162 research outputs found

    Solution of Homogeneous Linear Fractional Differential Equations Involving Conformable Fractional Derivative

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    In this paper, we have found the solution of linear sequential fractional differential equations involving conformable fractional derivatives of order  with constant coefficients. For this purpose, we first discussed fundamental properties of the conformable derivative and then obtained successive conformable derivatives of the fractional exponential function. After this, we determined the analytic solution of linear sequential fractional differential equations (L.S.F.D.E.) in terms of a fractional exponential function. We have demonstrated this developed method with a few examples of homogeneous linear fractional differential equations. This method gives a conjugation with the method to solve classical linear differential equations with constant coefficients

    Abundant Exact Soliton Solutions to the Space-Time Fractional Phi-Four Effective Model for Quantum Effects Through the Modern Scheme

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    The space-time fractional Phi-four (PF) equation is measured as a particular case of the familiar Klein-Fock-Gordon (KFG) model and plentiful quantum effects can be investigated through the PF model’s solutions. In this article, the auxiliary equation method (AEM) is employed to attain the traveling wave solutions and in this purpose, the complex wave transformation and Maple software are utilized. The constructed wave solutions are the form likely, hyperbolic, exponential, rational, and trigonometric functions as well as their integration. The physical significance of the obtained solutions for the specific values of the integrated parameters in the course of representing graphs and understood the physical phenomena. It is shown that the AEM is powerful, effective and simple and provide more general traveling wave solutions to the NLEEs

    General Closed Form Wave Solutions of Nonlinear Space-Time Fractional Differential Equation in Nonlinear Science

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    We have enucleated new and further exact general wave solutions, along with multiple exact traveling wave solutions of space-time nonlinear fractional Chan-Hillard equation, by applying a relatively renewed technique two variables -expansion method. Also, based on fractional complex transformation and the properties of the modified Riemann-Liouville fractional order operator, the fractional partial differential equations are transforming into the form of ordinary differential equation. This method can be rumination of as the commutation of well-appointed -expansion method introduced by M. Wang et al.. In this paper, it is mentioned that the two variables - expansion method is more legitimate, modest, sturdy and effective in the sense of theoretical and pragmatical point of view. Lastly, by treating computer symbolic program Mathematica, the uniqueness of our attained wave solutions are represented graphically and reveal a comparison in a submissive manner

    New structure for exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation via conformable fractional derivative

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    In this paper new fractional derivative and direct algebraic method are used to construct exact solutions of the nonlinear time fractional Sharma-Tasso-Olver equation. As a result, three families of exact analytical solutions are obtained. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations

    I. Complete and orthonormal sets of exponential-type orbitals with noninteger principal quantum numbers

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    The definition for the Slater-type orbitals is generalized. Transformation between an orthonormal basis function and the Slater-type orbital with non-integer principal quantum numbers is investigated. Analytical expressions for the linear combination coefficients are derived. In order to test the accuracy of the formulas, the numerical Gram-Schmidt procedure is performed for the non-integer Slater-type orbitals. A closed form expression for the orthogonalized Slater-type orbitals is achieved. It is used to generalize complete orthonormal sets of exponential-type orbitals obtained by Guseinov in [Int. J. Quant. Chem. 90, 114 (2002)] to non-integer values of principal quantum numbers. Riemann-Liouville type fractional calculus operators are considered to be use in atomic and molecular physics. It is shown that the relativistic molecular auxiliary functions and their analytical solutions for positive real values of parameters on arbitrary range are the natural Riemann-Liouville type fractional operators

    Abundant optical soliton solutions for an integrable (2+1)-dimensional nonlinear conformable Schrödinger system

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    Abstract The analytical solutions of the integrable generalized ( 2 + 1 ) -dimensional nonlinear conformable Schrodinger (NLCS) system of equations was explored in this paper with the aid of three novel techniques which consist of ( G ′ / G ) -expansion method, generalized Riccati equation mapping method and the Kudryashov method in the conformable sense. We have discovered a new and more general variety of exact traveling wave solutions by using the proposed methods with a variety of soliton solutions of several structures. With several plots illustrating the behavior of dynamic shapes of the solutions, the findings are highly applicable and detailed the physical dynamic of the considered nonlinear system

    SOLITARY WAVE SOLUTIONS FOR SPACE-TIME FRACTIONAL COUPLED INTEGRABLE DISPERSIONLESS SYSTEM VIA GENERALIZED KUDRYASHOV METHOD

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    In this article, space-time fractional coupled integrable dispersionless system is considered, and we use fractional derivative in the sense of modified Riemann-Liouville. The fractional system has reduced to an ordinary differential system by fractional transformation and the generalized Kudryashov method is applied to obtain exact solutions. We also testify performance as well as precision of the applied method by means of numerical tests for obtaining solutions. The obtained results have been graphically presented to show the properties of the solutions
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