Fractional System of Korteweg-De Vries Equations via Elzaki Transform

In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable.

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Novel Computations of the Time-Fractional Fisher’s Model via Generalized Fractional Integral Operators by Means of the Elzaki Transform

Fractal and Fractional ◽

10.3390/fractalfract5030094 ◽

2021 ◽

Vol 5 (3) ◽

pp. 94

Author(s):

Saima Rashid ◽

Zakia Hammouch ◽

Hassen Aydi ◽

Abdulaziz Garba Ahmad ◽

Abdullah M. Alsharif

Keyword(s):

Fractional Order ◽

Fixed Point Theory ◽

Series Representation ◽

Point Theory ◽

Hybrid Technique ◽

Transform Method ◽

Fractional Integral Operators ◽

Generalized Fractional Integral ◽

In Series ◽

Strongly Agree

The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics and genetics, specifically in chemistry. The Caputo and Antagana-Baleanu fractional derivatives in the Caputo sense are used to test the intricacies of this mechanism numerically. In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. Moreover, the stability of the approach was demonstrated using fixed point theory. Several illustrative cases are described that strongly agree with the precise solutions. Moreover, tables and graphs are included in order to conceptualize the influence of the fractional order and on the previous findings. The projected technique illustrates that only a few terms are sufficient for finding an approximate outcome, which is computationally appealing and accurate to analyze. Additionally, the offered procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, but it could be generalized to other complex physical phenomena.

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An efficient approach for solution of fractional-order Helmholtz equations

Advances in Difference Equations ◽

10.1186/s13662-020-03167-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Nehad Ali Shah ◽

Essam R. El-Zahar ◽

Mona D. Aljoufi ◽

Jae Dong Chung

Keyword(s):

Perturbation Method ◽

Fractional Order ◽

Homotopy Perturbation Method ◽

Hybrid Technique ◽

Science And Engineering ◽

Helmholtz Equations ◽

Transform Method ◽

Homotopy Perturbation ◽

Suggested Technique ◽

Present Technique

AbstractIn this article, a hybrid technique called the homotopy perturbation Elzaki transform method has been implemented to solve fractional-order Helmholtz equations. In the hybrid technique, the Elzaki transform method and the homotopy perturbation method are amalgamated. Three problems are solved to validate and demonstrate the efficacy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the results by other techniques. It is shown that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.

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Aboodh Transform Iterative Method for Spatial Diffusion of a Biological Population with Fractional-Order

Mathematics ◽

10.3390/math9020155 ◽

2021 ◽

Vol 9 (2) ◽

pp. 155

Author(s):

Gbenga O. Ojo ◽

Nazim I. Mahmudov

Keyword(s):

Iterative Method ◽

Fractional Order ◽

Population Model ◽

Numerical Solutions ◽

Computational Effort ◽

Spatial Diffusion ◽

Transform Method ◽

Biological Population ◽

The Laplace Transform ◽

New Iterative Method

In this paper, a new approximate analytical method is proposed for solving the fractional biological population model, the fractional derivative is described in the Caputo sense. This method is based upon the Aboodh transform method and the new iterative method, the Aboodh transform is a modification of the Laplace transform. Illustrative cases are considered and the comparison between exact solutions and numerical solutions are considered for different values of alpha. Furthermore, the surface plots are provided in order to understand the effect of the fractional order. The advantage of this method is that it is efficient, precise, and easy to implement with less computational effort.

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New numerical solutions of fractional-order Korteweg-de Vries equation

Results in Physics ◽

10.1016/j.rinp.2020.103326 ◽

2020 ◽

Vol 19 ◽

pp. 103326

Author(s):

Mustafa Inc ◽

Mohammad Parto-Haghighi ◽

Mehmet Ali Akinlar ◽

Yu-Ming Chu

Keyword(s):

Fractional Order ◽

Numerical Solutions ◽

Vries Equation ◽

De Vries

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Analytical Solution of Two-Dimensional Sine-Gordon Equation

Advances in Mathematical Physics ◽

10.1155/2021/6610021 ◽

2021 ◽

Vol 2021 ◽

pp. 1-15

Author(s):

Alemayehu Tamirie Deresse ◽

Yesuf Obsie Mussa ◽

Ademe Kebede Gizaw

Keyword(s):

Numerical Solutions ◽

Initial Conditions ◽

Nonlinear Partial Differential Equations ◽

Test Problems ◽

Two Dimensional ◽

Sine Gordon Equation ◽

Science And Engineering ◽

Transform Method ◽

Differential Transform ◽

Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

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A Novel Analytical View of Time-Fractional Korteweg-De Vries Equations via a New Integral Transform

Symmetry ◽

10.3390/sym13071254 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1254

Author(s):

Saima Rashid ◽

Aasma Khalid ◽

Sobia Sultana ◽

Zakia Hammouch ◽

Rasool Shah ◽

...

