FRACTIONAL CAUCHY TRANSFORMS
EDITORIAL FOR SPECIAL ISSUE FRACTIONAL CALCULUS OPERATORS AND THE MITTAG LEF ER FUNCTION SOME DYNAMICAL MODELS INVOLVING FRACTIONAL ORDER DERIVATIVESWITH THE MITTAG LEF ER TYPE KERNELSAND THEIR APPLICATIONSBASED UPON THE LEGENDRE SPECTRAL COLLOCATION METHOD CESARO LIMITS FOR FRACTIONAL DYNAMICS MITTAG LEF ER FUNCTION AS AN APPROXIMANT TO THE CONCENTRATED FERRO UID S MAGNETIZATION REDUCTION FORMULAS FOR GENERALIZED HYPERGEOMETRIC SERIES ASSOCIATED WITH NEW SEQUENCES HILFER HADAMARD FRACTIONAL BOUNDARY VALUE PROBLEMS WITH NONLOCAL MIXED BOUNDARY CERTAIN RECURRENCE RELATIONS OF TWO PARAMETRIC MITTAG LEF ER FUNCTION AND THEIR APPLICATION IN FRACTIONAL CALCULUS FEJ ER HADAMARD TYPE INEQUALITIES FOR H M P CONVEX FUNCTIONS VIA EXTENDED GENERALIZED THE MITTAG LEF ER FUNCTIONS FOR A CLASS OF FIRST ORDER FRACTIONAL INITIAL VALUE PROBLEMS DUAL SOLUTION VIA RIEMANN LIOUVILLE FRACTIONAL DERIVATIVE GENERALIZED K FRACTIONAL INTEGRAL OPERATORS ASSOCIATED WITH P OLYA SZEG O AND CHEBYSHEV TYPES CERTAIN INTEGRAL AND DIFFERENTIAL FORMULAS INVOLVING THE PRODUCT OF SRIVASTAVA S POLYNOMIALS AND EXTENDED WRIGHT FUNCTION THE MITTAG LEF ER FUNCTION FOR RE EVALUATING THE CHLORINE TRANSPORT MODEL COMPARATIVE HERMITE HADAMARD FRACTIONAL INTEGRAL INEQUALITIES VIA ABEL GONTSCHAROFF GREEN S FUNCTION DIRICHLET AVERAGES OF GENERALIZED MITTAG LEF ER TYPE FUNCTION FRACTIONAL INTEGRAL INEQUALITIES OF HERMITE HADAMARD TYPE FOR H G M CONVEX FUNCTIONS WITH EXTENDED MITTAG LEF ER FUNCTION A NEW FRACTIONAL POISSON PROCESS GOVERNED BY A RECURSIVE FRACTIONAL DIFFERENTIAL EQUATION EDITORIAL FOR SPECIAL ISSUE FRACTIONAL CALCULUS OPERATORS AND THE MITTAG LEF ER FUNCTION AMONG THE NUMEROUS APPLICATIONS OF THE THEORY OF FRACTIONAL CALCULUS IN ALMOST ALL APPLIED SCIENCES APPLICATIONS IN NUMERICAL ANALYSIS AND VARIOUS ELDS OF PHYSICS AND ENGINEERING STAND OUT APPLICATIONS OF INEQUALITIES INVOLVING FUNCTION INTEGRALS AND THEIR DERIVATIVES AS WELL AS APPLICATIONS OF FRACTIONAL DIFFERENTIATION INEQUALITIES HAVE MOTIVATED MANY RESEARCHERS TO INVESTIGATE EXTENSIONS AND GENERALIZATIONS USING VARIOUS FRACTIONAL DIFFERENTIAL AND INTEGRAL OPERATORS OF PARTICULAR IMPORTANCE IS THE MITTAG LEF ER FUNCTION WHICH WITH ITS GENERALIZATIONS APPEARS AS A SOLUTION TO DIFFERENTIAL OR INTEGRAL EQUATIONS OF FRACTIONAL ORDER THIS PRODUCED NEW RESULTS FOR MORE GENERALIZED FRACTIONAL INTEGRAL OPERATORS CONTAINING THE MITTAG LEF ER FUNCTION IN THEIR KERNELS THIS SPECIAL ISSUE CONTAINS 15 PUBLISHED PAPERS IN 1 THE AUTHORS USE DERIVATIVES OF FRACTIONAL ORDER WHICH ARE BASED ON GENERALIZED KERNELS OF THE MITTAG LEF ER TYPE TO PRESENT THREE MODELS OF FRACTIONAL ORDER THESE MODELS INVOLVE FRACTIONAL ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS THE ACCURACY OF THE SOLUTION WAS CHECKED IN COMPARISON WITH THE EXACT SOLUTION AND BY CALCULATING