3813:Discrete geometry
3779:
3778:
3424:(12): 3415–3420.
3294:Quantum Mechanics
3225:978-0-19-517324-6
3152:978-1-4614-5477-9
3110:978-1-4757-3069-2
2871:asymptotic series
2853:
2827:
2799:
2786:
2766:
2753:
2733:
2720:
2697:
2691:
2669:
2656:
2636:
2632:
2630:
2598:
2593:
2567:
2561:
2538:dynamical systems
2471:quantum mechanics
2380:quantum mechanics
2343:{\displaystyle D}
2312:quantum mechanics
2281:
2268:
2245:
2232:
2209:
2203:
2183:
2148:
2128:
2115:
2089:
2076:
2053:
2040:
2007:
1998:is inserted into
1987:
1952:
1927:
1892:
1872:
1866:
1846:
1833:
1813:
1800:
1777:
1765:
1745:
1729:
1709:
1696:
1676:
1670:
1647:
1641:
1621:
1615:
1577:(say, the Earth,
1567:Newtonian gravity
1563:Keplerian ellipse
1544:Keplerian ellipse
1540:orbit of the Moon
1504:
1488:
1468:
1437:
1425:asymptotic series
1406:
1400:
1374:
1364:
1351:
1287:Some authors use
1196:
1190:
1154:
1109:
1089:
1076:
1067:In this example,
935:
929:
901:quantum mechanics
817:
816:
708:Gottfried Leibniz
599:Finite difference
391:
390:
252:
251:
222:Dynamical systems
126:
125:
118:
100:
4474:
4209:Feynman integral
4192:
4152:Potential theory
4041:random variables
3931:Fourier analysis
3914:Operator algebra
3841:Clifford algebra
3793:Computer algebra
3719:
3626:
3619:
3612:
3603:
3598:
3572:
3550:
3548:
3523:
3521:
3514:
3493:
3492:
3482:
3464:
3440:
3434:
3433:
3413:
3407:
3406:
3400:
3395:
3393:
3385:
3374:
3363:
3361:
3351:
3334:(767): 243–265.
3320:(1 March 1927).
3314:
3308:
3307:
3303:978-0-58235691-7
3289:
3283:
3278:
3272:
3267:
3261:
3260:
3258:
3257:
3251:
3245:. Archived from
3244:
3236:
3230:
3229:
3207:
3201:
3196:
3190:
3189:
3171:
3165:
3164:
3136:
3123:
3122:
3094:
2893:two-body problem
2863:
2861:
2860:
2855:
2851:
2850:
2849:
2837:
2836:
2825:
2809:
2807:
2806:
2801:
2797:
2796:
2795:
2784:
2776:
2774:
2773:
2768:
2764:
2763:
2762:
2751:
2743:
2741:
2740:
2735:
2731:
2730:
2729:
2718:
2710:
2708:
2707:
2702:
2695:
2689:
2682:
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2679:
2674:
2667:
2666:
2665:
2654:
2646:
2644:
2643:
2638:
2634:
2633:
2631:
2628:
2624:
2623:
2611:
2610:
2596:
2594:
2591:
2590:
2589:
2577:
2576:
2565:
2563:
2559:
2511:Feynman diagrams
2432:Feynman diagrams
2410:linear equations
2349:
2347:
2346:
2341:
2316:Feynman diagrams
2294:
2292:
2291:
2286:
2279:
2278:
2277:
2266:
2258:
2256:
2255:
2250:
2243:
2242:
2241:
2230:
2222:
2220:
2219:
2214:
2207:
2201:
2193:
2191:
2190:
2185:
2181:
2180:
2179:
2164:
2163:
2146:
2138:
2136:
2135:
2130:
2126:
2125:
2124:
2113:
2102:
2100:
2099:
2094:
2087:
2086:
2085:
2074:
2066:
2064:
2063:
2058:
2051:
2050:
2049:
2038:
2030:
2028:
2027:
2022:
2020:
2019:
2005:
1997:
1995:
1994:
1989:
1985:
1984:
1983:
1968:
1967:
1950:
1940:
1938:
1937:
1932:
1925:
1924:
1923:
1908:
1907:
1890:
1882:
1880:
1879:
1874:
1870:
1864:
1856:
1854:
1853:
1848:
1844:
1843:
1842:
1831:
1823:
1821:
1820:
1815:
1811:
1810:
1809:
1798:
1790:
1788:
1787:
1782:
1775:
1763:
1755:
1753:
1752:
1747:
1743:
1742:
1741:
1727:
1719:
1717:
1716:
1711:
1707:
1706:
1705:
1694:
1686:
1684:
1683:
1678:
1674:
1668:
1657:
1655:
1654:
1649:
1645:
1639:
1631:
1629:
1628:
1623:
1619:
1613:
1514:
1512:
1511:
1506:
1502:
1501:
1500:
1486:
1478:
1476:
1475:
1470:
1466:
1465:
1464:
1463:
1459:
1455:
1435:
1416:
1414:
1413:
1408:
1404:
1398:
1388:
1387:
1385:
1384:
1379:
1372:
1371:
1370:
1362:
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1360:
1349:
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1347:
1341:
1340:
1331:
1330:
1315:
1314:
1283:
1281:
1280:
1275:
1263:
1262:
1249:
1248:
1233:
1232:
1206:
1204:
1203:
1198:
1194:
1188:
1164:
1162:
1161:
1156:
1152:
1145:
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1132:
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1118:
1107:
1099:
1097:
1096:
1091:
1087:
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1085:
1074:
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1055:
1047:
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1024:
1023:
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1013:
1001:
1000:
991:
990:
978:
977:
952:
948:
946:
945:
940:
933:
927:
886:
884:
883:
878:
866:
864:
862:
861:
856:
809:
802:
795:
773:Phyllis Nicolson
758:Rudolf Lipschitz
641:Green's function
617:Infinite element
608:
573:Solution methods
551:
409:
320:By variable type
274:
156:Natural sciences
149:
128:
121:
114:
110:
107:
101:
99:
58:
34:
26:
4482:
4481:
4477:
4476:
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4473:
4472:
4471:
4427:
4426:
4425:
4420:
4392:
4354:
4338:
4300:
4248:
4214:Poisson algebra
4190:
4072:
4065:
4023:
3919:Operator theory
3817:
3775:
3741:Tensor software
3717:
3666:Automata theory
3635:
3630:
3554:
3526:
3519:
3512:
3504:
3501:
3496:
3442:
3441:
3437:
3415:
3414:
3410:
3396:
3386:
3376:
3375:
3366:
3316:
3315:
3311:
3304:
3291:
3290:
3286:
3279:
3275:
3268:
3264:
3255:
3253:
3249:
3242:
3238:
3237:
3233:
3226:
3209:
3208:
3204:
3197:
3193:
3186:
3185:978-145378-1470
3173:
3172:
3168:
3153:
3138:
3137:
3126:
3111:
3096:
3095:
3080:
3076:
3071:
3007:
2991:
2983:coupled cluster
2961:
2948:
2888:
2841:
2828:
2820:
2819:
2787:
2779:
2778:
2754:
2746:
2745:
2721:
2713:
2712:
2684:
2683:
2657:
2649:
2648:
2615:
2602:
2595:
2581:
2568:
2564:
2554:
2553:
2507:Richard Feynman
2440:
2332:
2331:
2328:
2306:'s orbit, that
2269:
2261:
2260:
2233:
2225:
2224:
2196:
2195:
2171:
2155:
2141:
2140:
2116:
2108:
2107:
2077:
2069:
2068:
2041:
2033:
2032:
2011:
2000:
1999:
1975:
1959:
1945:
1944:
1915:
1899:
1885:
1884:
1859:
1858:
1834:
1826:
1825:
1801:
1793:
1792:
1758:
1757:
1756:for some small
1733:
1722:
1721:
1697:
1689:
1688:
1663:
1662:
1634:
1633:
1608:
1607:
1599:
1528:
1489:
1481:
1480:
1447:
1443:
1438:
1430:
1429:
1393:
1392:
1352:
1322:
1306:
1294:
1293:
1292:
1240:
1224:
1213:
1212:
1183:
1182:
1136:
1123:
1110:
1102:
1101:
1077:
1069:
1068:
1038:
1028:
1015:
1005:
992:
982:
969:
958:
957:
950:
922:
921:
909:
869:
868:
847:
846:
845:
813:
784:
783:
782:
713:Jacob Bernoulli
697:
684:
683:
665:
634:Petrov–Galerkin
602:
587:
574:
566:
565:
564:
546:
492:Boundary values
481:
473:
472:
448:
435:
434:
433:
407:
401:
393:
392:
380:
357:
315:
271:
258:
257:
253:
231:Social sciences
187:
165:
146:
122:
111:
105:
102:
59:
57:
47:
35:
24:
17:
12:
11:
5:
4480:
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4470:
4469:
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4459:
4454:
4449:
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4405:
4397:
4394:
4393:
4391:
4390:
4385:
4380:
4375:
4374:
4373:
4362:
4360:
4356:
4355:
4353:
4352:
4346:
4344:
4340:
4339:
4337:
4336:
4329:
4324:
4319:
4314:
4308:
4306:
4302:
4301:
4299:
4298:
4293:
4288:
4283:
4278:
4273:
4268:
4262:
4260:
4254:
4253:
4250:
4249:
4247:
4246:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4200:
4198:
4189:
4188:
4187:
4186:
4181:
4171:
4170:
4169:
4164:
4154:
4149:
4148:
4147:
4137:
4136:
4135:
4130:
4125:
4120:
4115:
4110:
4105:
4095:
4094:
4093:
4088:
4077:
4075:
4067:
4066:
4064:
4063:
4058:
4053:
4044:
4033:
4031:
4025:
4024:
4022:
4021:
4016:
4015:
4014:
4009:
4004:
3999:
3989:
3988:
3987:
3982:
3977:
3972:
3962:
3961:
3960:
3955:
3950:
3945:
3935:
3934:
3933:
3923:
3922:
3921:
3916:
3906:
3905:
3904:
3902:Control theory
3899:
3889:
3888:
3887:
3882:
3877:
3867:
3866:
3865:
3860:
3855:
3845:
3844:
3843:
3833:
3827:
3825:
3819:
3818:
3816:
3815:
3810:
3805:
3800:
3795:
3789:
3787:
3781:
3780:
3777:
3776:
3774:
3773:
3768:
3763:
3758:
3753:
3748:
3743:
3738:
3733:
3727:
3725:
3716:
3715:
3710:
3705:
3700:
3695:
3694:
3693:
3683:
3678:
3673:
3668:
3663:
3662:
3661:
3656:
3645:
3643:
3637:
3636:
3631:
3629:
3628:
3621:
3614:
3606:
3600:
3599:
3551:
3524:
3500:
3499:External links
3497:
3495:
3494:
3455:(1): L71–L75.
3435:
3408:
3364:
3309:
3302:
3284:
3273:
3262:
3231:
3224:
3218:. p. 34.
