The q difference calculus describes many phenomena in. Purchase fractional differential equations, volume 198 1st edition. For some recent developments on fractional qdifference calculus and boundary value problems of fractional qdifference equations, see 316 and the references therein. Boundary value problems of fractional qdifference schroinger. By using the guokrasnoselskii fixed point theorem and banach contraction mapping principle as well as schaefers fixed point theorem, we obtain the main results. The aim of this paper is to investigate the existence and uniqueness of solutions for nonlinear fractional q difference equations with threepoint boundary. At the end of the manuscript, we have an example that illustrates the key findings. Using the fourier transform, a general approximation for the mixed fractional derivatives is analyzed. In this paper, we discussed the problem of nonlocal value for nonlinear fractional q difference equation.
This ninechapter monograph introduces a rigorous investigation of qdifference operators in standard and fractional settings. Boundary value problems for fractional q difference. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Numerical methods for fractional differential equations. A finite difference method which is secondorder accurate in time and in space is proposed for twodimensional fractional percolation equations. On nonlocal fractional q integral boundary value problems of fractional q difference and fractional q integrodifference equations involving different numbers of order and q.
Pdf existence results for fractional qdifference equations with. Jul 11, 2012 this chapter is devoted to certain problems of fractional qdifference equations based on the basic riemannliouville fractional derivative and the basic caputo fractional derivative. Fractional derivatives, fractional integrals, and fractional. In this paper, we study the boundary value problem of a fractional qdifference equation with nonlocal conditions involving the fractional qderivative of the caputo type, and the nonlinear term contains a fractional qderivative of caputo type. Liouville fractional qdifference equations and their. The iterative positive solution for a system of fractional q. Fractional differential equations fdes involve fractional derivatives of the form d. The present paper deals with the existence and uniqueness of solutions of fractional difference equations. It is a generalization of the ordinar y differentiation and integration to noninteger arbitrary order. Recently numerical methods have been used approximate. Pdf on the fractional difference equations of order 2, q.
Positive solutions for nonlinear caputo type fractional q. In this paper, we study a class of caputo fractional q difference inclusions in banach spaces. The origin of the fractional q difference calculus can be traced back to the work in 5, 6 by alsalam and by agarwal. Pdf on the oscillation of qfractional difference equations.
Existence and uniqueness results for qfractional difference equations with plaplacian operators. In this paper, we consider a class of fractional qdifference schroinger equations precisely the timeindependent. Fractals and fractional calculus in continuum mechanics. Positive solutions of a nonlinear qfractional difference. Finally we give an illustrative example in the last section. Applications of fractional calculus semantic scholar. Pdf in this paper, we discuss the existence of positive solutions for nonlocal q integral boundary value problems of fractional qdifference. Existence and uniqueness of solutions for mixed fractional q. This ninechapter monograph introduces a rigorous investigation of q difference operators in standard and fractional settings. Anyone who has made a study of di erential equations will know that even supposedly elementary examples can be hard to solve. Du, a superfastpreconditioned iterative method for steadystate spacefractional diffusion equations, j. A sequence of real numbers, indexed by either z or n 0, is written in. Du, a fast finite difference method for threedimensional timedependent space fractional diffusion equations and its efficient implementation, j.
Recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional q difference calculus were made. By means of bananchs contraction mapping principle and schaefers fixedpoint theorem, some existence results for the solutions are obtained. The obtained inequality generalizes several existing results from the literature including the standard lyapunov inequality. Fractional derivatives, fractional integrals, and fractional differential equations in matlab ivo petra technical university of ko ice slovak republic 1. In this note, we study a type of nonlinear riemann. These results extend the corresponding ones of ordinary differential equations of. Pdf the quantum calculus deals with quantum derivatives and integrals, and has proven to be relevant for quantum mechanics. Di erence equations relate to di erential equations as discrete mathematics relates to continuous mathematics. Fractional order difference equations in the present section, we establish theorems on existence and uniqueness of solutions for various classes of fractional order di. A cauchy problem for some fractional qdifference equations. The q difference calculus is an interesting and old subject. Positive solutions of nonlinear boundary value problems for. The classical tools of fixed point theorems such as krasnoselskiis theorem and banachs contraction principle are used. Motivated by recent interest in the study of fractionalorder differential equations, the topic of qfractional equations has attracted the attention of many researchers.
Fractional differential equations consist of a fractional differential with specified value of the unknown function at more than one given point in the domain of the solution. New results on the existence and uniqueness of solutions for qfractional boundary value. Then we give an example for the illustration of the results obtained. Pdf weak solutions for nonlinear fractional differential. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with.
