Differential equations of fractional order:methods results and problem —I

References

• Al-Abedeen , A.Z. 1976 . Existence theorem on differential . Equation of generalized order , 1 : 95 – 104 .
   Google Scholar
• Al-Abedeen , A.Z. and Arora , H.L. 1978 . A global existence and Uniqueness theroem ordinary differential equation of generalized order . canad.Math.Bull , 21 ( 3 ) : 267 – 271 .
   Google Scholar
• Al-Bassam , M.A. 1965 . Some existence theorems on differential equation of generalized order . Reine Angew. Math. , 218 : 70 – 78 .
   Web of Science ®Google Scholar
• AL-bassam , M.A. 1982 . On fractional calculus and its applications to the theory of ordinary differential equations of generalized order . Lect.Notes in pure and Appl.Math , 80 : 305 – 331 .
   Google Scholar
• Al-bassam , M.A. 1986 . “ fractzonal analysis and its applications, Modern analysis and its applications ” . New Delhi : Prentice Hall of India . (New Delhi, 1983)
   Google Scholar
• Al-bassam , M.A. 1987 . “ On generalized power series and generalized operational calculus and its applications ” . In Nonlinear analysis, , 51 – 58 . Singapore : World Sci. Publishing .
   Google Scholar
• Al-bassam , M.A. and Luchko , YU.F . 1995 . generalized fractional calculus and its application to the solution of integro-differentia1 equations. . J.Fract.calc , 7 : 69 – 88 .
   Google Scholar
• Alonso , J. 1964 . On differential equations of fractional order . Dr.Diss. Univ. Cincinnati , : 64 – 11 . 70pp.in diss abstrs n 4, pos 2231 order ,947
   Google Scholar
• Arora , H.L. and Alshamani , J.G. 1980 . Stability of differential equation of non-integer order through fixed point in the large, Indian . J.pure Math , 11 ( 3 ) : 307 – 313 .
   Web of Science ®Google Scholar
• Babenko , YU.I . 1986 . “ Heat and Mass Trasfer ” . In The Method of Calclation for the Heat and Diffusion Flows , Moscow : Himaya .
   Google Scholar
• Barrett , J.H. 1954 . Differential equations of non-integer order, Canad. . J.Math , 6 ( 4 ) : 529 – 541 .
   Web of Science ®Google Scholar
• Berens , H. and Westphal , U. 1968 . A Cauchy problem for a generalized wave equation . Acta. Sci. Math. , 29 ( 1–2 ) : 93 – 106 . (Szeged)
   Web of Science ®Google Scholar
• Beyer , H. and Kempfle , S. 1995 . Definition of physically consistent damping laws with fractional derivatives . Z Angew. Math. Mech. , 75 ( 8 ) : 623 – 635 .
   Web of Science ®Google Scholar
• Bonilla , B. , Kilbas , A.A. , Rivero , M. , Rodriguez , l. and Trujillo , J.J. 2000 . Modified Bessel-type function and solution of differential and integral equations . Indian J.Pure Appl Math , 31 ( 1 ) : 93 – 109 .
   Web of Science ®Google Scholar
• Bonilla , B. , Kilbas , A.A. and Trujillo , J.J. 2000 . Systems of nonlinear fractional diffrential equations in the space of summable functions . Proc.Inst Math. Nats. Akad. Nauk Bciarusi , 6 : 38 – 46 .
   Google Scholar
• Campos , L.m.B.C . 1990 . on the solution of some sample fractional differentail equations Internat . .J Math Math Sci. , 13 ( 3 ) : 481 – 496 .
   Google Scholar
• Davis , H.T. 1927 . The applications of fractional operators to functional equations . Amer.j.Math , 49 ( 1 ) : 123 – 142 .
   Google Scholar
• Delbosoco , D. and Rodino , L. 1996 . Existence and uniqueness for a nonlinear fractional differential equation . J.Math.Anal.Appl , 204 ( 2 ) : 609 – 625 .
