Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology

Kumar, Dipankar; Seadawy, Aly R.; Joardar, Atish Kumar

doi:10.1016/j.cjph.2017.11.020

Cited by 250 publications

(120 citation statements)

References 41 publications

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“…As we all know Khalil et al introduced the concept of conformable (local version) fractional derivative, which coincides with the standard (nonlocal version) fractional derivatives on polynomials up to a constant multiple and also can be used to characterize fractional Newton mechanics and the model in mathematical biology . In particular, local version fractional derivative is well behaved and obeys the Leibniz rule and chain rule, which has been proved not well kept for nonlocal version fractional derivative like Riemann‐Liouville and Caputo derivatives .…”

Section: Introductionmentioning

confidence: 99%

“…As we all know Khalil et al 1 introduced the concept of conformable (local version) fractional derivative, which coincides with the standard (nonlocal version) fractional derivatives 2 on polynomials up to a constant multiple and also can be used to characterize fractional Newton mechanics 3 and the model in mathematical biology. 4 In particular, local version fractional derivative is well behaved and obeys the Leibniz rule and chain rule, which has been proved not well kept for nonlocal version fractional derivative like Riemann-Liouville and Caputo derivatives. [5][6][7] Meanwhile, it is a natural extension of the usual derivative and can be widely used to establish chain rule, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions, Grünwald-Letnikov approach, and calculus of variations for conformable version fractional calculus (see, for example, Abdeljawad 8 ).…”

Section: Introductionmentioning

confidence: 99%

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Convergence analysis for iterative learning control of conformable fractional differential equations

Wang

Shen

et al. 2018

Math Methods in App Sciences

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This paper mainly deals with iterative learning control for the conformable fractional differential equations. The standard P‐type, Dα‐type, and conformable PIαDα‐type learning updating laws are proposed to derive the convergence results for nonlinear and linear problems varying with the initial state is (not) coincident with the desired initial state. Finally, numerical examples are given to illustrate the results.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence analysis for iterative learning control of conformable fractional differential equations

Wang

Shen

et al. 2018

Math Methods in App Sciences

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“…biotechnology [7], electrodynamics [8], and many other fields [9][10][11]. The solution for differential equations having order arbitrary describing above phenomena plays a pivotal part in labelling the behaviour of complex problems arises in nature.…”

mentioning

confidence: 99%

A reliable technique for fractional modified Boussinesq and approximate long wave equations

et al. 2019

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In this paper, an efficient technique is employed to study the modified Boussinesq and approximate long wave equations of the Caputo fractional time derivative, namely -homotopy analysis transform method. These equations are playing a vital rule in describing the properties of shallow water waves through distinct dispersion relation. The convergence analysis and error analysis has been presented in the present investigation for the future scheme. We illustrate two examples to demonstrate the leverage and effectiveness of the proposed scheme, and the error analysis has been discussed to verify the accuracy. The numerical simulation has been conducted to ensure the exactness of the future technique. The obtained numerical and graphical results are divulge, the proposed scheme is computationally very accurate and straightforward to study and find the solution for fractional coupled nonlinear complex phenomena arised in science and technology.

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“…So, various methods have been investigated for this purpose. Among others,some of them are [1,2,3,4,5,6,7]. In this study, we are going to concern with obtaining numerical solutions of time fractional Klein Gordon equation in terms of Caputo sense derivative which is one of the fundamental equations seen in fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation

Karaağaç¹,

Uçar²,

Yağmurlu³

et al. 2018

Journal of Polytechnic

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ÖZSunulan çalışmada, zamana göre kesirli mertebeden türevli lineer olmayan Klein Gordon denklemini çözmek için yeni bir nümerik şema sunuldu. Kesirli mertebeden denklemin yaklaşık çözümleri kübik B-spline kollokasyon sonlu eleman yöntemi ve L2 algortimasına dayanmaktadır. Denklemde verilen kesirli türev ise Caputo anlamında ele alınmıştır. Yöntemler kullanılarak, kesirli mertebeden diferansiyel denklem bilgisayar kodlamasına elverişli cebirsel denklem sistemine dönüştürülür. Daha sonra, amaçlanan yöntemin güvenilirliğini ve etkisini göstermek amacı ile iki model problem ele alındı ve hata normları hesaplandı. Yeni hesaplanan hata normları saysısal çözümlerin tam çözümlerle uyum içinde olduğunu göstermektedir.Anahtar Kelimeler: Sonlu eleman yöntemi, kollokasyon, kesirli mertebeden Klein Gordon denklemi, Caputo türevi. A New Perspective on The Numerical Solution forFractional Klein Gordon Equation ABSTRACTIn the present manuscript, a new numerical scheme is presented for solving the time fractional nonlinear Klein-Gordon equation. The approximate solutions of the fractional equation are based on cubic B-spline collocation finite element method and L2 algorithm. The fractional derivative in the given equation is handled in terms of Caputo sense. Using the methods, fractional differential equation is converted into algebraic equation system that are appropriate for computer coding. Then, two model problems are considered and their error norms are calculated to demonstrate the reliability and efficiency of the proposed method. The newly calculated error norms show that numerical results are in a good agreement with the exact solutions.

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Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology

Cited by 250 publications

References 41 publications

Convergence analysis for iterative learning control of conformable fractional differential equations

Convergence analysis for iterative learning control of conformable fractional differential equations

A reliable technique for fractional modified Boussinesq and approximate long wave equations

A New Perspective on The Numerical Solution for Fractional Klein Gordon Equation

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