Exact Solution of Riccati Fractional Differential Equation

We deal with a singular nonlocal fractional differential equation with Riemann-Stieltjes integral conditions. The exact iterative solution is established under the iterative technique. The iterative sequences have been proved to converge uniformly to the exact solution, and estimation of the approximation error and the convergence rate have been derived. An example is also given to demonstrate the results.

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Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Mathematics ◽

10.3390/math8122231 ◽

2020 ◽

Vol 8 (12) ◽

pp. 2231

Author(s):

Vasily E. Tarasov

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Fractional Derivative ◽

Fractional Differential Equation ◽

Power Law ◽

Dynamic Processes ◽

Nonlinear Fractional Differential Equation ◽

First Order ◽

Fractional Differential ◽

Special Case

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

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DGJ Method for Linear and Nonlinear Third order Fractional Differential Equation

10.20944/preprints201812.0134.v1 ◽

2018 ◽

Author(s):

Mahmut Modanli

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Fractional Differential Equation ◽

Initial Boundary ◽

Third Order ◽

Partial Differential ◽

Approximation Solution ◽

The Third ◽

Numerical Result ◽

Fractional Differential

DGJ (Daftardar-Gejii-Jafaris) method is used to obtain numerical solution of the third order fractional differential equation. Providing the DGJ method converges, the approximate solution is a good and effective numerical result which is close to the exact solution or the exact solution. For this,the examples of the explaning the method are presented. The proposed method is implemented for the approximation solution of the third order nonlinear fractional partial differential equations. The method was shown to be unsuitable and inconsistent for an example of a nonlinear fractional partial differential equation depend on initial-boundary value conditions. The fact that these numerical results are not consistent can be explained by the fact that the method is not convergent.

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Exact Solution of Two-Term Nonlinear Fractional Differential Equation with Sequential Riemann-Liouville Derivatives

Lecture Notes in Electrical Engineering - Advances in the Theory and Applications of Non-integer Order Systems ◽

10.1007/978-3-319-00933-9_14 ◽

2013 ◽

pp. 161-170 ◽

Cited By ~ 3

Author(s):

Marek Błasik ◽

Małgorzata Klimek

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Fractional Differential Equation ◽

Nonlinear Fractional Differential Equation ◽

Fractional Differential

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Convergence Analysis from the Solution of Riccati’s Fractional Differential Equation by Using Polynomial Least Squares Method

EKSAKTA Berkala Ilmiah Bidang MIPA ◽

10.24036/eksakta/vol21-iss1/211 ◽

2020 ◽

Vol 21 (1) ◽

pp. 7-14

Author(s):

Dian Nuryani ◽

Endang Rusyaman ◽

Betty Subartini

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Least Squares ◽

Fractional Differential Equation ◽

Least Squares Method ◽

Approximate Solutions ◽

Numerical Approach ◽

Percentage Error ◽

Mathematical Form ◽

Fractional Differential

Riccati's Fractional Differential Equation (RFDE) has become a topic of study for researchers because RFDE can model variety of phenomenon in science such as random processes, optimal control and diffusion problems. Phenomena that can be modeled in a mathematical form can make it easier for humans to analyze several things from that phenomenon. RFDE generally does not have an exact solution, therefore a numerical approach solution is needed, one of the methods that gives good accuracy to the actual or exact solution is Polynomial Least Squares, where the errors calculated based on mean absolute percentage error (MAPE) produce a percentage below 1%. In addition, the convergence of a sequence from approximate solutions indicates that the sequence will converge to a solution.

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Analytical solutions for conformable fractional Bratu-type equations

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v7i1.8849 ◽

2018 ◽

Vol 7 (1) ◽

pp. 15 ◽

Cited By ~ 4

Author(s):

Mousa Ilie ◽

Jafar Biazar ◽

Zainab Ayati

Keyword(s):

Differential Equation ◽

Exact Solution ◽

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equation ◽

Fractional Differential Equations ◽

Analytical Solutions ◽

Analytical Method ◽

Conformable Fractional Derivative ◽

Fractional Differential

Solving fractional differential equations have a prominent function in different science such as physics and engineering. Therefore, are different definitions of the fractional derivative presented in recent years. The aim of the current paper is to solve the fractional differential equation by a semi-analytical method based on conformable fractional derivative. Fractional Bratu-type equations have been solved by the method and to show its capabilities. The obtained results have been compared with the exact solution.

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