scholarly journals Semi-Hyers–Ulam–Rassias Stability of a Volterra Integro-Differential Equation of Order I with a Convolution Type Kernel via Laplace Transform

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2181
Author(s):  
Daniela Inoan ◽  
Daniela Marian
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Convolution Type ◽  
Convolution Product ◽  
Polynomial Functions ◽  
Integro Differential Equation ◽  
The Laplace Transform ◽  
The Stability ◽  
Rassias Stability

In this paper, we investigate the semi-Hyers–Ulam–Rassias stability of a Volterra integro-differential equation of order I with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions. An important aspect that appears in the form of the studied equation is the symmetry of the convolution product.

scholarly journals A Meshless Method Based on the Laplace Transform for the 2D Multi-Term Time Fractional Partial Integro-Differential Equation

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1972
Author(s):  
Kamran Kamran ◽  
Zahir Shah ◽  
Poom Kumam ◽  
Nasser Aedh Alreshidi
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Meshless Method ◽  
Original Problem ◽  
Contour Integral ◽  
Weakly Singular Kernel ◽  
Integro Differential Equation ◽  
Time Stepping ◽  
Weakly Singular ◽  
The Laplace Transform

In this article, we propose a localized transform based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo’s sense with a weakly singular kernel. The purpose of coupling the localized meshless method with the Laplace transform is to avoid the time stepping procedure by eliminating the time variable. Then, we utilize the local meshless method for spatial discretization. The solution of the original problem is obtained as a contour integral in the complex plane. In the literature, numerous contours are available; in our work, we will use the recently introduced improved Talbot contour. We approximate the contour integral using the midpoint rule. The bounds of stability for the differentiation matrix of the scheme are derived, and the convergence is discussed. The accuracy, efficiency, and stability of the scheme are validated by numerical experiments.

scholarly journals Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Elhassan Eljaoui ◽  
Said Melliani ◽  
L. Saadia Chadli
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Convolution Type ◽  
Convolution Product ◽  
Fuzzy Mapping ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Improper Integral ◽  
Convolution Formula

We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.

The integral of geometric Brownian motion

2001 ◽  
Vol 33 (1) ◽  
pp. 223-241 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Finite Time ◽  
Geometric Brownian Motion ◽  
Time Interval ◽  
Finite Time Interval ◽  
Special Cases ◽  
The Laplace Transform

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

Stability and Controllability of Euler-Bernoulli Beams With Intelligent Constrained Layer Treatments

1996 ◽  
Vol 118 (1) ◽  
pp. 70-77 ◽  
Author(s):  
I. Y. Shen
Keyword(s):  
Laplace Transform ◽  
Closed Form ◽  
Characteristic Equation ◽  
Matrix Equation ◽  
Homogeneous Equation ◽  
Transform Domain ◽  
Threshold Control ◽  
Control Gain ◽  
The Laplace Transform ◽  
The Stability

This paper studies the stability and controllability of Euler-Bernoulli beams whose bending vibration is controlled through intelligent constrained layer (ICL) damping treatments proposed by Baz (1993) and Shen (1993, 1994). First of all, the homogeneous equation of motion is transformed into a first order matrix equation in the Laplace transform domain. According to the transfer function approach by Yang and Tan (1992), existence of nontrivial solutions of the matrix equation leads to a closed-form characteristic equation relating the control gain and closed-loop poles of the system. Evaluating the closed-form characteristic equation along the imaginary axis in the Laplace transform domain predicts a threshold control gain above which the system becomes unstable. In addition, the characteristic equation leads to a controllability criterion for ICL beams. Moreover, the mathematical structure of the characteristic equation facilitates a numerical algorithm to determine root loci of the system. Finally, the stability and controllability of Euler-Bernoulli beams with ICL are illustrated on three cantilever beams with displacement or slope feedback at the free end.

scholarly journals Aplicación del método de resumación de Borel en la ecuación diferencial de Euler

2020 ◽  
Vol 5 (2) ◽  
pp. 59
Author(s):  
Oswaldo José Larreal Barreto
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Relevant Information ◽  
Formal Series ◽  
Initial Problem ◽  
Divergent Series ◽  
Summarization Method ◽  
The Laplace Transform ◽  
Analytical Extension ◽  
Euler Series