   Web of Science ®Google Scholar
• Dimovski , I.H. 1966 . Operational calculus for a class oj diferential operators . C.R. Arad Bulgar. Sci. , 19 ( 12 ) : 1111 – 1114 .
   Google Scholar
• Dimovski , I.H. 1990 . Dimovski, Convolutional Calculus, Matherrmatics and its Applications . 43 East European Series
   Google Scholar
• Ditkin , V.A. 1957 . The theory of operator calculus . 116 : 15 – 17 . (Russian) Dokl. AN SSSR (N.S.),
   Google Scholar
• Ditkin , V.A. and prudnikov , A.P. 1963 . On the theory of operational calculus stemming from the Bessel equation. . English translation in USSR Comput. Math. Math. Phys. , 3 : 223,296 – 238,315 .
   Google Scholar
• Ditkin , V.A. and Prudnikov , A.P. 1965 . “ Integral Transfoms and Operational Calculus ” . Oxford : Pergamon Press .
   Google Scholar
• Ditkin , V.A. and Prudnikov , A.P. 1965 . “ Operational Calculus ” . Moscow : Vysshaja Shkola .
   Google Scholar
• Doetsch , G. 1955 . “ Handbuch der Laplace-Transformation ” . Basel-Stuttgart : Birkhä Verlag .
   Google Scholar
• Dorta , J.A. 1974 . “ Sobre ecuaciones diferenczales de orden no entro ” . Spain : Universidad de la Laguna .
   Google Scholar
• Dzhrbashyan , M.M. 1984 . Interpolatzion and spectral expansions associated with differential operators of fractional order . J.Contemporary Math. Anal. , 19 ( 2 ) : 81,1, – 181,116 . Russian,Izv.Akad.Nauk Armyan.SSR,Ser Mat.translation in Sov
   Google Scholar
• Dzhrbashyan , M.M. 1987 . Interpolatzion and spectral expansions associated with differential operators of fractional order . A Haar memorial conference. , I,II : 303 – 322 . (Budapest 1985) Colloq Math.Soc Janos Bolyai,49
   Google Scholar
• Dzharbashyan , M.M. 1988 . The basis property of systems of eigenfunctions that are generated by a boundary value problem in a complex domain. . translation in Soviet J. Contemporary Math. Anal. , 23 ( 5 ) : 409,1 – 421,14 . Izv. Akad. Nauk Armyan. SSR, Ser. Mat
   Google Scholar
• Dzhrbashyan , M.M. 1989 . differenlal operators of fractional order and boundary value.problem in the complax domain . The gohber anniversary collection , II : 153 – 172 . oper. Theroy.Adv.Appl
   Google Scholar
• Dzhrbasnyan , M.M. 1993 . “ Harmonic analysis and boundery value problems in the complex domain Operator theroy ” . In Advances and applications , Vol. 65 , Basel : Birkhäuser Verlag .
   Google Scholar
• Dzhrbashyan , M.M. and Nersesyan , A.B. 1960 . Expansions in special biorthogonal systems and boundary value problem for differential equations of fractional order. . translation in Soviet Math. , 132,1 ( 4 ) : 747,629 – 750,633 . (Russian), Dokl. Akad. Nauk SSSR,
   Google Scholar
• Dzhrbashyan , M.M. and Nersesyan . 1961 . Expansions in certain biorthogonai systems and boundary value problems for differntail equations of fractional order. . 10 : 89 – 179 . , Thdy Moskov Mat. Otm.
   Google Scholar
• Dzhrbashyan , M.M. and Nersesyan , A.B. 1968 . Fractional derivatives and the Cauchg problem for differntial equations of fractional order . Akad Nauk Armyan. SSR: Ser. Mat? , 3 ( 1 ) : 3 – 29 . Russaian
   Google Scholar
• Dzhrbashyan , M.M. and Rafaelyan . 1988 . Singular boundary value problems that geiierate systems of Fourier tgpe . translatiori in Soviet J. Contemporary Math. Anal. , 23 ( 2 ) : 109,1 – 122,16 .
