A Laplace substitution method of space‐time fractional order including jumbled partial derivatives

2021 ◽  
Author(s):  
Anas Arafa ◽  
Ibrahim Hanafy ◽  
Ahmed Hagag
Keyword(s):  
Fractional Order ◽  
Space Time ◽  
Substitution Method ◽  
Partial Derivatives

scholarly journals On the Question of Asymptotic Integration of Singularly Perturbed Fractional-Order Problems

2018 ◽  
Vol 6 (3) ◽  
Author(s):  
Burkhan Kalimbetov
Keyword(s):  
Differential Equations ◽  
Small Parameter ◽  
Fractional Order ◽  
Initial Problem ◽  
Singularly Perturbed ◽  
Asymptotic Integration ◽  
Partial Derivatives ◽  
Systems Of Differential Equations ◽  
Regularization Problem ◽  
Iterative Systems

In this paper we consider an initial problem for systems of differential equations of fractional order with a small parameter for the derivative. Regularization problem is produced, and algorithm for normal and unique solubility of general iterative systems of differential equations with partial derivatives is given. 

scholarly journals Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Tarikul Islam ◽  
Armina Akter
Keyword(s):  
Fractional Order ◽  
High Performance ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Space Time ◽  
Nonlinear Evolution Equations ◽  
Content Type

PurposeFractional order nonlinear evolution equations (FNLEEs) pertaining to conformable fractional derivative are considered to be revealed for well-furnished analytic solutions due to their importance in the nature of real world. In this article, the autors suggest a productive technique, called the rational fractional (DξαG/G)-expansion method, to unravel the nonlinear space-time fractional potential Kadomtsev–Petviashvili (PKP) equation, the nonlinear space-time fractional Sharma–Tasso–Olver (STO) equation and the nonlinear space-time fractional Kolmogorov–Petrovskii–Piskunov (KPP) equation. A fractional complex transformation technique is used to convert the considered equations into the fractional order ordinary differential equation. Then the method is employed to make available their solutions. The constructed solutions in terms of trigonometric function, hyperbolic function and rational function are claimed to be fresh and further general in closed form. These solutions might play important roles to depict the complex physical phenomena arise in physics, mathematical physics and engineering.Design/methodology/approachThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is of the form U(ξ)=∑i=0nai(DξαG/G)i/∑i=0nbi(DξαG/G)i.FindingsAchieved fresh and further abundant closed form traveling wave solutions to analyze the inner mechanisms of complex phenomenon in nature world which will bear a significant role in the of research and will be recorded in the literature.Originality/valueThe rational fractional (DξαG/G)-expansion method shows high performance and might be used as a strong tool to unravel any other FNLEEs. This method is newly established and productive.

scholarly journals Mathematical description of the bulk fluid flow and that of the contained impurity dispersion which uses Caputo or Riemann-Liouville fractional order partial derivatives is nonobjective

2020 ◽  
Vol 12 (3) ◽  
pp. 17-31
Author(s):  
Agneta M. BALINT ◽  
Stefan BALINT
Keyword(s):  
Fluid Flow ◽  
Integral Representation ◽  
Fractional Order ◽  
Mathematical Description ◽  
Reference Frames ◽  
Finite Interval ◽  
Bulk Fluid ◽  
Fractional Order Derivative ◽  
Partial Derivatives ◽  
Impurity Dispersion

In this paper it is shown that the mathematical description of a Newtonian, incompressible, viscous bulk fluid flow and that of the contained impurity dispersion which uses Caputo or Riemann-Liouville fractional order derivative, having integral representation on finite interval, is nonobjective. This means that, two different observers describing the flow or the contained impurity dispersion with these tools obtain two different results which cannot be reconciled i.e. transformed into each other using only formulas that link the coordinates of a point in two fixed orthogonal reference frames and formulas that link the numbers representing a moment of time in two different choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: which of the obtained results is correct?

Mathematical Description of Elastic Phenomena which Uses Caputo or Riemann-Liouville Fractional Order Partial Derivatives is Nonobjective

2020 ◽  
Author(s):  
Agneta M. Balint ◽  
Stefan Balint ◽  
Silviu Birauas
Keyword(s):  
Integral Representation ◽  
Fractional Order ◽  
Mathematical Description ◽  
Reference Frames ◽  
Finite Interval ◽  
Constitutive Law ◽  
Partial Derivatives ◽  
Integer Order ◽  
Direct Replacement

In this paper it is shown that mathematical description of strain, constitutive law and dynamics obtained by direct replacement of integer order derivatives with Caputo or Riemann-Liouville fractional order partial derivatives, having integral representation on finite interval, in case of a guitar string, is nonobjective. The basic idea is that different observers, using this type of descriptions, obtain different results which cannot be reconciled, i.e. transformed into each other using only formulas that link the coordinates of the same point in two fixed orthogonal reference frames and formulas that link the numbers representing the same moment of time in two different choices of the origin of time measuring. This is not an academic curiosity! It is rather a problem: which one of the obtained results is correct?

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