A Cauchy Problem for Some Fractional q-Difference Equations with Nonlocal Conditions

In this paper, we discussed the problem of nonlocal value for nonlinear fractional q-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.

Memory effect in time fractional Schrödinger equation

A significant obstacle impeding the advancement of the time fractional Schrodinger equation lies in the challenge of determining its precise mathematical formulation.In order to address this,we undertake an exploration of the time fractional Schrodinger equation within the context of a non-Markovian environment.By leveraging a two-level atom as an illustrative case,we find that the choice to raise i to the order of the time derivative is inappropriate.In contrast to the conventional approach used to depict the dynamic evolution of quantum states in a non-Markovian environment,the time fractional Schrodinger equation,when devoid of fractional-order operations on the imaginary unit i,emerges as a more intuitively comprehensible framework in physics and offers greater simplicity in computational aspects.Meanwhile,we also prove that it is meaningless to study the memory of time fractional Schrodinger equation with time derivative 1<α≤2.It should be noted that we have not yet constructed an open system that can be fully described by the time fractional Schrodinger equation.This will be the focus of future research.Our study might provide a new perspective on the role of time fractional Schrodinger equation.

A Novel Accurate Method forMulti-Term Time-Fractional Nonlinear Diffusion Equations in Arbitrary Domains

Anovel accuratemethod is proposed to solve a broad variety of linear and nonlinear(1+1)-dimensional and(2+1)-dimensional multi-term time-fractional partial differential equations with spatial operators of anisotropic diffusivity.For(1+1)-dimensional problems,analytical solutions that satisfy the boundary requirements are derived.Such solutions are numerically calculated using the trigonometric basis approximation for(2+1)-dimensional problems.With the aid of these analytical or numerical approximations,the original problems can be converted into the fractional ordinary differential equations,and solutions to the fractional ordinary differential equations are approximated by modified radial basis functions with time-dependent coefficients.An efficient backward substitution strategy that was previously provided for a single fractional ordinary differential equation is then used to solve the corresponding systems.The straightforward quasilinearization technique is applied to handle nonlinear issues.Numerical experiments demonstrate the suggested algorithm’s superior accuracy and efficiency.

THE SPARSE REPRESENTATION RELATED WITH FRACTIONAL HEAT EQUATIONS

This study introduces a pre-orthogonal adaptive Fourier decomposition(POAFD)to obtain approximations and numerical solutions to the fractional Laplacian initial value problem and the extension problem of Caffarelli and Silvestre(generalized Poisson equation).As a first step,the method expands the initial data function into a sparse series of the fundamental solutions with fast convergence,and,as a second step,makes use of the semigroup or the reproducing kernel property of each of the expanding entries.Experiments show the effectiveness and efficiency of the proposed series solutions.

Crank-Nicolson Quasi-Compact Scheme for the Nonlinear Two-Sided Spatial Fractional Advection-Diffusion Equations

The higher-order numerical scheme of nonlinear advection-diffusion equations is studied in this article, where the space fractional derivatives are evaluated by using weighted and shifted Grünwald difference operators and combining the compact technique, in the time direction is discretized by the Crank-Nicolson method. Through the energy method, the stability and convergence of the numerical scheme in the sense of L2-norm are proved, and the convergence order is . Some examples are given to show that our numerical scheme is effective.

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES

Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).

The Fractional Investigation of Some Nonlinear Partial Differential Equations by Using an Efficient Procedure

The nonlinearity inmany problems occurs because of the complexity of the given physical phenomena.The present paper investigates the non-linear fractional partial differential equations’solutions using the Caputo operator with Laplace residual power seriesmethod.It is found that the present technique has a direct and simple implementation to solve the targeted problems.The comparison of the obtained solutions has been done with actual solutions to the problems.The fractional-order solutions are presented and considered to be the focal point of this research article.The results of the proposed technique are highly accurate and provide useful information about the actual dynamics of each problem.Because of the simple implementation,the present technique can be extended to solve other important fractional order problems.

Analysis of anomalous transport with temporal fractional transport equations in a bounded domain

Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations.The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α(α ∈(0, 1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result,if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.