THE RESIDUAL ERROR FUNCTION THE ASYMPTOTIC BEHAVIOR OF RANDOM TIME CHANGES IN DYNAMICAL SYSTEMS IS STUDIED IN 2 AS TIME VARIES RANDOMLY THREE CLASSES ARE PROPOSED THAT SHOW DIFFERENT PATTERNS OF IN 3 THE AUTHORS SHOWED THAT THE STATIC MAGNETIZATION CURVE OF HIGHLY CONCENTRATED FERRO UIDS CAN BE ACCURATELY APPROXIMATED BY THE MITTAG LEF ER FUNCTION OF THE INVERSE EXTERNAL MAGNETIC ELD BY INTRODUCING TWO SERIES OF NEW NUMBERS AND THEIR DERIVATIVES 4 AND CHOOSING TO USE SIX WELL KNOWN GENERALIZED KUMMER ADDITION FORMULAS THE AUTHORS ESTABLISH SIX CLASSES OF GENERALIZED ADDITION FORMULAS AND DEMONSTRATE SIX IDENTITIES REGARDING NITE SUMS THE PAPER 5 DEALS WITH THE EXISTENCE AND UNIQUENESS OF THE SOLUTION FOR THE HILFER HADAMARD FRACTIONAL DIFFERENTIAL EQUATION SUPPLEMENTED WITH MIXED NON LOCAL BOUND IN 6 THE AUTHORS DEVELOPED SOME NEW RECURRENCE RELATIONS FOR TWO PARAMETRIC MITTAG LEF ER FUNCTIONS AND DISCUSSED SOME APPLICATIONS OF THESE RECURRENCE RELATIONS BY APPLYING RIEMANN LIOUVILLE FRACTIONAL INTEGRALS AND DIFFERENTIAL OPERATORS FOUR NEW RELATIONS BETWEEN FOX WRIGHT FUNCTIONS WITH CERTAIN SPECIAL CASES ARE ESTABLISHED THE GOAL OF THE PAPER 7 WAS TO ND NEW VERSIONS OF FEJ R HADAMARD TYPE INEQUALITIES FOR H M P CONVEX FUNCTIONS BY MEANS OF EXPANDED FRACTIONAL OPERATORS WITH MITTAG LEF ER FUNCTIONS THESE INEQUALITIES HOLD AT THE SAME TIME FOR DIFFERENT TYPES OF CONVEXITIES AND DIFFERENT TYPES OF FRACTIONAL INTEGRALS IN 8 A NEW APPROACH IS GIVEN FOR SOLVING A CLASS OF RST ORDER FRACTIONAL INITIAL VALUE PROBLEMS THE RESULTS ARE BASED ON THE RIEMANN LIOUVILLE FRACTIONAL DERIVATIVE AND THE SOLUTIONS ARE EXPRESSED IN TERMS OF MITTAG LEF ER EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS THE PAPER 9 AIMS TO DE NE AN OPERATOR THAT CONTAINS THE MITTAG LEF ER FUNCTION IN ITS KERNEL WHICH LEADS TO THE DEDUCTION OF MANY ALREADY EXISTING OPERATORS THE RESULTS OF THIS WORK REPRODUCE CHEBYSHEV AND P LYA SZEG TYPE INEQUALITIES FOR VARIOUS FRACTIONAL IN 10 THE AUTHORS INVESTIGATE CERTAIN IMAGE FORMULAS RELATED TO THE PRODUCT OF SRIVAS TAVA POLYNOMIALS AND EXTENDED WRIGHT FUNCTIONS USING FRACTIONAL INTEGRAL AND DIFFERENTIAL OPERATORS INCLUDING MARICHEV SAIGO MAEDA LAVOIE TROTTIER AND OBERHETTINGER THE PAPER 11 REVIEWS THE MATHEMATICAL MODEL OF CHLORINE TRANSPORT USED AS A WATER TREATMENT MODEL WHEN VARIABLE ORDER PARTIAL DERIVATIVES ARE INCLUDED TO DESCRIBE THE CHLORINE TRANSFER SYSTEM IN THIS PAPER A NEW ANALYTICAL SOLUTION USING MITTAG LEF ER FUNCTIONS IS PRESENTED HERMITE HADAMARD INEQUALITIES FOR RIEMANN LIOUVILLE FRACTIONAL INTEGRALS ARE PRESENTED IN 12 USING A NEWER APPROACH BASED ON THE ABEL GONTSCHAROFF GREEN FUNCTION THE AUTHORS ESTABLISHED CERTAIN INTEGRAL IDENTITIES AND OBTAINED NEW RESULTS FOR MONOTONE FUNCTIONS THE PAPER 13 INVESTIGATES THE DIRICHLET AND MODI ED DIRICHLET AVERAGE OF