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## Research Article Solution of Some Types of Differential Equations: Operational Calculus and Inverse Differential Operators

### Citations

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Citation Context ...ts ... |

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Citation Context ...For the operator ̂ 2 − ... |

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Citation Context ...2... |

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Operational formulas connected with two generalizations of Hermite polynomials
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Citation Context ... exp(... |

10 |
Generalized polynomials, operational identities and their applications
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Citation Context ...of Hermite, Laguerre, and other relevant polynomials [1] is useful when solvingmany physical problems (see, e.g., [2, 3]). Generalised forms of these polynomials exist with many variables and indices =-=[4, 5]-=-. In what follows, we develop an analytical method to obtain solutions for various types of partial differential equations on the base of operational identities, employing expansions in series of Herm... |

6 |
Kampé de Feriet, Fonctions Hypergéométriques et Hypersphériques. Polynomes D’Hermite (Gauthiers-Villars
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(Show Context)
Citation Context ...few types of differential equations, allowing explicit and straightforward analytical solutions. It is common knowledge that expansion into series of Hermite, Laguerre, and other relevant polynomials =-=[1]-=- is useful when solvingmany physical problems (see, e.g., [2, 3]). Generalised forms of these polynomials exist with many variables and indices [4, 5]. In what follows, we develop an analytical method... |

5 |
A representation theory for solutions of a higher order heat equation,
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- 1992
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Citation Context ...raightforward analytical solutions. It is common knowledge that expansion into series of Hermite, Laguerre, and other relevant polynomials [1] is useful when solvingmany physical problems (see, e.g., =-=[2, 3]-=-). Generalised forms of these polynomials exist with many variables and indices [4, 5]. In what follows, we develop an analytical method to obtain solutions for various types of partial differential e... |

3 |
Оrthogonality properties of the hermite and related polynomials,
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(Show Context)
Citation Context ...of Hermite, Laguerre, and other relevant polynomials [1] is useful when solvingmany physical problems (see, e.g., [2, 3]). Generalised forms of these polynomials exist with many variables and indices =-=[4, 5]-=-. In what follows, we develop an analytical method to obtain solutions for various types of partial differential equations on the base of operational identities, employing expansions in series of Herm... |

3 |
Solutions of integrodifferential equations and operational methods
- Dattoli, Migliorati, et al.
- 2007
(Show Context)
Citation Context ...odified forms [1, 6]. The key for building these solutions will be an operational approach and development of the formalism of inverse functions and inverse differential operators, already touched in =-=[7, 8]-=-. We will demonstrate in what follows that when used properly and combined, in particular, with integral transforms, such an approach leads to elegant analytical solutions with transparent physical me... |

2 |
Operational methods and Differential Equations with Applications to Initial-Value problems,
- Dattoli, Srivastava, et al.
- 2007
(Show Context)
Citation Context ...tion is useful for the solution of a broad spectrumof partial differential equations, related to heat and diffusion processes. Some of them have been already studied by operational method (see, e.g., =-=[10, 12]-=-). Below we will focus on the generalities of the solution of the following problem: ... |

2 | K.V.Zhukovsky, Appèl Polynomial Series Expansion,
- Dattoli
- 2010
(Show Context)
Citation Context ...would like to underline that operators ?̂? and ?̂?may not commute, so, dependently on the value of their commutator, we will obtain different sequences of operators in (23), disentangling ?̂? and ?̂? =-=[13, 14]-=-. In the simplest case, when operators ?̂? and ?̂? are multiplication and differentiation operators, respectively, they can be easily disentangled in the exponent with account for [?̂?, ?̂?] = 1, (22)... |

2 | Summation Formulae and Stirling Numbers,
- Dattoli, Migliorati, et al.
- 2009
(Show Context)
Citation Context ...would like to underline that operators ?̂? and ?̂?may not commute, so, dependently on the value of their commutator, we will obtain different sequences of operators in (23), disentangling ?̂? and ?̂? =-=[13, 14]-=-. In the simplest case, when operators ?̂? and ?̂? are multiplication and differentiation operators, respectively, they can be easily disentangled in the exponent with account for [?̂?, ?̂?] = 1, (22)... |

1 |
Negative derivatives and special functions,”AppliedMathematics
- Dattoli, Germano, et al.
- 2010
(Show Context)
Citation Context ...odified forms [1, 6]. The key for building these solutions will be an operational approach and development of the formalism of inverse functions and inverse differential operators, already touched in =-=[7, 8]-=-. We will demonstrate in what follows that when used properly and combined, in particular, with integral transforms, such an approach leads to elegant analytical solutions with transparent physical me... |

1 |
A new family of integral transforms and their applications
- Zhukovsky
- 2006
(Show Context)
Citation Context ...√1 + 4... |