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ALA 20210 On the operational solution of the system of fractional differential equations Đurđica Takači Department of Mathematics and Informatics Faculty of Science, University of Novi Sad Novi Sad, Serbia djtak@dmi.uns.ac.rs
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The Mikusinski operator field The set of continuous functions with supports in with the usual addition and the multiplication given by the convolution is a commutative ring without unit element. By the Titchmarsh theorem, it has no divisors of zero; its quotient field is called the Mikusinski operator field
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The Mikusinski operator field The elements of the Mikusinski operator field are convolution quotients of continuous functions
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The Mikusinski operator field The Wright function The character of the operational function
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The matrices with operators, square matrix, is a given vector, is the unknown vector
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Example
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The matrices with operators, square matrix, is a given vector, is the unknown vector
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The matrices with operators The exact solution of The approximate solution
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Fractional calculus The origins of the fractional calculus go back to the end of the 17th century, when L'Hospital asked in a letter to Leibniz about the sense of the notation the derivative of order Leibniz replied: “An apparent paradox, from which one day useful consequences will be drawn"
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Fractional calculus The Riemann-Liouville fractional integral operator of order Fractional derivative in Caputo sense
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Fractional calculus Basic properties of integral operators
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Fractional calculus Relations between fractional integral and differential operators
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Relations between the Mikusiński and the fractional calculus
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On the character of solutions of the time-fractional diffusion equation to appear in Nonlinear Analysis Series A: Theory, Methods & Applications Djurdjica Takači, Arpad Takači, Mirjana Štrboja
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The time-fractional diffusion equation
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with the conditions
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The time-fractional diffusion equation
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In the field of Mikusinski operators the time-fractional diffusion equation has the form
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The time-fractional diffusion equation The solution is The character of operational functions The Wright function
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The time-fractional diffusion equation The exact solution
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A numerical example The exact solution In the Mikusinski field
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The solution has the form A numerical example
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The exact solution
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A numerical example
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The system of fractional differential equations Initial value problem (IVP) Caputo fractional derivative, order
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The initial value problem (IVP) has a unique continuous solution
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References Caputo, M., Linear models of dissipation whose Q is almost frequency independent- II, Geophys. J. Royal Astronom. Soc., 13, No 5 (1967), 529-539 (Reprinted in: Fract. Calc. Appl. Anal.,11, No 1 (2008), 3-14.) Mainardi, F., Pagnini, G., The Wright functions as the solutions of time-fractional diffusion equation, Applied Math. and Comp., Vol.141, Iss.1, 20 August 2003, 51-62. Podlubny, I., Fractional Differential Equations, Acad. Press, San Diego (1999). Ross, B., A brief history and exposition of fundamental theory of fractional calculus, In: "Fractional Calculus and Its Applications" (Proc. 1st Internat. Conf. held in New Haven, 1974; Ed. B. Ross), Lecture Notes in Math. 457, Springer Verlag, N. York (1975), pp. 1-37.
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Podlubny, I., Fractional Differential Equations, Acad. Press, San Diego (1999). Ross, B., A brief history and exposition of fundamental theory of fractional calculus, In: "Fractional Calculus and Its Applications" (Proc. 1st Internat. Conf. held in New Haven, 1974; Ed. B. Ross), Lecture Notes in Math. 457, Springer-Verlag, N. York (1975), pp. 1-37.
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Thank you for your attention!
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