   Google Scholar
• Elizarraraz , D. and Verde-Star , L. 2000 . Star, Fractional divided differences and the solution of differential and equations of fractional order . Adv. in Appl. Mat,h., , 24 ( 3 ) : 260 – 283 .
   Web of Science ®Google Scholar
• Ei-Sayed , A.M.A. 1988 . Fractional differential equations . Kyunypook Math. , 28 ( 2 ) : 119 – 122 .
   Google Scholar
• El-Sayed , A.M.A. 1992 . On the fractional differential equations . Appl. Math.Comput. , 49 ( 2-3 ) : 205 – 213 .
   Web of Science ®Google Scholar
• El-Sayed , A.M.A. 1993 . Linear differential equations of fvactional order . Appi. Math. Cornput , 55 ( 1 ) : 1 – 12 .
   Web of Science ®Google Scholar
• El-Sayed , A.M.A. 1996 . Finite Weyl fractional calculus and abstract fractional differential equation . J.Fract.Calc. , 9 : 59 – 68 .
   Google Scholar
• El-Sayed , A.M.A. and Gafer , F.M. 1997 . Exact solution of linear differential equations of fractional order . Kyungpook Math .J , 37 ( 2 ) : 181 – 189 .
   Google Scholar
• El-Saved , A.M.A. and Gafar , F.M. Sturm-Liouville equation of fractional order . Kyungpook Math , 37 ( 1 ) 9 – 17 .
   Google Scholar
• El-Sayed , A.M.A. and Ibragim , A.-G . 1995 . Multivalued fractional differential equations . Appl. Math. Comput. , 68 ( 1 ) : 15 – 25 .
   Web of Science ®Google Scholar
• Erdelyi , A. , Magnus , W. , Oberhettinger , F. and Tricomi , F.G. 1981 . “ Higher Transcendental functions ” . Vol. 1 , Florida , Melbourne : McGraw-Hill Book Corp . Reprinted Krieger Melbourne ,Florida,1981
   Google Scholar
• Erdelyi , A. , Magnus , W. , Oberhettinger , F. and Tricomi , F.G. 1981 . “ Higher Transcendental functions ” . Vol. 2 , Florida , Melbourne
   Google Scholar
• Erdelyi , A. , Magnus , W. , Oberhettinger , F. and Tricomi , F.G. 1981 . “ Higher Transcendental functions ” . Vol. 3 , Florida , Melbourne
   Google Scholar
• Fedosov , V.P. and Yanenko , N.N. 1982 . Solution of a mixed problem for the wave equation . 262 ( 3 ) : 549 – 553 . Dokl. Akad. Nauk SSSR
   Google Scholar
• Fujita , Y. 1990 . Cauchy problems of fractional order and stable procsses,japan . J. Appl. Math , 7 ( 3 ) : 459 – 476 .
   Google Scholar
• Fujiwara , M. 1933 . On the integration and differen.tintion of an arbitary order . Tohoku Math. J. , 37 : 110 – 121 .
   Google Scholar
• Gaul , L. , Klin , P. and Kempfle , S. 1991 . Damping description involuing fractional operators, Mech. . Systerrrs Signal Processing , 5 : 81 – 88 .
   Web of Science ®Google Scholar
• Gelfand , I.M. and Shilov , G.F. “ Generalized Functions, ” .
   Google Scholar
• Gerasimov , A.N. 1948 . A generalization of linear laws of defomation and its application to internal friction problem. . Akad. Nauk SSSR. Prikl. Mat. Mekh. , 12 : 251 – 260 . Russian
   Google Scholar
• Giona , M. and Roman , H.E. 1992 . Fractional diffusion equation on fractals one-dimensional case and asymptotic behaviour . J . Phys. A. , 25 ( 8 ) : 2093 – 2105 .