Keyword(s):

Nonlinear Waves ◽

Analytical Procedure ◽

Integral Transform ◽

Kdv Equation ◽

Analytical Techniques ◽

Transform Method ◽

Fractional Pdes ◽

Approximate Analytical Solutions ◽

The Kdv Equation ◽

De Vries

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs.

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New Iterative Method for the Solution of Fractional Damped Burger and Fractional Sharma-Tasso-Olver Equations

Complexity ◽

10.1155/2018/3249720 ◽

2018 ◽

Vol 2018 ◽

pp. 1-7

Author(s):

Mohammad Jibran Khan ◽

Rashid Nawaz ◽

Samreen Farid ◽

Javed Iqbal

Keyword(s):

Exact Solution ◽

Iterative Method ◽

Fractional Order ◽

Approximate Solutions ◽

Differential Transform Method ◽

Transform Method ◽

Order Solution ◽

Differential Transform ◽

New Iterative Method ◽

Good Agreement

The new iterative method has been used to obtain the approximate solutions of time fractional damped Burger and time fractional Sharma-Tasso-Olver equations. Results obtained by the proposed method for different fractional-order derivatives are compared with those obtained by the fractional reduced differential transform method (FRDTM). The 2nd-order approximate solutions by the new iterative method are in good agreement with the exact solution as compared to the 5th-order solution by the FRDTM.

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The Exponential Cubic B-Spline Algorithm for Korteweg-de Vries Equation

Advances in Numerical Analysis ◽

10.1155/2015/367056 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8 ◽

Cited By ~ 7

Author(s):

Ozlem Ersoy ◽

Idris Dag

Keyword(s):

Numerical Experiments ◽

Numerical Solutions ◽

Vries Equation ◽

Kdv Equation ◽

Algebraic Equations ◽

Thomas Algorithm ◽

B Spline ◽

Suggested Algorithm ◽

De Vries

The exponential cubic B-spline algorithm is presented to find the numerical solutions of the Korteweg-de Vries (KdV) equation. The problem is reduced to a system of algebraic equations, which is solved by using a variant of Thomas algorithm. Numerical experiments are carried out to demonstrate the efficiency of the suggested algorithm.

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KdV Equation Model in Open Cylindrical Channel under Precession

Journal of Nonlinear Mathematical Physics ◽

10.1007/s44198-021-00007-8 ◽

2021 ◽

Vol 28 (4) ◽

pp. 466-491

Author(s):

Hajar Alshoufi

Keyword(s):

Vries Equation ◽

Kdv Equation ◽

Additional Term ◽

Vertical Axis ◽

Resonant Mode ◽

Equation Model ◽

Concentric Cylinders ◽

De Vries ◽

Forced Term ◽

Good Agreement

AbstractA new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope $$\tau$$ τ from the inertial vertical $$z$$ z , in uniform rate $${\Omega }_{1}=\tau \Omega$$ Ω 1 = τ Ω , and the whole tank is elevated over other table rotating at rate $$\Omega$$ Ω . Under these conditions, a set of Kelvin waves is formed on the free surface depending on the angle of tilt, characterized by the slope $$\tau$$ τ , volume of water, and rotation rate. The resonant mode in the system appears in the form of a single Kelvin solitary wave, whose amplitude satisfies the Korteweg-de Vries equation with forced term. The equation was derived following classical perturbation methods, the additional term made the equation a non-integrable one, that cannot be solved without the help of numerical methods. Invoking the simple finite difference scheme method, it was found that the numerical results are in a good agreement with the experiment.

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Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

Fractal and Fractional ◽

10.3390/fractalfract5030131 ◽

2021 ◽

Vol 5 (3) ◽

pp. 131

Author(s):

Hari M. Srivastava ◽

Abedel-Karrem N. Alomari ◽

Khaled M. Saad ◽

Waleed M. Hamanah

Keyword(s):

Collocation Method ◽

Fractional Order ◽

Numerical Solutions ◽

Spectral Collocation Method ◽

Hybrid Technique ◽

Algebraic Equations ◽

Spectral Collocation ◽

Finite Differences Method ◽

Nonlinear Algebraic Equations ◽

Raphson Method

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

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