Solution of Partial Differential Equations by Infinite Series (By the Fourier Method)

1994 ◽  
pp. 1250-1261
Author(s):  
Karel Rektorys
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Infinite Series ◽  
Fourier Method ◽  
Partial Differential

scholarly journals A fuzzy solution of nonlinear partial differential equations

2021 ◽  
Vol 5 (1) ◽  
pp. 51-63
Author(s):  
Mawia Osman ◽  
◽  
Zengtai Gong ◽  
Altyeb Mohammed Mustafa ◽  
◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Infinite Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Series Expansions ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Fuzzy Solution

In this paper, the reduced differential transform method (RDTM) is applied to solve fuzzy nonlinear partial differential equations (PDEs). The solutions are considered as infinite series expansions which converge rapidly to the solutions. Some examples are solved to illustrate the proposed method.

XIII.—“On the Singular Solutions of Partial Differential Equations of the First Order.”

1913 ◽  
Vol 32 ◽  
pp. 150-163
Author(s):  
H. Levy
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Theorem ◽  
Infinite Series ◽  
Singular Solutions ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Original Case

The complete integral of the differential equationφ(xyzpq) = 0is a relation among the variables, which includes as many arbitrary constants as there are independent variables. But it is important to distinguish carefully between differential equations which have been formed by the elimination of constants from some complete primitive, and those whose origin is quite unknown, or which may have been constructed by some method totally different from the first.In the original case, the differential equation can always be integrated in finite terms, while in the latter, only under the conditions laid down in Cauchy's Existence Theorem can an integral be obtained, and even then usually as an infinite series.

scholarly journals ON INFINITE SERIES OF NONLOCAL CONSERVATION LAWS FOR PARTIAL DIFFERENTIAL EQUATIONS

2018 ◽  
Vol 21 (3) ◽  
pp. 150-159
Author(s):  
N. G. Khor’kova
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Infinite Series ◽  
Conservation Law ◽  
Integrability Condition ◽  
Normal System ◽  
Partial Differential ◽  
Nonlocal Symmetries ◽  
Nonlocal Conservation Laws

Популярное в математике понятие интегрируемости дифференциальных уравнений (и столь же разнообразно трактуемое) тесно связано с существованием симметрий и законов сохранения. Все известные интегрируемые дифференциальные уравнения обладают бесконечными сериями симметрий и (или) законов сохранения. Однако также имеется целый ряд уравнений, важных для приложений, но имеющих крайне скудный запас симметрий или законов сохранения. Попытки расширить понятия симметрии и закона сохранения предпринимались разными авторами, и на эту тему имеется обширная литература. В данной статье представлен следующий результат. Если ℓ-нормальная система дифференциальных уравнений в частных производных имеет когомологически нетривиальный закон сохранения, то этот закон сохранения порождает бесконечную серию нелокальных законов сохранения. Этот факт обобщает аналогичный результат статьи автора для дифференциальных уравнений (не систем). Результат получен в рамках геометрической теории дифференциальных уравнений в частных производных. Согласно геометрическому подходу, многообразие, снабженное конечномерным распределением, удовлетворяющим условиям интегрируемости Фробениуса, называется диффеотопом (diffiety), если локально оно имеет вид бесконечно продолженного уравнения Ɛ∞. Диффеотопы являются объектами категории дифференциальных уравнений, введенной А.М. Виноградовым. Под симметриями уравнения понимают преобразования (конечные или инфинитизимальные) бесконечного продолжения уравнения, которые сохраняют распределение Картана, а под законами сохранения – (n-1)-e классы когомологий горизонтального комплекса де Рама уравнения, где n – число независимых переменных уравнения. Накрытием называется эпиморфизм  τ:Ɛ⟶ Ɛ∞ в категории дифференциальных уравнений, порождающий изоморфизм распределений. Симметрии и законы сохранения диффеотопа ࣟƐ называются нелокальными симметриями и законами сохранения уравнения ࣟƐ  Выбор подходящего накрытия позволяет получать новые (нелокальные) симметрии и законы сохранения исследуемого уравнения. В работе приведена конструкция одного накрытия и доказано существование бесконечных серий нелокальных законов сохранения у широкого класса систем дифференциальных уравнений в частных производных.системы дифференциальных уравнений в частных производных; накрытия дифференциальных уравнений; нелокальные симметрии и законы сохранения  The notion of integrability of differential equations is closely connected with the existence of symmetries and conservation laws. All known integrable differential equations have infinite series of symmetries and (or) conservation laws. However, there is also a number of equations that are important for applications, but with an extremely scarce stock of symmetries or conservation laws. Attempts to extend the concepts of symmetry and conservation law were made by different authors. This article presents the following result. If a ℓ-normal system of partial differential equations has a cohomologically nontrivial conservation law, then this conservation law generates an infinite series of non-local conservation laws. This fact generalizes the analogous result of the author for differential equations (not systems). The result is obtained within the framework of geometrical theory of partial differential equations (PDE). A manifold supplied with an infinite-dimensional distribution satisfying the Frobenius complete integrability condition is called a diffiety, if it is locally in the form of  Ɛ∞. Diffieties are objects of the category of differential equations introduced by A.M. Vinogradov. Symmetries of PDE are transformations (finite or infinitesimal) of the infinite prolongation  Ɛ∞ preserving the Cartan distribution, while conservation laws are (n-1)-cohomology classes of the horizontal de Rham cohomology. If a covering τ:Ɛ⟶ Ɛ∞ is given, then symmetries and conservation laws of the diffiety Ɛ are called nonlocal symmetries and conservation laws of the equation Ɛ .In appropriate coverings one can get new (nonlocal) symmetries and conservation laws for an equation under consideration. In this paper we investigate one covering and prove the existence of infinite series of nonlocal conservation laws.   

scholarly journals Basicity of Systems of Sines with Linear Phase in Weighted Sobolev Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
V. F. Salmanov ◽  
A. R. Safarova
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Fourier Method ◽  
Linear Phase ◽  
Partial Differential ◽  
Perturbed Systems

The perturbed systems of sines, which appear when solving some partial differential equations by the Fourier method, are considered in this paper. Basis properties of these systems in weighted Sobolev spaces of functions are studied.

scholarly journals Solution of the First Boundary-Value Problem for a System of Autonomous Second-Order Linear Partial Differential Equations of Parabolic Type with a Single Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Josef Diblík ◽  
Denis Khusainov ◽  
Oleksandra Kukharenko ◽  
Zdeněk Svoboda
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Parabolic Type ◽  
Second Order ◽  
Partial Differential ◽  
Order Linear ◽  
Linear Partial Differential Equations

The first boundary-value problem for an autonomous second-order system of linear partial differential equations of parabolic type with a single delay is considered. Assuming that a decomposition of the given system into a system of independent scalar second-order linear partial differential equations of parabolic type with a single delay is possible, an analytical solution to the problem is given in the form of formal series and the character of their convergence is discussed. A delayed exponential function is used in order to analytically solve auxiliary initial problems (arising when Fourier method is applied) for ordinary linear differential equations of the first order with a single delay.

Sign in / Sign up

Export Citation Format

Share Document