Analog of the Gellerstedt problem for a loaded equation of the third order

Abstract

In this paper, we study an analog of the Gellerstedt problem for the third‐order loaded equation with boundary conditions on nonparallel characteristics in the hyperbolic domain of the equation.

A generalized $(G'/G)$-expansion method for the loaded modified Korteweg-de Vries equation

“…Step 5. Substituting the values a j (j = 1, 2, ... , m) and ξ as well as the solutions of equation ( 6) into (5) we have the exact solutions of equation ( 2).…”

“…It is known that loaded differential equations have great practical applications. In the literature [1,2,3,4,5], loaded differential equations are typically called equations containing in the coefficients or in the right-hand side any functionals of the solution, in particular the values of the solution or its derivatives on manifolds of lower dimension. These types of equations were explored in the works of N.N.…”

Cauchy problem for a high‐order loaded integro‐differential equation

“…Let . In this case, the domain of definition will be and we will find the solution of Problems (), (), considering loaded part as known, 36 …”

“…and we will find the solution of Problems (2.10), (2.11), considering loaded part as known, 36 Let us introduce the notations v 0 (x, t, 𝜏) = U(x, t, 𝜏) and v k (x, t, 𝜏) = G k v 0 (x, t, 𝜏). In these notation, Problems (2.10), (2.11) are equivalent to the following problem about v k (x, t, 𝜏), k = 0, m − 1, finding functions of equations…”

Extension of the Tricomi problem for a loaded parabolic–hyperbolic equation with a characteristic line of change of type

“…Problem $$$ {T}_1 $$$ in case $$$ n=1 $$$ was first posed and investigated by Gellerstedt for the classical mixed‐type equation. For a loaded equation, the Gellerstedt problem was studied in Baltaeva and Pedro and Abdullaev and Islomov 26,27 . The present generalized type of problem $$$ {T}_1 $$$ for the classical equation of elliptic–hyperbolic type was proposed in the works of Djuraev and Sopuev, Mirsaburov et al, and Bitsadze 25,28,29 …”