3202:
3191:
3184:
3166:
3151:
3124:
3109:
3077:
3075:
3072:
3070:
3069:
3064:
3059:
3054:
3049:
3044:
3039:
3034:
3029:
3024:
3019:
3014:
3012:Boundary layer
3008:
3006:
3003:
2995:shell-crossing
2990:
2989:Shell-crossing
2987:
2960:
2957:
2947:
2944:
2887:
2884:
2848:
2844:
2840:
2835:
2831:
2794:
2790:
2761:
2757:
2728:
2724:
2700:
2694:
2672:
2664:
2660:
2627:
2622:
2618:
2614:
2609:
2605:
2601:
2588:
2584:
2580:
2575:
2571:
2439:
2436:
2364:wave equations
2339:
2327:
2324:
2318:, which allow
2284:
2276:
2272:
2248:
2240:
2236:
2212:
2206:
2178:
2174:
2170:
2167:
2162:
2158:
2154:
2151:
2123:
2119:
2092:
2084:
2080:
2056:
2048:
2044:
2018:
2014:
2010:
1982:
1978:
1974:
1971:
1966:
1962:
1958:
1955:
1930:
1922:
1918:
1914:
1911:
1906:
1902:
1898:
1895:
1869:
1841:
1837:
1808:
1804:
1780:
1774:
1771:
1768:
1740:
1736:
1732:
1704:
1700:
1673:
1644:
1618:
1598:
1595:
1527:
1524:
1499:
1496:
1492:
1462:
1458:
1454:
1450:
1446:
1441:
1403:
1377:
1369:
1359:
1355:
1346:
1339:
1334:
1329:
1325:
1321:
1318:
1313:
1309:
1305:
1302:
1289:big O notation
1285:
1284:
1273:
1270:
1267:
1261:
1258:
1255:
1247:
1243:
1239:
1236:
1231:
1227:
1223:
1220:
1193:
1165:represent the
1151:
1148:
1143:
1139:
1135:
1130:
1126:
1122:
1117:
1113:
1084:
1080:
1065:
1064:
1053:
1050:
1045:
1041:
1035:
1031:
1027:
1022:
1018:
1012:
1008:
1004:
999:
995:
989:
985:
981:
976:
972:
968:
965:
938:
932:
908:
905:
876:
854:
815:
814:
812:
811:
804:
797:
789:
786:
785:
781:
780:
775:
770:
765:
763:Ernst Lindelöf
760:
755:
750:
745:
740:
735:
733:Joseph Fourier
730:
725:
720:
718:Leonhard Euler
715:
710:
705:
699:
698:
695:
694:
691:
690:
686:
685:
682:
681:
676:
671:
664:
663:
658:
653:
648:
643:
638:
637:
636:
626:
621:
620:
619:
612:Finite element
609:
605:Crank–Nicolson
596:
591:
585:
580:
576:
575:
572:
571:
568:
567:
563:
562:
557:
552:
544:
539:
526:
524:Phase portrait
521:
516:
515:
514:
512:Cauchy problem
509:
504:
499:
489:
483:
482:
480:General topics
479:
478:
475:
474:
471:
470:
465:
460:
455:
449:
446:
445:
442:
441:
437:
436:
432:
431:
426:
425:
424:
413:
412:
411:
402:
399:
398:
395:
394:
389:
388:
387:
386:
379:
378:
373:
367:
364:
363:
359:
358:
356:
355:
353:Nonhomogeneous
346:
341:
338:
332:
331:
330:
322:
321:
317:
316:
314:
313:
308:
303:
298:
293:
288:
283:
277:
272:
269:
268:
265:
264:
263:Classification
260:
259:
250:
249:
248:
247:
242:
234:
233:
227:
226:
225:
224:
219:
214:
206:
205:
199:
198:
197:
196:
191:
185:
180:
175:
167:
166:
164:
163:
158:
152:
147:
144:
143:
140:
139:
135:
134:
124:
123:
38:
36:
29:
15:
13:
10:
9:
6:
4:
3:
2:
4479:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4434:
4432:
4417:
4413:
4409:
4406:
4404:
4403:
4399:
4398:
4395:
4389:
4386:
4384:
4381:
4379:
4376:
4372:
4369:
4368:
4367:
4364:
4363:
4361:
4359:Organizations
4357:
4351:
4348:
4347:
4345:
4341:
4334:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4309:
4307:
4303:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4263:
4261:
4259:
4255:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4220:
4219:Quantum group
4217:
4215:
4212:
4210:
4207:
4205:
4202:
4201:
4199:
4197:
4193:
4185:
4182:
4180:
4177:
4176:
4175:
4174:Supersymmetry
4172:
4168:
4165:
4163:
4160:
4159:
4158:
4157:String theory
4155:
4153:
4150:
4146:
4143:
4142:
4141:
4138:
4134:
4131:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4109:
4106:
4104:
4101:
4100:
4099:
4096:
4092:
4089:
4087:
4084:
4083:
4082:
4079:
4078:
4076:
4074:
4068:
4062:
4059:
4057:
4056:Path integral
4054:
4052:
4048:
4045:
4042:
4038:
4037:Distributions
4035:
4034:
4032:
4030:
4026:
4020:
4017:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3994:
3993:
3990:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3967:
3966:
3963:
3959:
3956:
3954:
3951:
3949:
3946:
3944:
3941:
3940:
3939:
3936:
3932:
3929:
3928:
3927:
3924:
3920:
3917:
3915:
3912:
3911:
3910:
3907:
3903:
3900:
3898:
3895:
3894:
3893:
3890:
3886:
3883:
3881:
3878:
3876:
3873:
3872:
3871:
3868:
3864:
3861:
3859:
3856:
3854:
3851:
3850:
3849:
3846:
3842:
3839:
3838:
3837:
3834:
3832:
3829:
3828:
3826:
3824:
3820:
3814:
3811:
3809:
3806:
3804:
3803:Combinatorics
3801:
3799:
3796:
3794:
3791:
3790:
3788:
3786:
3782:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3737:
3734:
3732:
3729:
3728:
3726:
3724:
3720:
3714:
3711:
3709:
3706:
3704:
3701:
3699:
3696:
3692:
3689:
3688:
3687:
3684:
3682:
3679:
3677:
3676:Coding theory
3674:
3672:
3669:
3667:
3664:
3660:
3657:
3655:
3652:
3651:
3650:
3647:
3646:
3644:
3642:
3641:Computational
3638:
3634:
3627:
3622:
3620:
3615:
3613:
3608:
3607:
3604:
3596:
3592:
3588:
3584:
3580:
3576:
3571:
3566:
3562:
3558:
3552:
3547:
3542:
3538:
3534:
3530:
3525:
3518:
3511:
3507:
3503:
3502:
3498:
3490:
3486:
3481:
3476:
3472:
3468:
3463:
3458:
3454:
3450:
3446:
3439:
3436:
3431:
3427:
3423:
3419:
3412:
3409:
3404:
3399:|people=
3391:
3383:
3379:
3373:
3371:
3369:
3365:
3359:
3355:
3350:
3345:
3341:
3337:
3333:
3329:
3328:
3323:
3319:
3318:Dirac, P.A.M.
3313:
3310:
3305:
3299:
3295:
3288:
3285:
3282:
3277:
3274:
3271:
3266:
3263:
3252:on 2018-04-17
3248:
3241:
3235:
3232:
3227:
3221:
3217:
3213:
3206:
3203:
3200:
3195:
3192:
3187:
3181:
3177:
3170:
3167:
3162:
3158:
3154:
3148:
3144:
3143:
3135:
3133:
3131:
3129:
3125:
3120:
3116:
3112:
3106:
3102:
3101:
3093:
3091:
3089:
3087:
3085:
3083:
3079:
3073:
3068:
3065:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3018:
3015:
3013:
3010:
3009:
3004:
3002:
3000:
2996:
2988:
2986:
2984:
2980:
2975:
2971:
2966:
2958:
2956:
2954:
2945:
2943:
2941:
2937:
2933:
2929:
2925:
2921:
2917:
2912:
2909:
2905:
2901:
2896:
2894:
2885:
2883:
2881:
2877:
2873:
2872:
2867:
2846:
2842:
2838:
2833:
2829:
2817:
2813:
2792:
2788:
2759:
2755:
2726:
2722:
2698:
2692:
2670:
2662:
2658:
2620:
2616:
2612:
2607:
2603:
2586:
2582:
2578:
2573:
2569:
2551:
2547:
2543:
2539:
2534:
2532:
2528:
2524:
2520:
2515:
2512:
2508:
2504:
2499:
2497:
2496:hydrogen atom
2493:
2489:
2488:Zeeman effect
2484:
2480:
2476:
2472:
2467:
2465:
2461:
2457:
2453:
2449:
2445:
2437:
2435:
2433:
2428:
2426:
2421:
2419:
2415:
2411:
2406:
2404:
2400:
2396:
2392:
2388:
2383:
2381:
2377:
2373:
2369:
2365:
2362:and commonly
2361:
2357:
2353:
2337:
2325:
2323:
2321:
2317:
2313:
2309:
2305:
2301:
2296:
2282:
2274:
2270:
2246:
2238:
2234:
2210:
2204:
2176:
2172:
2168:
2165:
2160:
2156:
2152:
2149:
2121:
2117:
2106:
2090:
2082:
2078:
2054:
2046:
2042:
2016:
2012:
2008:
1980:
1976:
1972:
1969:
1964:
1960:
1956:
1953:
1941:
1928:
1920:
1916:
1912:
1909:
1904:
1900:
1896:
1893:
1867:
1839:
1835:
1806:
1802:
1791:The solution
1778:
1772:
1769:
1766:
1738:
1734:
1730:
1702:
1698:
1671:
1659:
1642:
1616:
1604:
1596:
1594:
1592:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1555:simple enough
1551:
1549:
1545:
1541:
1537:
1533:
1525:
1523:
1521:
1519:
1497:
1494:
1490:
1460:
1456:
1452:
1448:
1444:
1439:
1427:
1426:
1420:
1401:
1389:
1375:
1357:
1353:
1332:
1327:
1323:
1319:
1316:
1311:
1307:
1303:
1300:
1290:
1271:
1265:
1245:
1241:
1237:
1234:
1229:
1225:
1221:
1218:
1211:
1210:
1209:
1191:
1180:
1176:
1172:
1168:
1149:
1146:
1141:
1137:
1133:
1128:
1124:
1120:
1115:
1111:
1082:
1078:
1051:
1048:
1043:
1039:
1033:
1029:
1025:
1020:
1016:
1010:
1006:
1002:
997:
993:
987:
983:
979:
974:
970:
966:
963:
956:
955:
954:
936:
930:
918:
914:
906:
904:
902:
898:
894:
889:
874:
852:
843:
838:
834:
830:
826:
822:
810:
805:
803:
798:
796:
791:
790:
788:
787:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
700:
693:
692:
687:
680:
677:
675:
672:
670:
667:
666:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
635:
632:
631:
630:
627:
625:
624:Finite volume
622:
618:
615:
614:
613:
610:
606:
600:
597:
595:
592:
590:
586:
584:
581:
578:
577:
570:
569:
561:
558:
556:
553:
549:
545:
543:
540:
538:
534:
530:
527:
525:
522:
520:
517:
513:
510:
508:
505:
503:
500:
498:
495:
494:
493:
490:
488:
485:
484:
477:
476:
469:
466:
464:
461:
459:
456:
454:
451:
450:
444:
443:
438:
430:
427:
423:
420:
419:
418:
415:
414:
410:
404:
403:
397:
396:
385:
382:
381:
377:
374:
372:
369:
368:
366:
365:
360:
354:
350:
347:
345:
342:
339:
337:
334:
333:
329:
326:
325:
324:
323:
318:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
278:
276:
275:
267:
266:
261:
256:
246:
243:
241:
238:
237:
236:
235:
232:
228:
223:
220:
218:
215:
213:
210:
209:
208:
207:
204:
200:
195:
192:
190:
186:
184:
181:
179:
176:
174:
171:
170:
169:
168:
162:
159:
157:
154:
153:
150:
142:
141:
136:
133:
129:
120:
117:
109:
106:February 2017
98:
95:
91:
88:
84:
81:
77:
74:
70:
67: –
66:
62:
61:Find sources:
55:
51:
45:
44:
39:This article
37:
33:
28:
27:
22:
4414: /
4410: /
4400:
4276:Optimization
4239:Superalgebra
4139:
4098:Field theory
4071:Mathematical
4049: /
3897:Chaos theory
3880:Gauge theory
3808:Graph theory
3703:Cryptography
3560:
3556:
3539:(10): 1617.
