On Factorization and Solution of Multidimensional Linear Partial Differential Equations

Tsarëv, S. P.

doi:10.1142/9789812778857_0011

Cited by 15 publications

(34 citation statements)

References 20 publications

(56 reference statements)

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“…Пусть r ∈ Ker X (L), q -соответствующий r Y -инвариант операто-ра L, z -соответствующий элемент ядра L. Тогда M(z) ∈ Ker L 1 и X-инвариант оператора L 1 можно построить по формуле (13). Полученное выражение,r = − (ab + a x + bq 0 + q 0x )z + z x q 0 + bz y + z x a + z xy (a + q 0 )z + z y (17) не упрощается так же просто как выражение для X-инварианта в дока-зательстве Теоремы 4.1.…”

Section: X-и Y-инвариантыunclassified

“…Получены многочисленные обоб-щения классической теории Дарбу как для гиперболических задач (для систем уравнений на плоскости [18,19,21,12], для уравнений с более чем двумя независимыми переменными [4,5,13], так и для негипербо-лических: [11,10,9,22]. (1)…”

Section: Introductionunclassified

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X- and Y-invariants of partial differential operators in the plane

Shemyakova

2011

Program Comput Soft

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X-и Y -инварианты дифференциальных операторов с частными производными на плоскости Екатерина ШемяковаАннотация В работе рассматривается классическая проблема компьютер-ной алгебры -символьное решение дифференциальных уравнений. А именно, широко используемые теоремы Дарбу о преобразовани-ях гиперболических операторов на плоскости с помощью диффе-ренциальных подстановок переносятся в пространство инвариан-тов. Вводятся X-и Y -инварианты для таких операторов как ре-шения некоторых уравнений записанных в терминах инвариантов Лапласа самого оператора L относительно калибровочных преоб-разований. Получены явные формулы преобразований множеств X-и Y -инвариантов при преобразованиях Дарбу.

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Section: X-и Y-инвариантыunclassified

Section: Introductionunclassified

X- and Y-invariants of partial differential operators in the plane

Shemyakova

2011

Program Comput Soft

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“…One of the important issues is a complete implementation of the algorithms described in Section 2 and in [21]. The current partial implementation developed by the authors for the computer algebra system REDUCE (cf.…”

Section: Vol 1 (2008) Exact Solutions Of Kinetic Equations Verhulstmentioning

confidence: 99%

“…This paper is devoted to a novel application of methods of explicit integration of hyperbolic linear systems of PDEs recently developed in [19][20][21] to an important class of dynamical nonlinear systems driven by a coloured noise.…”

Section: Introductionmentioning

confidence: 99%

“…In that paper a much more general method of explicit integration, applicable to arbitrary hyperbolic higher-order linear systems (or a single higher-order linear PDE) in the plane was developed. Later another generalization was proposed in [21], it gives closed-form complete solutions for some special class of second-order linear hyperbolic equations with more than two independent variables. We give a brief account of the classical Laplace method as well as its new generalizations in Section 2.…”

Section: Introductionmentioning

confidence: 99%

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Exact Solutions of Hyperbolic Systems of Kinetic Equations. Application to Verhulst Model with Random Perturbation

2008

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For hyperbolic first-order systems of linear partial differential equations (master equations), appearing in description of kinetic processes in physics, biology and chemistry we propose new procedures to obtain their complete closed-form non-stationary solutions. The methods used include the classical Laplace cascade method as well as its recent generalizations for systems with more than 2 equations and more than 2 independent variables. As an example we present the complete non-stationary solution (probability distribution) for Verhulst model driven by Markovian coloured dichotomous noise. Mathematics Subject Classification (2000). 60-08, 65C20, 68W30.

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Orbits of Darboux Groupoid for Hyperbolic Operators of Order Three

Shemyakova

2015

Trends in Mathematics

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Darboux transformations are viewed as morphisms in a Darboux category. Darboux transformations of type I which we defined previously, make an important subgroupoid consists of Darboux transformations of type I. We describe the orbits of this subgroupoid for hyperbolic operators of order three.We consider the algebras of differential invariants for our operators. In particular, we show that the Darboux transformations of this class can be lifted to transformations of differential invariants (which we calculate explicitly). (2010). Primary 70H06; Secondary 34A26. J 1 = I 1 − T x , J 2 = I 2 + T xy , J 3 = I 3 + I 2 + T xy , J 4 = 0 , J 5 = f − I 4y /2 + I 3x − I 2y + T xxy /2 , where f = I 5 − I 1 I 2 − I 4y /2, T = ln(f ). Here f = 0 and L = CM + f for some C ∈ K[D]. First results on orbits of Darboux groupoid With M of the form M = ∂ y + m: J 1 = I 1 − T y , J 2 = I 2 − T xy , J 3 = 0 , J 4 = I 4 − I 2 + T xy , J 5 = I 1 I 2 − I 1 T xy − I 2 T y − I 3x /2 + I 2x + I 4y + T xyy /2 + T xy T y + f , where f = I 5 − I 3x , T = ln(f ). Here f = 0 and L = CM + f for some C ∈ K[D]. With M of the form M = ∂ x + ∂ y + m: J 1 = I 1 + T x + T y , J 2 = I 2 , J 3 = −I 4 + 2I 3 + I 1y + T xy + T yy , J 4 = I 1x + 2I 4 − I 3 + T xx + T xy , J 5 = I 4y + I 3x + I 1xy /2 − I 1 I 4 − T x I 4 − T y I 4 + T xxy /2 + T xyy /2 + f , Mathematics Subject Classification

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On Factorization and Solution of Multidimensional Linear Partial Differential Equations

Cited by 15 publications

References 20 publications

X- and Y-invariants of partial differential operators in the plane

X- and Y-invariants of partial differential operators in the plane

Exact Solutions of Hyperbolic Systems of Kinetic Equations. Application to Verhulst Model with Random Perturbation

Orbits of Darboux Groupoid for Hyperbolic Operators of Order Three

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