  El propósito de este artículo es mostrar que a partir de la series divergentes se puede obtener información relevante que permite resolver algunos problemas, para lograr este cometido, inicialmente se hace una breve introducción a la teoría resurgente de Écalle, se establecen las definiciones básicas como: resumación de Borel, serie clase Gevrey1 e introducimos las herramientas necesarias, entre ellas la transformada de Borel y Laplace, además se hace un esquema de los pasos que se deben seguir para usar el método de resumación de Borel. Se muestra como ejemplo la ecuación diferencial de Euler, de la cual se halla una solución en forma de serie formal divergente. Siguiendo el esquema del método se debe calcular en primer lugar la transformada de Borel y asociar esta con una función que es analítica en un dominio, para así definir el dominio de la transformada de Laplace y obtener por extensión analítica las soluciones al problema inicial. Luego de este procedimiento las soluciones al problema inicial no deben estar dado por una serie divergente y en su lugar puede ser representado por integrales con caminos distintos, esto último puede permitir establecer relaciones entre las soluciones..   Palabras claves: resumación de Borel, ecuación diferencial de Euler, series divergentes.   Abstract The purpose of this article is to show that from the divergent series it is possible to obtain relevant information that allows solving some problems, to achieve this task, initially a brief introduction to the resurgent theory of Écalle is made, the basic definitions are established such as: Borel summarization, Gevrey1 class series and we introduce the necessary tools, among them the Borel and Laplace transform, we also outline the steps that must be followed to use the Borel summarization method. Euler’s differential equation is shown as an example, of which a solution is found in the form of a divergent formal series. Following the scheme of the method, the Borel transform must first be calculated and associated with a function that is analytic in a domain, in order to define the domain of the Laplace transform and obtain by analytical extension the solutions to the initial problem. After this procedure, the solutions to the initial problem should not be given by a divergent series and instead can be represented by integrals with different paths, the latter can allow establishing relationships between the solutions.   Keywords: Borel’s summary, Euler differential equation, series divergent.  

scholarly journals A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

2021 ◽  
Vol 5 (3) ◽  
pp. 85
Author(s):  
Tayyaba Akram ◽  
Zeeshan Ali ◽  
Faranak Rabiei ◽  
Kamal Shah ◽  
Poom Kumam
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Numerical Study ◽  
Stability And Convergence ◽  
Weakly Singular Kernel ◽  
Integro Differential Equation ◽  
Suggested Technique ◽  
Collocation Approach ◽  
The Stability ◽  
Efficiency And Reliability

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

scholarly journals A Convolution Type Nonlinear Integro-Differential Equation with a Variable Coefficient and an Inhomogeneity in the Linear Part

2020 ◽  
pp. 16-27
Author(s):  
С.Н. Асхабов
Keyword(s):  
Differential Equation ◽  
Variable Coefficient ◽  
Linear Part ◽  
Convolution Type ◽  
Integro Differential Equation

Изучается вольтерровское интегро-дифференциальное уравнение типа свертки со степенной нелинейностью, переменным коэффициентом $a(x)$ и неоднородностью $f(x)$ в линейной части, которое тесно связано с соответствующим нелинейным интегральным уравнением, возникающим при исследовании инфильтрации жидкости из цилиндрического резервуара в изотропную однородную пористую среду, при описании процесса распространения ударных волн в трубах, наполненных газом, при решении задачи о нагревании полубесконечного тела при нелинейном теплопередаточном процессе, в моделях популяционной генетики и других. Важно отметить, что в связи с указанными и другими приложениями особый интерес представляют непрерывные положительные при $x>0$ решения интегрального уравнения. На основе полученных точных нижней и верхней априорных оценок решения интегрального уравнения мы строим весовое полное метрическое пространство $P_b$, инвариантное относительно нелинейного интегрального оператора свертки, порожденного этим уравнением, и, применяя метод весовых метрик (аналог метода Белицкого), доказываем глобальную теорему о существовании и единственности решения изучаемого нелинейного интегро-дифференциального уравнения как в пространстве $P_b$, так и во всем классе $Q_0^1$ непрерывно дифференцируемых положительных при $x>0$ функций. Показано, что решение может быть найдено в пространстве $P_b$ методом последовательных приближений пикаровского типа. Для последовательных приближений получены оценки скорости их сходимости к точному решению в терминах весовой метрики пространства~$P_b$. В частности, при $f(x)=0$ из этой теоремы вытекает, что соответствующее однородное нелинейное интегро-дифференциальное уравнение, в отличие от линейного случая, имеет нетривиальное решение. Приведены также примеры, иллюстрирующие полученные результаты.

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