3536:
3533:Scholarpedia
3532:
3452:
3448:
3438:
3421:
3417:
3411:
3381:
3331:
3325:
3312:
3293:
3287:
3276:
3265:
3254:. Retrieved
3247:the original
3234:
3211:
3205:
3194:
3175:
3169:
3141:
3099:
2994:
2992:
2963:Many of the
2962:
2959:In chemistry
2949:
2913:
2897:
2889:
2880:chaos theory
2875:
2869:
2816:normal modes
2545:
2541:
2535:
2519:chaos theory
2516:
2500:
2468:
2441:
2429:
2425:interactions
2422:
2407:
2403:ground state
2398:
2386:
2384:
2329:
2307:
2300:Isaac Newton
2297:
2104:
1942:
1660:
1602:
1600:
1587:Solar System
1554:
1552:
1531:
1529:
1517:
1423:
1418:
1390:
1286:
1178:
1174:
1171:second-order
1170:
1166:
1066:
916:
910:
890:
842:power series
828:
818:
768:Émile Picard
753:Martin Kutta
743:George Green
703:Isaac Newton
655:
535: /
531: /
351: /
217:Chaos theory
112:
103:
93:
86:
79:
72:
60:
48:Please help
43:verification
40:
4416:topics list
4350:Mathematics
4266:Game theory
4167:Topological
4133:Topological
4128:Statistical
4091:Hamiltonian
2999:singularity
2930:in 1848 by
2358:(e.g., the
2295:and so on.
1175:third-order
1167:first-order
915:known as a
907:Description
821:mathematics
661:Runge–Kutta
406:Difference
349:Homogeneous
161:Engineering
4431:Categories
4322:Psychology
4286:Statistics
4086:Lagrangian
3713:Statistics
3649:Algorithms
3462:2010.12584
3256:2017-04-30
3074:References
2940:J.G. Galle
2932:Le Verrier
2475:Paul Dirac
2391:trajectory
778:John Crank
579:Inspection
533:Asymptotic
417:Stochastic
336:Autonomous
311:Non-linear
301:Fractional
76:newspapers
4327:Sociology
4317:Chemistry
4113:Effective
4108:Conformal
4103:Classical
3975:Geometric
3948:Geometric
3595:117362666
3570:1110.0723
3563:(1): 22.
3489:0035-8711
3401:ignored (
3161:821883201
3119:851704808
2866:"blow up"
2843:ω
2839:−
2830:ω
2789:ω
2756:ω
2723:ϕ
2659:ψ
2617:ω
2613:−
2604:ω
2583:ϕ
2570:ψ
2527:KAM torus
2235:ε
2169:ε
2153:≈
2139:and thus
2009:ε
1973:ε
1957:≈
1913:ε
1770:≪
1767:ε
1731:ε
1495:−
1491:ε
1440:ε
1402:ε
1354:ε
1320:ε
1269:→
1266:ε
1238:ε
1222:≈
1192:ε
1150:…
1052:⋯
1030:ε
1007:ε
984:ε
967:≡
875:ε
853:ε
519:Wronskian
497:Dirichlet
240:Economics
183:Chemistry
173:Astronomy
4402:Category
4051:analysis
3970:Exterior
3943:Exterior
3823:Analysis
3785:Discrete
3659:analysis
3517:Archived
3390:cite web
3005:See also
2985:method.
2904:Lagrange
2550:Poincaré
2460:Lagrange
2350:include
2326:Examples
1565:. Under
1518:singular
837:solution
629:Galerkin
529:Lyapunov
440:Solution
384:Notation
376:Operator
362:Features
281:Ordinary
4412:outline
4343:Related
4312:Biology
4162:Bosonic
4123:Quantum
4073:physics
4039: (
3771:Solvers
3575:Bibcode
3467:Bibcode
3336:Bibcode
2920:Poisson
2916:Laplace
2908:Laplace
2494:in the
2490:to the
2464:Laplace
2438:History
1419:regular
502:Neumann
286:Partial
194:Geology
189:Biology
178:Physics
90:scholar
3985:Vector
3980:Tensor
3958:Vector
3953:Tensor
3654:design
3593:
3487:
3356:
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