   Google Scholar
• Glaeske , H.-J. , Kilbas , A.A. and Saigo , M. 2000 . Modifed Bessel type integral transform and its compositions with fractional calculus operator on spaces Fp,u and F'p,u . J Comput.Appl.Math , 118 ( 1-2 ) : 151 – 168 .
   Web of Science ®Google Scholar
• Gorenflo , R. and Mainardi , F. 1996 . “ Fractional oscillations and Mittag-Lefler functions ” . In Fachbercich Math. , Vol. 14 , 1 – 22 . Berlin : und Inf., Freie Univ .
   Google Scholar
• Gorenflo , R. and Mainardi , F. 1997 . Fractional calculus integral and differential equations of fractional order . Fractal and Fractional Calculus in Continuum Mechanics CISM Courses and Lectures, 37 , 378 : 223 – 276 . Udina, 1996
   Google Scholar
• Gorenflo , R. and Mainardia , F. 1998 . Fractional calculus and stable probability distributions, Fourth Meeting on Current Ideas in Mechanics and Related Fields . Arc.Mec , 50 ( 3 ) : 377 – 388 . krakow,1997 Mech. Stos. 50,
   Google Scholar
• Gorenflo , R. and Rutman , R. 1995 . ultraslow and intermediate processes, Transform Methods and Special Functions, Sofia . SCTP Singapore, , 94 : 61 – 81 . proc 1st Intern. Workshop
   Google Scholar
• Grin'ko , A.P. 1991 . Solution a nonlinear differental equation with a generalized fractional derivalnve. (R . Dokl. Akad. Pllavltk BSSR, , , 35 ( 1 ) : 27 – 31 .
   Web of Science ®Google Scholar
• Hadid , S.B. 1995 . Local and global existence theorems on differential equations of non-integer order . J.Frac.Calc , 7 : 101 – 105 .
   Google Scholar
• Hadid , S.B. and Alshamani , J.G. 1986 . Liapunov Stability of differential equations of non-integer order . Arab J.Math , 7 ( 1-2 ) : 5 – 17 .
   Google Scholar
• Hadid , S.B. and Luchko , Yu.F . 1996 . operational method for solving fractional differential equations of an arbitrary real order . Panamer. Math. J. , 6 ( 1 ) : 57 – 73 .
   Google Scholar
• Hadid , S.B. , Masaedeh , B. and Momani , S. 1996 . On the existence of maximal and minimal solution. of differential equations of non-integer order. . J. Fract.calc , 9 : 41 – 44 .
   Google Scholar
• Hayek , N. , Trujillo , J. , Rivero , M. , Bonilla , B. and Moreno , J.C. 1999 . An extension of Picard-Lindeloff theorem to fractional diflerential equations. . Appl.Anal , 70 ( 3-4 ) : 347 – 361 .
   Google Scholar
• Hille , E. and Tamarkin , J.D. 1930 . the theory of linear integral equations . Ann.Math , 31 : 479 – 528 .
   Google Scholar
• Karapetyants , N.K. , Kilbas , A.A. and Saigo , M. 1996 . On the solution of nonlinear Volterra convolution equation with power nonlinearity, J. . J.Integral Equations Appl , 8 : 4,429 – 445 .
   Google Scholar
• Kilbas , A.A. , Bonilla , B. , Rodriguez , J. , Trujillo , J.J. and Rivero , M. 1998 . Compositions of fractional integral and derivatives with a Bessnel-type function and the solution of differential and integral equations. . Dokl Nats. Akad Nauk Belarusi, , 42 : 2,25 – 29 . Russian
   Web of Science ®Google Scholar
• Kilbas , A.A. , Bonilla , B. and Trujillo , J.J. 2000 . Existence and uniqueness theorems for nonlinear fractional differential equations . Demonstratio Math., , 33 : 3,583 – 602 .
