Fractional integration of the H-function of several variables

While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX>. Each of these three triple hypergeometric functions <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX> has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function <TEX>$H_B$</TEX>.

On Boundary Values of Hypergeometric Functions of Several Variables

On the Euler integral representations of hypergeometric functions in several variables

Many studies have proved that the fractional-order system is more accurate than the integer-order system. The theory of fractional calculus has received extensive attention in many fields. The research on flux-controlled memristor has been carried out for several years; however, fractional-order flux-controlled memristor has not received widespread attention. In light of the theory of fractional calculus, a generalized fractional-order dynamic model of flux-controlled memristor is proposed for the first time in this paper. The mathematical model is established and its characteristics of the pinched hysteresis loops are analyzed by changing the order and the frequency. The specific computational formulas of areas of pinched hysteresis loops are given. The equivalent circuit of the fractional-order flux-controlled memristor is constructed, and circuit simulations are given to illustrate the effectiveness of theoretical analysis and numerical calculation. In this study, the flux-controlled memristor model is extended to the fractional-order model that is closer to the real device, and the analysis method of areas of pinched hysteresis loops is further expanded.

In this study, a new non‐linear recursive mechanism for Volterra least mean square (VLMS) algorithm is proposed in the domain of non‐linear adaptive signal processing and control. The proposed adaptive scheme is developed by applying concepts and theories of fractional calculus in weight adaptation structure of standard VLMS approach. The design scheme based on fractional VLMS (F‐VLMS) algorithm is applied to parameter estimation problem of non‐linear Hammerstein Box‐Jenkins system for different noise and step size variations. The adaptive variables of F‐VLMS are compared from actual parameters of the system as well as with the results of conventional VLMS for each case to verify its correctness. Comprehensive statistical analyses are conducted based on sufficient large number of independent runs and performance indices in terms of mean square error, variance account for and Nash–Sutcliffe efficiency establish the worth and effectiveness of the scheme.

Implementation of fractional-order electromagnetic potential through a genetic algorithm

In this paper, we study convergence characteristics of a class of continuous time fractional-order cellular neural network containing delay. Using the method of Lyapunov and Mittag-Leffer functions, we derive sufficient condition for global Mittag-Leffer stability, which further implies global asymptotic stability of the system equilibrium. Based on the theory of fractional calculus and the generalized Gronwall inequality, we approximate the solution of the corresponding neural network model using discretization method by piecewise constant argument and obtain some easily verifiable conditions, which ensures that the discrete-time analogues preserve the convergence dynamics of the continuous-time networks. In the end, we give appropriate examples to validate the proposed results, illustrating advantages of the discrete-time analogue over continuous-time neural network for numerical simulation.

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

Aiming at the existing problems that the image denoising algorithm based on integer-order partial differential equation could lost part of edge and texture information. This image denoising algorithm based on fractional variational theory was proposed by the theory of fractional calculus and partial differential equation. The denoising model proposed in this paper introduces and implements the numeric computation of the fractional variation partial differential equations by constructing the fractional differential mask operators along eight directions of image. The simulation data prove that the image denoising algorithm based on fractional variation theory compared with the traditional image denoising algorithm could better retain the edge and texture detail information, obtain visual effect, and properly improve the signal-to-noise ratio.

The advantages of using fractional calculus in designing control algorithms has been long discussed and demonstrated. In this paper, the theory of fractional calculus is used in combination with the Internal Model Control methodology to offer an alternative control solution for the concentration of biodegradable organic matter in the aeration tank of a waste water treatment facility. In contrast to the integer order Internal Model Controller, the fractional order controller has two tuning parameters, which allow for an increased flexibility in the tuning procedure. The fractional order Internal Model Controller is designed based on a simplified model of the process that occurs in the aeration tank. The dynamics of this controller is then analyzed using the nonlinear model of the process. Simulation results are provided including set point tracking, disturbance rejection and overall robustness.

This paper presents a new digital potential transformer (PT) model. Due to the effect of hysteresis and saturation phenomenon, the magnetization curve of iron core has a strong nonlinear. From the perspective of hysteresis effect of iron core, this article presents a PT core model using the theory of fractional calculus, then propose a new PT circuit model. The model simulates the nonlinear part of iron core using the fractional order expressions and can reflect the feature of saturation. Finally it shows the correctness and validity of the model through the theoretical analysis and simulation results.

We consider the solvability of fractional differential equations involving the Riesz fractional derivative. Our approach basically relies on the reduction of the problem considered to the equivalent nonlinear mixed Volterra and Cauchy‐type singular integral equation and on the theory of fractional calculus. By establishing a compactness property of the Riemann–Liouville fractional integral operator on Lebesgue spaces and using the well‐known Krasnoselskii's fixed point theorem, an existence of at least one solution is gleaned. An example is finally included to show the applicability of the theory.