THE R FUNCTION I E THE MITTAG LEF ER TYPE FUNCTION THEY ARE GIVEN IN TERMS OF RIEMANN LIOUVILLE INTEGRALS AND HYPERGEOMETRIC FUNCTIONS OF SEVERAL VARIABLES HERMITE HADAMARD FRACTIONAL INTEGRAL INEQUALITIES FOR THE CLASS OF H G M CONVEX FUNCTIONS ARE PRESENTED IN 14 THE EXPANDED GENERALIZED MITTAG LEF ER FUNCTION IS CONTAINED IN THE KERNEL OF APPLIED FRACTIONAL INTEGRAL OPERATORS IN 15 A NEW FRACTIONAL POISSON PROCESS IS PROPOSED USING A RECURSIVE FRACTIONAL DIFFERENTIAL EQUATION THE ARRIVAL TIME DISTRIBUTION FUNCTIONS ARE DERIVED WHILE THE INTER ARRIVAL TIMES ARE NO LONGER INDEPENDENT AND IDENTICALLY DISTRIBUTED SRIVASTAVA H ALOMARI A SAAD K HAMANAH W SOME DYNAMICAL MODELS INVOLVING FRACTIONAL ORDER DERIVATIVES WITH THE MITTAG LEF ER TYPE KERNELS AND THEIR APPLICATIONS BASED UPON THE LEGENDRE SPECTRAL COLLOCATION METHOD FRACTAL FRACT 5 131 CROSSREF KONDRATIEV Y DA SILVA J CESARO LIMITS FOR FRACTIONAL DYNAMICS FRACTAL FRACT 2021 5 133 CROSSREF RYAPOLOV P POSTNIKOV E MITTAG LEF ER FUNCTION AS AN APPROXIMANT TO THE CONCENTRATED FERRO UID S MAGNETIZATION CURVE FRACTAL FRACT 2021 5 147 CROSSREF CHOI J QURESHI M BHAT A MAJID J REDUCTION FORMULAS FOR GENERALIZED HYPERGEOMETRIC SERIES ASSOCIATED WITH NEW SEQUENCES AND APPLICATIONS FRACTAL FRACT 2021 5 150 CROSSREF AHMAD B NTOUYAS S HILFER HADAMARD FRACTIONAL BOUNDARY VALUE PROBLEMS WITH NONLOCAL MIXED BOUNDARY CONDITIONS FRACTAL FRACT 2021 5 195 CROSSREF SACHAN D JALOREE S CHOI J CERTAIN RECURRENCE RELATIONS OF TWO PARAMETRIC MITTAG LEF ER FUNCTION AND THEIR APPLICATION IN FRACTIONAL CALCULUS FRACTAL FRACT 2021 5 215 CROSSREF FARID G YUSSOUF M NONLAOPON K FEJ R HADAMARD TYPE INEQUALITIES FOR H M P CONVEX FUNCTIONS VIA EXTENDED GENERALIZED FRACTIONAL INTEGRALS FRACTAL FRACT 2021 5 253 CROSSREF EBAID A AL JEAID H THE MITTAG LEF ER FUNCTIONS FOR A CLASS OF FIRST ORDER FRACTIONAL INITIAL VALUE PROBLEMS DUAL SOLUTION VIA RIEMANN LIOUVILLE FRACTIONAL DERIVATIVE FRACTAL FRACT 2022 6 85 CROSSREF ZHANG Z FARID G MEHMOOD S NONLAOPON K YAN T GENERALIZED K FRACTIONAL INTEGRAL OPERATORS ASSOCIATED WITH P LYA SZEG AND CHEBYSHEV TYPES INEQUALITIES FRACTAL FRACT 2022 6 90 CROSSREF NAHEED S MUBEEN S RAHMAN G KHAN Z NISAR K CERTAIN INTEGRAL AND DIFFERENTIAL FORMULAS INVOLVING THE PRODUCT OF SRIVASTAVA S POLYNOMIALS AND EXTENDED WRIGHT FUNCTION FRACTAL FRACT 2022 6 93 CROSSREF ALJOHANI A EBAID A ALGEHYNE E MAHROUS Y CATTANI C AL JEAID H THE MITTAG LEF ER FUNCTION FOR RE EVALUATING THE CHLORINE TRANSPORT MODEL COMPARATIVE ANALYSIS FRACTAL FRACT 2022 6 125 CROSSREF LI Y SAMRAIZ M GUL A VIVAS CORTEZ M RAHMAN G HERMITE HADAMARD FRACTIONAL INTEGRAL INEQUALITIES VIA ABEL GONTSCHAROFF GREEN S FUNCTION FRACTAL FRACT 2022 6 126 CROSSREF KUMAR D RAM J CHOI J DIRICHLET AVERAGES OF GENERALIZED MITTAG LEF ER TYPE FUNCTION FRACTAL FRACT 2022 6 297 CROSSREF FRACTIONAL CAUCHY TRANSFORMS