   Google Scholar
• Kilbas , A.A. 2000 . Factional integrals and derivatives and differential equations of fractzonal order in weighted spaces of continous functions . Dokl. Nats. Akad. Nauk Belarusi, , 44 ( 6 )
   Google Scholar
• Kilbas , A. A. and Saigo , M. 1995 . On solution of integral equation of Abel Volterra type . Differential Integral Equations , 8 ( 5 ) : 993 – 1011 .
   Google Scholar
• Kilbas , A. A. and Saigo , M. 1995 . Fractional integrals and derivatives of functions of Mittag Leffler type . Dokl. Akad. Nauk Belarusi , 39 ( 4 ) : 22 – 26 . Russian
   Web of Science ®Google Scholar
• Kilbas , A. A. and Saigo , M. 1995 . Solution of Abel integral equations of the second kind and of differential equations of fractional order . Dokl. Akad. Nauk Belarusi , 39 ( 5 ) : 29 – 34 . Russian
   Web of Science ®Google Scholar
• Kilbas , A. A. and Saigo , M. 1996 . On Mittag Leffler type function, fractional calculus operators and solutions of integral equations . Integral Transform. Spec. Funct , 4 ( 4 ) : 355 – 370 .
   Web of Science ®Google Scholar
• Kilbas , A. A. and Saigo , M. 1996 . On solution in closed form of nonlinear integral and differential equations of fractional order . Univalent functions and the Briot Bouquet Differential Equations , 963 : 39 – 50 . Japanese Kyoto, 1996
   Google Scholar
• Kilbas , A. A. and Saigo , M. 1997 . Solution in closed form of a class of linear differential equations of of fractional order . translation in Differential Equations , 33 : 2,194 – 204 . Russian
   Web of Science ®Google Scholar
• Kilbas , A. A. and Saigo , M. 1998 . Application of fractional calculus to solve Abel Volterra nonlinear and linear integral equations . Transform Methods and Special Functions , 96 : 236 – 263 . Bulgarian Acad. Sci
   Google Scholar
• Kiryakova , V. S. 1994 . “ Generalized Fractional Calculus and Applications ” . In Pitman Research Notes in Mathematics , Vol. 301 , New York : John Wiley and Sons .
   Google Scholar
• Kochubei , A. N. 1990 . Diffusion of fractional order . translation in Differential Equations , 26 ( 4 ) : 485 – 492 . Russian
   Web of Science ®Google Scholar
• Koeller , R. C. 1984 . Applications of fractional calculus to the theory of viscoelasticity . Trans. ASME Ser. E. J. Appl. Mech , 51 ( 2 ) : 299 – 307 .
   Web of Science ®Google Scholar
• Kolwankar , K. M. and Gangal , A. P. 1998 . Local fractional Fokker Planck equation . Phys. Rev. Lett , 80 ( 2 ) : 214 – 217 .
   Web of Science ®Google Scholar
• Kratzel , E. 1979 . Integral transformations of Bessel type, Generalized Functions and Operational Calculus . Bulgar. Acad. Sci , : 148 – 155 . Proc. Conf. Varna 1975
   Google Scholar
• Leskovskii , I. P. 1977 . On the theory of fractional differentiation . Differentsial'nye Uravnenija , 13 ( 1 ) : 170 – 173 . Russian
   Google Scholar
• Leskovskii , I. P. 1980 . “ On the solution of the linear homogeneous differential equations with fractional derivatives and constant coefficients ” . In In. Some questions of diff. equat. in the solution of appl. problems , 85 – 88 . Tula, Tul'sk Politech. Inst . Russian
   Google Scholar
• Luchko , Yu. F. 1999 . Operational method in fractional calculus . Frac. Calc. Appl. Anal , 4 : 2,463 – 488 .
   Google Scholar
• Luchko , Yu. F. and Gorenflo , R. 1999 . An operational method for solving fractional differential equations with the Caputo derivatives . Acta Math. Vict nam , 24 ( 2 ) : 207 – 233 .
