scholarly journals On the strauss index of semilinear tricomi equation

2020 ◽  
Vol 19 (10) ◽  
pp. 4817-4838
Author(s):  
Daoyin He ◽  
◽  
Ingo Witt ◽  
Huicheng Yin ◽  
◽  
...  
Keyword(s):  
Tricomi Equation

scholarly journals Constructing analytic solutions on the Tricomi equation

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Emran Khoshrouye Ghiasi ◽  
Reza Saleh
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Tricomi Equation ◽  
Homotopy Analysis ◽  
Optimal Values

AbstractIn this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

scholarly journals Remark on the Tricomi Equation

1972 ◽  
Vol 48 ◽  
pp. 91-97
Author(s):  
Tadato Matsuzawa
Keyword(s):  
Open Set ◽  
Tricomi Equation

As an application of the Garleman-type estimation Hörmander [4], p. 221, has proved the following:A solution (distribution) of the Tricomi equationin an open set Ω in belongs to C∞(Ω) if it is in C∞(Ω-) where .

scholarly journals On the existence and cusp singularity of solutions to semilinear generalized Tricomi equations with discontinuous initial data

2015 ◽  
Vol 17 (03) ◽  
pp. 1450028 ◽  
Author(s):  
Zhuoping Ruan ◽  
Ingo Witt ◽  
Huicheng Yin
Keyword(s):  
Initial Data ◽  
Local Existence ◽  
Piecewise Smooth ◽  
Degree Zero ◽  
Essential Difference ◽  
Existence Of A Solution ◽  
Discontinuous Initial Data ◽  
Tricomi Equation ◽  
Low Regularity Solution ◽  
Straight Lines

In this paper, we are concerned with the local existence and singularity structures of low regularity solution to the semilinear generalized Tricomi equation [Formula: see text] with typical discontinuous initial data (u(0, x), ∂tu(0, x)) = (0, φ(x)), where m ∈ ℕ, x = (x1,…,xn), n ≥ 2, and f(t, x, u) is C∞ smooth on its arguments. When the initial data φ(x) is homogeneous of degree zero or piecewise smooth along the hyperplane {t = x1 = 0}, it is shown that the local solution u(t, x) ∈ L∞([0, T] × ℝn) exists and is C∞ away from the forward cuspidal conic surface [Formula: see text] or the cuspidal wedge-shaped surfaces [Formula: see text] respectively. On the other hand, for n = 2 and piecewise smooth initial data φ(x) along the two straight lines {t = x1 = 0} and {t = x2 = 0}, we establish the local existence of a solution [Formula: see text] and further show that [Formula: see text] in general due to the degenerate character of the equation under study, where [Formula: see text]. This is an essential difference to the well-known result for solution [Formula: see text] to the two-dimensional semilinear wave equation [Formula: see text] with (v(0, x), ∂tv(0, x)) = (0, φ(x)), where Σ0 = {t = |x|}, [Formula: see text] and [Formula: see text].

scholarly journals Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip

2018 ◽  
pp. 1-12
Author(s):  
O. D. Algazin
Keyword(s):  
Dirichlet Problem ◽  
Half Plane ◽  
Mixed Type ◽  
Polynomial Growth ◽  
Polynomial Solution ◽  
Transform Method ◽  
Polynomial Solutions ◽  
Tricomi Equation ◽  
Equation Of Mixed Type ◽  
Class Of Functions

In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.

scholarly journals On a semi-nonlocal boundary value problem for the three-dimensional Tricomi equation of an unbounded prismatic domain

2021 ◽  
pp. 8-16
Author(s):  
С.З. Джамалов ◽  
Р.Р. Ашуров ◽  
Х.Ш. Туракулов
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Three Dimensional ◽  
Generalized Solution ◽  
Nonlocal Boundary Value Problem ◽  
Tricomi Equation ◽  
The Fourier Transform

В данной статье изучаются методами «ε-регуляризации» и априорных оценок с применением преобразования Фурье однозначная разрешимость и гладкость обобщенного решения одной полунелокальной краевой задачи для трехмерного уравнения Трикоми в неограниченной призматической области. In this article, the methods of «ε-regularization» and a priori estimates using the Fourier transform are studied the unique solvability and smoothness of the generalized solution of one semi-nonlocal boundary value problem for the three-dimensional Tricomi equation in an unbounded prismatic domain.

scholarly journals A nonlocal problem for a generalized Trikomi equation with a spectral parameter in the unbounded domain

2021 ◽  
pp. 17-26
Author(s):  
Р.Т. Зуннунов
Keyword(s):  
Half Plane ◽  
Spectral Parameter ◽  
Unbounded Domain ◽  
Nonlocal Problem ◽  
Existence Of A Solution ◽  
Tricomi Equation ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Upper Half Plane ◽  
Generalized Tricomi Equation

В данной статье изучена нелокальная задача для обобщенного уравнения Трикоми со спектральным параметром в неограниченной области эллиптическая часть которой является верхней полуплоскостью. Единственность решения поставленной задачи доказана методом интегралов энергии. Существование решения поставленной задачи доказана методом функций Грина и интегральных уравнений. In this article, we study a nonlocal problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane. The uniqueness of the solution to the problem posed is proved by the method of energy integrals. The existence of a solution to the problem is proved by the method of Green’s functions and integral equations.

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