On Time Fractional Partial Differential Equations and Their Solution by Certain Formable Transform Decomposition Method

This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach.

Conformable fractional heat equation with fractional translation symmetry in both time and space

We investigate the fractional heat equation with fractional translation in both time and position with different fractional orders.As examples,we consider a rod and anα-disk with an initial constant temperature and discuss their cooling processes in the examined formalism.

Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One-and Two-Dimensions

In this paper,finite difference schemes for solving time-space fractional diffusion equations in one dimension and two dimensions are proposed.The temporal derivative is in the Caputo-Hadamard sense for both cases.The spatial derivative for the one-dimensional equation is of Riesz definition and the two-dimensional spatial derivative is given by the fractional Laplacian.The schemes are proved to be unconditionally stable and convergent.The numerical results are in line with the theoretical analysis.

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper,a local discontinuous Galerkin(LDG)scheme for the time-fractional diffusion equation is proposed and analyzed.The Caputo time-fractional derivative(of orderα,with 0<α<1)is approximated by a finite difference method with an accuracy of order3-α,and the space discretization is based on the LDG method.For the finite difference method,we summarize and supplement some previous work by others,and apply it to the analysis of the convergence and stability of the proposed scheme.The optimal error estimate is obtained in the L2norm,indicating that the scheme has temporal(3-α)th-order accuracy and spatial(k+1)th-order accuracy,where k denotes the highest degree of a piecewise polynomial in discontinuous finite element space.The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces.Furthermore,we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated withSλand consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations.We also establish certain interesting examples to illustrate the usability of our results.

The Fractional Investigation of Fornberg-Whitham Equation Using an Efficient Technique

In the last few decades,it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences.For this reason,fractional partial differential equations(FPDEs)are of more importance to model the different physical processes in nature more accurately.Therefore,the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution.In the current work,the idea of fractional calculus has been used,and fractional FornbergWhithamequation(FFWE)is represented in its fractional view analysis.Awell-knownmethod which is residual power series method(RPSM),is then implemented to solve FFWE.TheRPSMresults are discussed through graphs and tables which conform to the higher accuracy of the proposed technique.The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution.Because the RPSM procedure is simple and straightforward,it can be extended to solve other FPDEs and their systems.

POSITIVE SOLUTIONS WITH HIGH ENERGY FOR FRACTIONAL SCHRODINGER EQUATIONS

In this paper, we study the Schrodinger equations (-△)^(s)u + V(x)u = a(x)|u|^(p-2)u + b(x)|u|^(q-2)u, x∈R^(N),where 0 < s < 1, 2 < q < p < 2_(s)^(*), 2_(s)^(*) is the fractional Sobolev critical exponent. Under suitable assumptions on V, a and b for which there may be no ground state solution, the existence of positive solutions are obtained via variational methods.

Efficient Finite Difference/Spectral Method for the Time Fractional Ito Equation Using Fast Fourier Transform Technic

A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.

The Natural Transform Decomposition Method for Solving Fractional Klein-Gordon Equation

In this paper, a coupling of the natural transform method and the Admoian decomposition method called the natural transform decomposition method (NTDM), is utilized to solve the linear and nonlinear time-fractional Klein-Gordan equation. The (NTDM), is introduced to derive the approximate solutions in series form for this equation. Solutions have been drawn for several values of the time power. To identify the strength of the method, three examples are presented.

Variational Iteration Method for Solving Time Fractional Burgers Equation Using Maple

The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.

Adomian Decomposition Method for Solving Time Fractional Burgers Equation Using Maple

In this paper, the Adomian decomposition method was used to solve the Time Fractional Burger equation using Mabel program. This method was applied to a number of examples of the Time Fractional Burger Equation. The obtained numerical results were presented in the form of tables and graphics. The difference between the exact solutions and the numerical solutions shows us the effectiveness of the solution using the Mabel program and that this method gave accurate results and was close to the exact solution, in addition to its ability to obtain the numerical solution quickly and efficiently using the Mabel program.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.