Kadomtsev–Petviashvili Hierarchy: Negative Times

The Kadomtsev–Petviashvili equation is known to be the leading term of a semi-infinite hierarchy of integrable equations with evolutions given by times with positive numbers. Here, we introduce new hierarchy directed to negative numbers of times. The derivation of such systems, as well as the corresponding hierarchy, is based on the commutator identities. This approach enables introduction of linear differential equations that admit lifts up to nonlinear integrable ones by means of the special dressing procedure. Thus, one can construct not only nonlinear equations, but corresponding Lax pairs as well. The Lax operator of this evolution coincides with the Lax operator of the “positive” hierarchy. We also derive (1 + 1)-dimensional reductions of equations of this hierarchy.

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The variational iteration method for solving n-th order fuzzy differential equations

Open Physics ◽

10.2478/s11534-011-0083-7 ◽

2012 ◽

Vol 10 (1) ◽

Cited By ~ 19

Author(s):

Hossein Jafari ◽

Mohammad Saeidy ◽

Dumitru Baleanu

Keyword(s):

Differential Equations ◽

Nonlinear Equations ◽

Analytical Method ◽

Iteration Method ◽

Initial Conditions ◽

Variational Iteration Method ◽

Linear Differential Equations ◽

Variational Iteration ◽

Linear And Nonlinear ◽

Linear Differential

AbstractThe variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

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Existence and Stability Results on Hadamard Type Fractional Time-Delay Semilinear Differential Equations

Mathematics ◽

10.3390/math8081242 ◽

2020 ◽

Vol 8 (8) ◽

pp. 1242

Author(s):

Nazim Mahmudov ◽

Areen Al-Khateeb

Keyword(s):

Time Delay ◽

Differential Equations ◽

Explicit Form ◽

Nonlinear Equations ◽

Matrix Function ◽

Linear Differential Equations ◽

Existence And Stability ◽

Linear Differential ◽

Stability Results

A delayed perturbation of the Mittag-Leffler type matrix function with logarithm is proposed. This combines the classic Mittag–Leffler type matrix function with a logarithm and delayed Mittag–Leffler type matrix function. With the help of this introduced delayed perturbation of the Mittag–Leffler type matrix function with a logarithm, we provide an explicit form for solutions to non-homogeneous Hadamard-type fractional time-delay linear differential equations. We also examine the existence, uniqueness, and Ulam–Hyers stability of Hadamard-type fractional time-delay nonlinear equations.

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I. On the calculus of symbols.— Fourth memoir. With applications to the theory of non-linear differential equations

Proceedings of the Royal Society of London ◽

10.1098/rspl.1863.0030 ◽

1864 ◽

Vol 13 ◽

pp. 126-126

Keyword(s):

Differential Equations ◽

Linear Function ◽

Nonlinear Equations ◽

Linear Differential Equations ◽

Linear Functions ◽

Non Linear ◽

Linear Differential

In the preceding memoirs on the Calculus of Symbols, systems have been constructed for the multiplication and division of non-commutative symbols subject to certain laws of combination; and these systems suffice for linear differential equations. But when we enter upon the consideration of nonlinear equations, we see at once that these methods do not apply. It becomes necessary to invent some fresh mode of calculation, and a new notation, in order to bring non-linear functions into a condition which admits of treatment by symbolical algebra. This is the object of the following memoir. Professor Boole has given, in his ‘Treatiseon Differential Equations/ a method due to M. Sarrus, by which we ascertain whether a given non-linear function is a complete differential. This method, as will be seen by anyone who will refer to Professor Boole’s treatise, is equivalent to finding the conditions that a non-linear function may be externally divisible by the symbol of differentiation. In the following paper I have given a notation by which I obtain the actual expressions for those conditions, and for the symbolical remainders arising in the course of the division, and have extended my investigations to ascertaining the results of the symbolical division of non linear functions by linear functions of the symbol of differentiation.

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