SOLUTIONS OF THE BENJAMIN-ONO EQUATION IN FRACTIONAL ORDER SOBOLEV SPACES

We show for the Benjamin-Ono equation an existence uniqeness theorem in Sobolev spaces of arbitrary fractional order s greater-than-or-equal-to 2, provided the initial data is given in the same space.

Exact solution for the rotational flow of a generalized second grade fluid in a circular cylinder

This paper deals with the rotational flow of a generalized second grade fluid, within a circular cylinder, due to a torsional shear stress. The fractional calculus approach in the constitutive relationship model of a second grade fluid is introduced. The velocity field and the resulting shear stress are determined by means of the Laplace and finite Hankel transforms to satisfy all imposed initial and boundary conditions. The solutions corresponding to second grade fluids as well as those for Newtonian fluids are obtained as limiting cases of our general solutions. The influence of the fractional coefficient on the velocity of the fluid is also analyzed by graphical illustrations.

On Skew Triangular Matrix Rings

In this paper, we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations. The system is used to represent constitutive relation for viscoelastic model of fractional differential equations. Our results are based on the fixed point theorems of increasing operator and the cone theory, some illustrative examples are also presented.

Dynamic determination reserves of the underground gas storage

One of the key problems in the use of underground gas storage is frequent leakage. It can lead to the actual gas storage amount being less than that accounted for. Combining numerical simulation and parameter auto fit, this paper ascertains the dynamic variation of the pressure in the storage reservoir, adjusts the actual injecting and producing gas to fit the accounted pressure with the tested pressure, obtains the gas leakage of the storage, and then determines the difference between accounted amount and leakage amount. The result is the actual reserves of the storage. The simulation result shows that the method presented can provide a theoretic foundation for estimating the leakage amount, thereby ensuring the actual reserves, searching the leakage route, and reducing leakage by adjusting the storage method.

Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition

The unified displacement function(UDF)is presented to describe the deformation behaviours of the tunnel profile along with time under the surface slope condition.Based on the discrete Fourier method,the third-order UDF in the physical plane is expanded to the Laurent series in the complex variable plane.The complex variable method is employed to derive the elastic analytical solution of stra-tum displacement,when the third-order UDF is taken as the displacement boundary condition of tunnel cross-section(DBCTC).The proposed elastic solution agrees well with the results of the finite element method for the consistent model,which verifies the correctness of the proposed analytical solution.Combining the corresponding principle and fractional Generalized Kelvin viscoelastic constitutive model,the fractional viscoelastic solution under the surface slope condition is determined.The time effect of stratum displacement is presented in two aspects:time-dependent DBCTC and time-dependent material parameters.The parameter analysis is performed to investigate influences of deformation modes of the third-order UDF,slope angle,tunnel radius and fractional order on the time effect of stratum vertical and horizontal displacement.

linear and nonlinear fractional differential equation modified Riemann–Liouville derivatives exact solutions fractional auxiliary sub-equation expansion method Mittag–Leffler function method

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the（3＋1）-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.

A collocation method for numerical solutions of fractional-order logistic population model

In this study, a collocation technique is presented for approximate solution of the fractional-order logistic population model. Actually, we develop the Bessel collocation method by using the fractional derivative in the Caputo sense to obtain the approximate solutions of this model problem. By means of the fractional derivative in the Caputo sense, the collocation points, the Bessel functions of the first kind, the method transforms the model problem into a system of nonlinear algebraic equations. Numerical applications are given to demonstrate efficiency and accuracy of the method. In applications, the reliability of the scheme is shown by the error function based on the accuracy of the approximate solution.

Solutions in the Discrete Fractional Forms of Two Differential Equations with Singular Points

In present note,we apply the Leibniz formula with the nabla operator in discrete fractional calculus(DFC)due to obtain the discrete fractional solutions of a class of associated Bessel equations(ABEs)and a class of associated Legendre equations(ALEs),respectively.Thus,we exhibit a new solution method for such second order linear ordinary differential equations with singular points.

Dark Soliton Solutions of Space-Time Fractional Sharma–Tasso–Olver and Potential Kadomtsev–Petviashvili Equations

Dark soliton solutions for space-time fractional Sharma–Tasso–Olver and space-time fractional potential Kadomtsev–Petviashvili equations are determined by using the properties of modified Riemann–Liouville derivative and fractional complex transform. After reducing both equations to nonlinear ODEs with constant coefficients, the tanh ansatz is substituted into the resultant nonlinear ODEs. The coefficients of the solutions in the ansatz are calculated by algebraic computer computations. Two different solutions are obtained for the Sharma–Tasso–Olver equation as only one solution for the potential Kadomtsev–Petviashvili equation. The solution profiles are demonstrated in 3D plots in finite domains of time and space.

Solving Space-Time Fractional Differential Equations by Using Modified Simple Equation Method

In this article,we establish new and more general traveling wave solutions of space-time fractional Klein–Gordon equation with quadratic nonlinearity and the space-time fractional breaking soliton equations using the modified simple equation method.The proposed method is so powerful and effective to solve nonlinear space-time fractional differential equations by with modified Riemann–Liouville derivative.

Assessment of Potential Nutrient Release from Phosphate Rock and Dolostone for Application in Acid Soils

Finding alternative local sources of plant nutrients is a practical, low-cost, and long-term strategy. In this study, laboratory column experiments were conducted in a completely randomized design to evaluate the feasibility of using phosphate rock and dolostone as fertilizers or acid-neutralizing agents for application in tropical acid soils. The dissolution rates of different particle-size fractions(0.063–0.25, 0.25–0.5, and 0.5–2 mm) of both rocks were studied by citric acid solution at p H 4 and 2 and water, with extraction times of 1, 3, 5, 7, 12, 24, 72, 144, 240, and 360 h. The results showed that the dissolution of both rocks depended on the particle size,leaching solution, and extraction time. The dissolution rate of rock-forming minerals increased as the specific surface area increased,corresponding to a decrease in particle size. In all cases, the release kinetics was characterized by two phases: 1) a first stage of rapid release that lasted 24 h and would ensure short-term nutrient release, and 2) a second stage of slow release after 24 h, representing the long-term nutrient release efficiency. Both rocks were suitable as slow-release fertilizers in strongly acid soils and would ensure the replenishment of P, Ca, and Mg. A combination of fine and medium particle-size fractions should be used to ensure high nutrient-release efficiency. Much work could remain to determine the overall impact of considerable amounts of fresh rocks in soils.

Multiobjective Fractional Variational Problem on Higher-Order Jet Bundles

The main goal of this paper is to introduce necessary efficiency conditionsfor a class of multi-time vector fractional variational problems with nonlinear equal-ity and inequality constraints involving higher-order partial derivatives.We considerthe multi-time multiobjective variational problem(MFP)of minimizing a vector ofpath-independent curvilinear integral functionals quotients subject to PDE and/or PDIconstraints,developing an optimization theory on the higher-order jet bundles.

EFFECT OF FRACTIONAL ORDER PARAMETER ON THERMOELASTIC BEHAVIORS IN INFINITE ELASTIC MEDIUM WITH A CYLINDRICAL CAVITY

The thermal shock problems involved with fractional order generalized theory is studied by an analytical method. The asymptotic solutions for thermal responses induced by transient thermal shock are derived by means of the limit theorem of Laplace transform. An infinite solid with a cylindrical cavity subjected to a thermal shock at its inner boundary is studied. The propagation of thermal wave and thermal elastic wave, as well as the distributions of displacement, temperature and stresses are obtained from these asymptotic solutions. The investigation on the effect of fractional order parameter on the propagation of two waves is also conducted.