$l$-Problem of moments for one-dimensional systems defined by integro-differential equations with Erdélyi - Kober operators

2018 ◽  
Author(s):  
Sergey Sergeevich Postnov
Keyword(s):  
Differential Equations ◽  
One Dimensional ◽  
Problem Of Moments

scholarly journals l-problem of moments for one-dimensional integro-differential equations with Erdélyi-Kober operators

2019 ◽  
Vol 486 (6) ◽  
pp. 659-662
Author(s):  
Sergey S. Postnov
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Linear Equations ◽  
Integral Operators ◽  
Analytical Investigation ◽  
Explicit Solutions ◽  
One Dimensional ◽  
Problem Of Moments

Purpose: to investigate the possibility of statement of l-problem of moments for one-dimensional linear equations of three types, which contain Erdélyi-Kober differential and integral operators of fractional order. Methods: formulation of l-problem of moments for each type of investigated equations, analytical investigation and solution of problem formulated Results. Conditions derived that determine the possibility and solvability of the problem stated. In some cases an explicit solutions of l-problem of moments obtained. Conclusions. The possibility of statement of formulated l-problem of moments shown in cases that defined by conditions obtained in paper. Some analytical solutions of investigated problem obtained.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

Some differential equations for problems of wave scattering in a one-dimensional medium

Astrophysics ◽  
1999 ◽  
Vol 42 (3) ◽  
pp. 316-321 ◽  
Author(s):  
D. M. Sedrakian ◽  
A. Zh. Khachatrian
Keyword(s):  
Differential Equations ◽  
Wave Scattering ◽  
One Dimensional

A convenient formulation of Sadowsky's model for elastic ribbons

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Sébastien Neukirch ◽  
Basile Audoly
Keyword(s):  
Differential Equations ◽  
Constitutive Relation ◽  
Classical Problem ◽  
Constitutive Law ◽  
System Of Differential Equations ◽  
Dimensional Model ◽  
Simple Method ◽  
One Dimensional ◽  
Aspect Ratios ◽  
Nonlinear Constitutive Relation

Elastic ribbons are elastic structures whose length-to-width and width-to-thickness aspect ratios are both large. Sadowsky proposed a one-dimensional model for ribbons featuring a nonlinear constitutive relation for bending and twisting: it brings in both rich behaviours and numerical difficulties. By discarding non-physical solutions to this constitutive relation, we show that it can be inverted; this simplifies the system of differential equations governing the equilibrium of ribbons. Based on the inverted form, we propose a natural regularization of the constitutive law that eases the treatment of singularities often encountered in ribbons. We illustrate the approach with the classical problem of the equilibrium of a Möbius ribbon, and compare our findings with the predictions of the Wunderlich model. Overall, our approach provides a simple method for simulating the statics and the dynamics of elastic ribbons.

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