scholarly journals On Gronwall’s type integral inequalities with singular kernels

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1041-1049 ◽  
Author(s):  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Fractional Differential Equations ◽  
Parabolic Type ◽  
Integral Inequalities ◽  
Stability Estimates ◽  
Type Integral ◽  
Singular Kernels ◽  
Fractional Differential ◽  
Integral Differential Equation

In this paper, the generalizations of Gronwall?s type integral inequalities with singular kernels are established. In applications, theorems on stability estimates for the solutions of the nonliner integral equation and the integral-differential equation of the parabolic type are presented. Moreover, these inequalities can be used in the theory of fractional differential equations.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than One

2005 ◽  
Vol 1 (2) ◽  
pp. 178-185 ◽  
Author(s):  
Pankaj Kumar ◽  
Om P. Agrawal
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Volterra Type ◽  
Entire Domain ◽  
Type Integral ◽  
Fractional Differential

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1<α<2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra-type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra-type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve two problems, linear and nonlinear, using two different types of polynomials, cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results for problem 1 and the numerical results obtained using another scheme for problem 2. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

scholarly journals A Note on the Stability of the Integral-Differential Equation of the Parabolic Type in a Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Unique Solvability ◽  
Parabolic Type ◽  
Stability Estimates ◽  
The Difference ◽  
The Stability ◽  
Integral Differential Equation

The integral-differential equation of the parabolic type in a Banach space is considered. The unique solvability of this equation is established. The stability estimates for the solution of this equation are obtained. The difference scheme approximately solving this equation is presented. The stability estimates for the solution of this difference scheme are obtained.

Numerical Scheme for the Solution of Fractional Differential Equations of Order Greater Than 1

2005 ◽  
Author(s):  
Pankaj Kumar ◽  
Om P. Agrawal
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Numerical Scheme ◽  
Numerical Solutions ◽  
Volterra Type ◽  
Entire Domain ◽  
Type Integral ◽  
Fractional Differential

This paper presents a numerical scheme for the solutions of Fractional Differential Equations (FDEs) of order α, 1 &lt; α &lt; 2 which have been expressed in terms of Caputo Fractional Derivative (FD). In this scheme, the properties of the Caputo derivative are used to reduce an FDE into a Volterra type integral equation. The entire domain is divided into several small domains, and the distribution of the unknown function over the domain is expressed in terms of the function values and its slopes at the node points. These approximations are then substituted into the Volterra type integral equation to reduce it to algebraic equations. Since the method enforces the continuity of variables at the node points, it provides a solution that is continuous and with a slope that is also continuous over the entire domain. The method is used to solve a simple FDE using two different types of polynomials, namely cubic order and fractional order. Results obtained using both types of polynomials agree well with the analytical results. However, the fractional order polynomials give more accurate results than the cubic order polynomials do. This suggests that for the numerical solutions of FDEs fractional order polynomials may be more suitable than the integer order polynomials. A series of numerical studies suggests that the algorithm is stable.

scholarly journals Fractional differential equations and Volterra–Stieltjes integral equations of the second kind

2019 ◽  
Vol 38 (4) ◽  
Author(s):  
Avyt Asanov ◽  
Ricardo Almeida ◽  
Agnieszka B. Malinowska
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Equivalence Relation ◽  
Numerical Experiments ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Stieltjes Integral ◽  
Fractional Differential

Abstract In this paper, we construct a method to find approximate solutions to fractional differential equations involving fractional derivatives with respect to another function. The method is based on an equivalence relation between the fractional differential equation and the Volterra–Stieltjes integral equation of the second kind. The generalized midpoint rule is applied to solve numerically the integral equation and an estimation for the error is given. Results of numerical experiments demonstrate that satisfactory and reliable results could be obtained by the proposed method.

scholarly journals Henry-Gronwall Integral Inequalities with “Maxima” and Their Applications to Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Phollakrit Thiramanus ◽  
Jessada Tariboon ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Integral Inequalities ◽  
Weakly Singular ◽  
Type Integral ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

Some new weakly singular Henry-Gronwall type integral inequalities with “maxima” are established in this paper. Applications to Caputo fractional differential equations with “maxima” are also presented.

scholarly journals Some generalized Gronwall-Bellman-Bihari type integral inequalities with application to fractional stochastic differential equation

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 815-824
Author(s):  
Sabir Hussain ◽  
Halima Sadia ◽  
Sidra Aslam
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Integral Inequalities ◽  
Type Integral ◽  
Singular Kernels

In this paper some new Gronwall-Bellman-Bihari type integral inequalities with singular as well as non-singular kernels have been discussed, generalizing some already existing results. As an application of the derived results, the behaviour of solution of the fractional stochastic differential equation has been discussed.

scholarly journals Fractional spectral integral methods for valuing cryptocurrency asset flow fractional differential equations.

2021 ◽  
Author(s):  
Claude Moutsinga ◽  
Edson Pindza ◽  
Eben Mare
Keyword(s):  
Differential Equation ◽  
Numerical Simulations ◽  
Fractional Differential Equations ◽  
Emerging Market ◽  
Three Dimensional ◽  
Operator Approach ◽  
Integral Methods ◽  
Spectral Integration ◽  
Spectral Integral ◽  
Fractional Differential

Since its inception in 2009, the cryptocurrency market has grown considerably. Several authors have proposed models to explain the price movements of assets in this new emerging market. However, only few researches have been done using the dynamical approach. This paper proposes a robust time fractional spectral method for studying a three dimensional fractional differential equation that models cryptocurrency asset flow obtained by utilizing the concept of liquidity price. The method relies on fractional spectral integration matrix operator approach. Numerical simulations are conducted to show efficiency of the numerical method on the fractional cryptocurrency model compared to existing methods.

scholarly journals Fuzzy Conformable Fractional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Atimad Harir ◽  
Said Melliani ◽  
Lalla Saadia Chadli
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Uniqueness Theorem ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Existence And Uniqueness Theorem ◽  
Fractional Differential ◽  
Fractional Differentiability ◽  
Derivatives Of ◽  
Fuzzy Fractional Differential Equation

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order q ∈ 0,1 .

scholarly journals Solution of Fractional Differential Equations Utilizing Symmetric Contraction

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Aftab Hussain
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fractional Differential Equations ◽  
Metric Space ◽  
Metric Spaces ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
Continuous Mappings ◽  
Fractional Differential ◽  
Fixed Point Technique

The aim of this paper is to present another family of fractional symmetric α - η -contractions and build up some new results for such contraction in the context of ℱ -metric space. The author derives some results for Suzuki-type contractions and orbitally T -complete and orbitally continuous mappings in ℱ -metric spaces. The inspiration of this paper is to observe the solution of fractional-order differential equation with one of the boundary conditions using fixed-point technique in ℱ -metric space.

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