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