   Google Scholar
• Luchko , Yu. F. and Srivastava , H. M. 1995 . The exact solution of certain differential equations of fractional order by using operational calculus . Comput. Math. Appl , 29 ( 8 ) : 73 – 85 .
   Web of Science ®Google Scholar
• Luchko , Yu. F. and Yakubovich , S. B. 1993 . Operational calculus for the generalized fractional differential operator and applications . Math. Balkanica , 7 ( 2 ) : 119 – 130 . N. S
   Google Scholar
• Luchko , Yu. F. and Yakubovich , S. B. 1994 . An operational method for solving some classes of integro differential equations . translation in Differential Equations , 30 ( 2 ) : 247 – 256 . Russian
   Web of Science ®Google Scholar
• Luszcki , Z. and Rzepecki , B. 1987, 1988 . An existence theorem for ordinary differential equations of order α ε (0,1] . Demostratio Math , 20 ( 3 ) : 471 – 4,475 .
   Google Scholar
• Mainardi , F. 1996 . Fractional relaxation oscillation and fractional diffusionwave phenomena . Chaos, Solitons and Fractals , 7 ( 9 ) : 1461 – 1477 .
   Web of Science ®Google Scholar
• Mainardi , F. 1997 . Fractional calculus: some basic problems in continuum and statistical mechanics . CISM Courses and Lectures , 378 : 291 – 348 . Udine, 1996
   Google Scholar
• Mandelbrojt , S. 1925 . Sulla generalizzazione del calcolo delle variazione . Atti Reale Accad. Naz. Lincei. Rend. Cl. sci. fiz. mat. s natur , 1 : 151 – 156 . French
   Google Scholar
• Maravall , D. 1971 . Linear differential equations of non integer order and fractional oscilations . Rev. Ac. Ci , 65 : 245 – 258 . Spanish
   Google Scholar
• Matignon , D. 1996 . Stability results for fractional differential equations with applications to control processing . Computational Engineering in Systems and Applications, multiconference , 2 July : 952 – 956 . IMACC, IEEE SMC
   Google Scholar
• Mbodje , B. and Montseny , G. 1995 . Boundary fractional derivative control of the wave equation . IEEE Trans. Automat. Control , 40 ( 2 ) July : 378 – 382 .
   Web of Science ®Google Scholar
• Meller , N. A. 1963 . Some aplications of operational calculus to problems of analysis . J. Vychisl. Mat. i Mat. Fiz , 3 ( 3 ) July : 71 – 78 . Russian
   Google Scholar
• Metzler , R. , Glöckle , W. G. and Nonnenmacher , T. F. 1994 . Fractional model equation for anomalous diffusion . Physica A , 211 July : 13 – 24 .
   Web of Science ®Google Scholar
• Metzler , R. and Nonnenmacher , T. F. 1997 . Fractional diffusion: exact representations of spectral functions . J. Phys. A , 30 ( 4 ) July : 1089 – 1093 .
   Google Scholar
• Metzler , R. and Nonnenmacher , T. F. 1998 . Fractional diffusion, waiting time distributions, and Cattaneo type equations . Phys. Rev. E , 57 ( 3,6 ) July : 6409 – 6414 .
   Web of Science ®Google Scholar
• Mikusinski , J. 1959 . “ Operational Calculus ” . New York : Pergamon Press .
   Google Scholar
• Miller , K. S. 1993 . Fractional differential equations . J. Frac. Calc , 3 : 49 – 57 .
   Google Scholar
• Miller , K. S. 1995 . Derivatives of noninteger order . Math. Mag , 68 : 183 – 192 .
   Google Scholar
• Miller , K. S. and Ross , B. 1993 . “ An Introduction to the Fractional Calculus and Fractional Differential Equations ” . New York : John Wiley and Sons .
