scholarly journals Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Ming Li ◽  
Wei Zhao
Keyword(s):  
Integral Equation ◽  
Fractional Order ◽  
Operational Calculus ◽  
Point Of View ◽  
Alternative Method ◽  
Type Integral

This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.

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.

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 isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Bo Xu ◽  
Yufeng Zhang ◽  
Sheng Zhang
Keyword(s):  
Spectral Problem ◽  
Fractional Order ◽  
Point Of View ◽  
Adjoint Equations ◽  
Scattering Data ◽  
Bilinear Method ◽  
Soliton Equation ◽  
Scattering Transform ◽  
First Time ◽  
Hirota Bilinear

AbstractAblowitz–Kaup–Newell–Segur (AKNS) linear spectral problem gives birth to many important nonlinear mathematical physics equations including nonlocal ones. This paper derives two fractional order AKNS hierarchies which have not been reported in the literature by equipping the AKNS spectral problem and its adjoint equations with local fractional order partial derivative for the first time. One is the space-time fractional order isospectral AKNS (stfisAKNS) hierarchy, three reductions of which generate the fractional order local and nonlocal nonlinear Schrödinger (flnNLS) and modified Kortweg–de Vries (fmKdV) hierarchies as well as reverse-t NLS (frtNLS) hierarchy, and the other is the time-fractional order non-isospectral AKNS (tfnisAKNS) hierarchy. By transforming the stfisAKNS hierarchy into two fractional bilinear forms and reconstructing the potentials from fractional scattering data corresponding to the tfnisAKNS hierarchy, three pairs of uniform formulas of novel N-fractal solutions with Mittag-Leffler functions are obtained through the Hirota bilinear method (HBM) and the inverse scattering transform (IST). Restricted to the Cantor set, some obtained continuous everywhere but nondifferentiable one- and two-fractal solutions are shown by figures directly. More meaningfully, the problems worth exploring of constructing N-fractal solutions of soliton equation hierarchies by HBM and IST are solved, taking stfisAKNS and tfnisAKNS hierarchies as examples, from the point of view of local fractional order derivatives. Furthermore, this paper shows that HBM and IST can be used to construct some N-fractal solutions of other soliton equation hierarchies.

Existence, uniqueness and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
JinRong Wang ◽  
Chun Zhu ◽  
Michal Fečkan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Fractional Order ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Existence Results ◽  
Limit Property ◽  
Type Integral

AbstractIn this paper, we apply certain measure of noncompactness and fixed point theorem of Darbo type to derive the existence and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order with three parameters. Moreover, we also present the uniqueness and another existence results of the solutions to the above equations. Finally, two examples are given to illustrate the obtained results.

EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

2021 ◽  
Vol 10 (6) ◽  
pp. 2687-2710
Author(s):  
F. Akutsah ◽  
A. A. Mebawondu ◽  
O. K. Narain
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Existence Of Solution ◽  
Volterra Type ◽  
Complete Metric Spaces ◽  
Type Integral ◽  
New Type

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

scholarly journals On Solutions of Fractional Order Boundary Value Problems with Integral Boundary Conditions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Hussein A. H. Salem ◽  
Mieczysław Cichoń
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Point Of View ◽  
Integral Boundary ◽  
Special Cases

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

Abel-Type Integral Equation

2020 ◽  
pp. 325-334
Author(s):  
A.G. Ramm ◽  
A.I. Katsevich
Keyword(s):  
Integral Equation ◽  
Type Integral

scholarly journals On the Wolff-type Integral System with Negative Exponents

2021 ◽  
Author(s):  
Rong Zhang ◽  
Ling Li
Keyword(s):  
Integral Equation ◽  
Positive Solutions ◽  
Equation System ◽  
Type Condition ◽  
Radial Solutions ◽  
Entire Solutions ◽  
Integral System ◽  
Type Integral ◽  
Bounded Functions ◽  
Type Potential

In this paper, we are concerned with the positive continuous entire solutions of the Wolff-type integral system \begin{equation*} \left\{ \begin{array}{ll} &u(x) =C_{1}(x)W_{\beta,\gamma} (v^{-q})(x), \\[3mm] &v(x) =C_{2}(x)W_{\beta,\gamma} (u^{-p})(x), \end{array} \right. \end{equation*} where $n\geq1$, $\min\{p,q\}>0$, $\gamma>1$, $\beta>0$ and $\beta\gamma\neq n$. In addition, $C_{i}(x) \ (i=1,2)$ are some double bounded functions. If $\beta\gamma\in (0,n)$, the Serrin-type condition is critical for existence of the positive solutions for some double bounded functions $C_{i}(x)$ $(i=1,2)$. Such an integral equation system is related to the study of the $\gamma$-Laplace system and $k$-Hessian system with negative exponents. Estimated by the integral of the Wolff type potential, we obtain the asymptotic rates and the integrability of positive solutions, and studied whether the radial solutions exist.

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