(NOTE: Every chapter ends with Questions to Guide Your Review, Practice Exercises, and Additional Exercises.) P. Preliminaries. Real Numbers and the Real Line. Coordinates, Lines, and Increments. Functions. Shifting Graphs. Trigonometric Functions. 1. Limits and Continuity. Rates of Change and Limits. Rules for Finding Limits. Target Values and Formal Definitions of Limits. Extensions of the Limit Concept. Continuity. Tangent Lines. 2. Derivatives. The Derivative of a Function. Differentiation Rules. Rates of Change. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation and Rational Exponents. Related Rates of Change. 3. Applications of Derivatives. Extreme Values of Functions. The Mean Value Theorem. The First Derivative Test for Local Extreme Values. Graphing with y e and y . Limits as x a a, Asymptotes, and Dominant Terms. Optimization. Linearization and Differentials. Newton's Method. 4. Integration. Indefinite Integrals. Differential Equations, Initial Value Problems, and Mathematical Modeling. Integration by Substitution--Running the Chain Rule Backward. Estimating with Finite Sums. Riemann Sums and Definite Integrals. Properties, Area, and the Mean Value Theorem. Substitution in Definite Integrals. Numerical Integration. 5. Applications of Integrals. Areas Between Curves. Finding Volumes by Slicing. Volumes of Solids of Revolution--Disks and Washers. Cylindrical Shells. Lengths of Plan Curves. Areas of Surfaces of Revolution. Moments and Centers of Mass. Work. Fluid Pressures and Forces. The Basic Pattern and Other Modeling Applications. 6. Transcendental Functions. Inverse Functions and Their Derivatives. Natural Logarithms. The Exponential Function. ax and logax. Growth and Decay. L'Hopital's Rule. Relative Rates of Growth. Inverse Trigonomic Functions. Derivatives of Inverse Trigonometric Functions Integrals. Hyperbolic Functions. First Order Differential Equations. Euler's Numerical Method Slope Fields. 7. Techniques of Integration. Basic Integration Formulas. Integration by Parts. Partial Fractions. Trigonometric Substitutions. Integral Tables and CAS. Improper Integrals. 8. Infinite Series. Limits of Sequences of Numbers. Theorems for Calculating Limits of Sequences. Infinite Series. The Integral Test for Series of Nonnegative Terms. Comparison Tests for Series of Nonnegative Terms. The Ratio and Root Tests for Series of Nonnegative Terms. Alternating Series, Absolute and Conditional Convergence. Power Series. Taylor and Maclaurin Series. Convergence of Taylor Series Error Estimates. Applications of Power Series. 9. Conic Sections, Parametrized Curves, and Polar Coordinates. Conic Sections and Quadratic Equations. Classifying Conic Sections by Eccentricity. Quadratic Equations and Rotations. Parametrizations of Plan Curves. Calculus with Parametrized Curves. Polar Coordinates. Graphing in Polar Coordinates. Polar Equations for Conic Sections. Integration in Polar Coordinates. 10. Vectors and Analytic Geometry in Space. Vectors in the Plane. Cartesian (Rectangular) Coordinates and Vectors in Space. Dot Products. Cross Products. Lines and Planes in Space. Cylinders and Quadric Surfaces. Cylindrical and Spherical Coordinates. 11. Vector-Valued Functions and Motion in Space. Vector-Valued Functions and Space Curves. Modeling Projectile Motion. Arc Length and the Unit Tangent Vector T. Curvature, Torison, and the TNB Frame. Planetary Motion and Satellites. 12. Multivariable Functions and Partial Derivatives. Functions of Several Variables. Limits and Continuity. Partial Derivatives. Differentiability, Linearization, and Differentials. The Chain Rule. Partial Derivatives with Constrained Variables. Directional Derivatives, Gradient Vectors, and Tangent Planes. Extreme Values and Saddle Points. Lagrange Multipliers. Taylor's Formula. 13. Multiple Integrals. Double Integrals. Areas, Moments, and Centers of Mass. Double Integrals in Polar Form. Triple Integrals in Rectangular Coordinates. Masses and Moments in Three Dimensions. Triple Integrals in Cylindrical and Spherical Coordinates. Substitutions in Multiple Integrals. 14. Integration in Vector Fields. Line Integrals. Vector Fields, Work, Circulation, and Flux. Path Independence, Potential Functions, and Conservative Fields. Green's Theorem in the Plane. Surface Area and Surface Integrals. Parametrized Surfaces. Stokes's Theorem. The Divergence Theorem and a Unified Theory. Appendices. Mathematical Induction. Proofs of Limit Theorems in Section 1.2. Complex Numbers. Simpson's One-Third Rule. Cauchy's Mean Value Theorem and the Stronger Form of L'Hopital's Rule. Limits that Arise Frequently. The Distributive Law for Vector Cross Products. Determinants and Cramer's Rule. Euler's Theorem and the Increment Theorem.

On integral formulas and their applications in functional calculus

The determination of the transient wall pipe temperature by fractional calculus theory