   Google Scholar
• Nkhushev , A. M. 1996 . Mathematical Problems in Biology , Moscow : nAUKA . Russian
   Google Scholar
• Nigmatullin , R. R. 1984 . To the theoretical explanation of the "universal" response . Phys. Sta. Sol (b) , 123 : 739 – 745 .
   Web of Science ®Google Scholar
• Nigmatullin , R. R . 1986 . The realization of the generalized transfer equation in a medium with fractal geometry . Phys. Sta. Sol (b) , 133 : 425 – 430 .
   Web of Science ®Google Scholar
• Nishimoto , K. 1984 . “ Fractional Calculus (Integration and Differentiation of Arbitrary Order) ” . Koriyama : Descartes Press Cor .
   Google Scholar
• Nishimoto , K. 1987 . “ Fractional Calculus (Integration and Differentiation of Arbitrary Order) ” . Vol. 2 , Koriyama : Descartes Press Cor .
   Google Scholar
• Nonnenmacher , T. F. 1991 . Fractional relaxation equations for viscoelasticity and related phenomena . Lect. Notes in Physics , 381 : 309 – 320 .
   Google Scholar
• Nonnenmacher , T. F. and Glöckle , W. G. 1991 . A fractional model for mechanical stress relaxation . Philosophical Magazine Lettters , 64 ( 2 ) : 89 – 93 .
   Web of Science ®Google Scholar
• Nonnenmacher , T. F. and Nonnenmacher , D. J. F. 1989 . Towards the formulation of a non linear fractional extended irreversible thermodynamics . Acta Phys. Hungar , 66 ( 1, 4 ) : 145 – 154 .
   Google Scholar
• Oldham , K. B. 1973 . Diffusive transport to planar, cylindrical, and spherical electrodes . J. Electroanal. Chem. and Interfacial Electrohem , 41 : 351 – 352 .
   Web of Science ®Google Scholar
• Oldham , K. B. and Spanier , J. 1970 . The replacement of Fick's law by a formulation involving semidifferentiation . J. Electroanal. Chem. and Interfacial Electrochem , 26 : 331 – 341 .
   Web of Science ®Google Scholar
• Oldham , K. B. and Spanier , J. 1974 . “ The Fractional Calculus ” . London : Academic Press .
   Google Scholar
• O'Shaughnessy , L. 1918 . Problem # 433 . Amer. Math. Month , 25 : 172 – 173 .
   Google Scholar
• Pitcher , E. and Sewell , W. E. 1938 . Existence theorem for solutions of differential equations of non integral order . Bull. Amer. Math. Soc , 44 : 2,100 – 107 . A correction in n. 12, p.888
   Google Scholar
• Podlubny , I. 1994 . The Laplace transform method for linear differential equations of the fractional order, Inst . Exper. Physics, Slovak Acad. Sci , : 32 Prepr. UEF 02 94
   Google Scholar
• Podlubny , I. 1995 . Solution of linear fractional differential equations with constant coefficients . Transform Methods and Special Functions, Sofia'94 , : 227 – 237 . Proc 1st Intern. Workshop
   Google Scholar
• Podlubny , I. 1999 . “ Fractional Differential Equations ” . In Mathematics in Sciences and Engineering , Vol. 198 , San Diego : Academic Press .
   Google Scholar
• Post , E. L. 1919 . Discussion of the solution of (d/dx)1/2y=y/x. . Amer. Math. Month , 26 : 37 – 39 . (#problem 433)
   Google Scholar
• Rafaelyan , S. G. 1984 . The basis property of certain biorthogonal systems in L2(-σ,σ) with weights . translation in Soviet J. Contemporary Math. Anal , 19 ( 3 ) : 21 – 32 . Russian
   Google Scholar
• Rodriguez , J. 1989 . Operational calculus for the generalized Bessel operator . 15 ( 3 ) : 179 – 186 .