It starts with elementary calculus of qdifferences and integration of jacksons type before turning to qdifference equations. Introduction the term fractional calculus is more than 300 years old. The details of some recent work on the topic can be found in 1420. Existence results for nonlinear fractional qdifference equations. We discuss the existence of weak solutions for a nonlinear boundary value problem of fractional differential equations in banach space. In this paper, we consider the following twopoint boundary value problem for qfractional plaplace difference equations. The origin of the fractional qdifference calculus can be traced back to the works in 4 by alsalam and.
Pdf boundary value problems of fractional qdifference. The differential equations involving riemannliouville differential operators of fractional order 0 q difference equations with fourpoint boundary problems. In this work, we investigate the following system of fractional q difference equations with fourpoint boundary problems. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Advances in difference equations boundary value problems for fractional q difference equations with nonlocal conditions xinhui li zhenlai han shurong sun hongling lu in this paper, we study the boundary value problem of a fractional q difference equation with nonlocal conditions involving the fractional q derivative of the caputo type, and the nonlinear term contains a fractional q derivative. A lyapunovtype inequality for a fractional qdifference.
On the fractional difference equations of order 2, q article pdf available in abstract and applied analysis 2011 january 2011 with 39 reads how we measure reads. Basic theory of fractional differential equations sciencedirect. We now point out two formulas that will be used later idq denotes the. Positive solutions for boundary value problem of nonlinear. This chapter is devoted to the use of the qlaplace, qmellin, and q 2fourier transforms to find explicit solutions of certain linear qdifference equations, linear fractional qdifference. They are generalizations of the ordinary differential equations to a random noninteger order. Oscillation theorems for nonlinear fractional difference.
The q difference calculus or quantum calculus was first developed by jackson 1, 2, while basic definitions and properties can be found in the papers 3, 4. The differential equations involving riemannliouville differential operators of fractional order 0 difference equation is a relation between the differences of a function at one or more general values of the independent variable. For notions and basic concepts of qfractional calculus, we refer to a recent text. Pdf on the fractional difference equations of order 2. Mar 18, 2020 lyapunov inequalities of left focal q difference boundary value problems and applications. They have attracted considerable interest due to their ability to model complex phenomena. After giving the basic properties we define the q derivative and q integral. In this paper, we establish a lyapunovtype inequality for a fractional q difference equation subject to dirichlettype boundary conditions. These equations usually describe the evolution of certain phenomena over the course of time.
Diethelm, an algorithm for the numerical solution of differential equations of fractional order, electronic transactions on numerical analysis 5 1997 16. The unique solution for a fractional qdifference equation with three. Pdf in this paper, sufficient conditions are established for the oscillation of solutions of q fractional difference equations of the form q. We initiate the study of fractional q difference inclusions on. Derivation of the fractional flow equation for a onedimensional oilwater system consider displacement of oil by water in a system of dip angle. Recent progress in differential and difference equations. In this study, we discuss some theorems related to the oscillatory behavior of nonlinear fractional difference equations equipped with wellknown fractional riemannliouville difference operator. Boundary value problems for fractional qdifference equations. In this paper, we establish some lyapunovtype inequalities for a class of linear and nonlinear fractional q difference boundary value problems under cauchy boundary conditions. A difference equation is a relation between the differences of a function at one or more general values of the independent variable. This chapter is devoted to certain problems of fractional qdifference equations based on the basic riemannliouville fractional derivative and the. In this paper, we consider the existence of positive solutions to nonlinear q di. Main formula for qfractional difference equation is investigated and in the aid of this definition we prove the uniqueness and existence of solution.
Advances in fractional differential equations and their real. Simpson, the numerical solution of linear multiterm fractional differential equations. Our analysis relies on the monchs fixed point theorem combined with the technique of measures of weak noncompactness. Boundary value problems of fractional q difference equations on the. Riemannliouville fractional difference equation 266 8.
Pdf existence results for fractional qdifference equations of order. Fractional qdifference equations, integral boundary conditions, fixed point theorem, existence and uniqueness. It starts with elementary calculus of q differences and integration of jacksons type before turning to q difference equations. This chapter is devoted to the use of the q laplace, q mellin, and q 2fourier transforms to find explicit solutions of certain linear q difference equations, linear fractional q difference. By means of generalized riccati transformation techniques, we establish some new oscillation criteria for fractional order nonlinear difference equations with damping. We use that result to provide an interval, where a certain mittagleffler function has no real zeros. Fractional differential equations, volume 198 1st edition.
A cauchy problem for some fractional q difference equations with nonlocal conditions. As in the classical theory of ordinary fractional differential equations, q difference equations of fractional order are divided into linear, nonlinear, homogeneous, and inhomogeneous equations with constant and variable coefficients. In this paper, we investigate the existence and uniqueness of solutions for mixed fractional qdifference boundary value problems involving the riemannliouville and the caputo fractional derivative. Positive solution for a class of plaplacian fractional qdifference.
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