   Google Scholar
• Rodriguez , J. , Trujillo , J. J. and Rivero , M. 1989 . Operational fractional calculus of Kratzel integral transformation . Differential equation Lecture Notes in Pure and Appl. Math , 118 : 613 – 620 . Xanthi, 1987
   Google Scholar
• Roman , H. E. and Giona , M. 1992 . Fractional diffusion equation on fractals: three dimensional case and scattering function . J. Phys. A , 25 ( 8 ) : 2107 – 2117 .
   Google Scholar
• Saigo , M. and Kilbas , A. A. 1998 . On Mittag Leffler type function and applications . Integral Transform. Spec. Funct , 7 ( 1, 2 ) : 97 – 112 .
   Web of Science ®Google Scholar
• Samko , S. G. , Kilbas , A. A. and Marichev , O. I. 1987 . Integral and Derivatives of Fractional Order and Some of Their Applications . Nauka i Tekhnika , Russian
   Google Scholar
• Samko , S. G. , Kilbas , A. A. and Marichev , O. I. 1993 . “ Fractional Integrals and Derivatives Theory and Applications ” . Yverdon : Gordon and Breach .
   Google Scholar
• Schneider , W. R. 1990 . Fractional diffusion . Dynamics and Stochastic Processes, Lecture Notes in Phys , 355 : 276 – 286 . Lisbon, 1988
   Google Scholar
• Schneider , W. R. 1990 . Fractional diffusion . Dynamics and Stochastic Processes, Lecture Notes in Phys , 355 : 276 – 286 . Lisbon, 1988
   Google Scholar
• Schneider , W. R. and Wyss , W. 1989 . Fractional diffusion and wave equations . J. Math. Phys , 30 ( 1 ) : 134 – 144 .
   Web of Science ®Google Scholar
• Seitkazieva , A. 1980 . Integro differential equations with a kernel that has a weak singularity . Studies in integro differential equations , 13 : 132 – 135 . Russian
   Google Scholar
• Semenchuk , N. P. 1982 . A class of differential equations of nonintegral order . Differentsial'nye Uravnenija , 18 ( 10 ) : 1831 – 1833 . Russian
   Google Scholar
• Tazali , A. Z.-A. M. 1982 . Local existence theorems for ordinary differential equations of fractional order . Ordinary and Patila Differential Equations, Lecture Notes in Math , 964 : 652 – 665 . Dundee, 1982
   Web of Science ®Google Scholar
• Tazali , A. Z. and Karim , I. F. 1994 . Continuous dependence on a parameter and continuation of solutions in the extended sense in ordinary differential equations of fractional order . Dirasat Ser. B Pure Apl. Sci , 21 ( 1 ) : 270 – 280 .
   Google Scholar
• Titchmarsh , E. C. 1986 . “ Introduction to the Theory of Fourier Transforms ” . New York : Chelsea Publishing Company .
   Google Scholar
• Torvik , P. J. and Bagley , R. L. 1984 . On the appearence of the fractional derivative in the behavior of real materials . J. Appl. Mech , 51 : 294 – 298 .
   Web of Science ®Google Scholar
• West , B. J. , Grigolini , P. , Metzler , R. and Nonnenmacher , T. F. 1997 . Fractional diffusion and Levy stable processes . Phys. Rew. E(3) , 55 ( 1 ) : 99 – 106 .
   Web of Science ®Google Scholar
• Westerlund , S. 1991 . Dead matter has memory! . Physica Scripta , 43 : 174 – 179 .
   Web of Science ®Google Scholar
• Wiener , K. 1983 . On solutions of a differential equation of nonintegral order that occurs in the theory of polarography . Wiss. Z. Martin Luther Univ. Halle Wittenberg Math Natur Reiche , 32 ( 1 ) : 41 – 46 . German
   Google Scholar
• Wyss , W. 1986 . The fractional diffusion equation . J. Math. Phys , 27 ( 11 ) : 2782 – 2785 .
   Web of Science ®Google Scholar