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A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations
(Amer Inst Mathematical Sciences-Aims, 2022) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Jarad, Fahd
It is important to deal with the exact solution of nonlinear PDEs of non-integer orders. Integral transforms play a vital role in solving differential equations of integer and fractional orders. 'o obtain analytical solutions to integer and fractional-order DEs, a few transforms, such as Laplace transforms, Sumudu transforms, and Elzaki transforms, have been widely used by researchers. We propose the Yang transform homotopy perturbation (YTHP) technique in this paper. We present the relation of Yang transform (YT) with the Laplace transform. We find a formula for the YT of fractional derivative in Caputo sense. We deduce a procedure for computing the solution of fractional-order nonlinear PDEs involving the power-law kernel. We show the convergence and error estimate of the suggested method. We give some examples to illustrate the novel method. We provide a comparison between the approximate solution and exact solution through tables and graphs.
A Novel Homotopy Perturbation Method with Applications to Nonlinear Fractional Order KdV and Burger Equation with Exponential-Decay Kernel
(Hindawi Ltd, 2021) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; De la Sen, Manuel
In this paper, we introduce the Yang transform homotopy perturbation method (YTHPM), which is a novel method. We provide formulae for the Yang transform of Caputo-Fabrizio fractional order derivatives. We derive an algorithm for the solution of Caputo-Fabrizio (CF) fractional order partial differential equation in series form and show its convergence to the exact solution. To demonstrate the novel approach, we include some examples with detailed solutions. We use tables and graphs to compare the exact and approximate solutions.
A novel method for analysing the fractal fractional integrator circuit
(Elsevier, 2021) Akgul, Ali; Ahmad, Shabir; Ullah, Aman; Baleanu, Dumitru; Akgul, Esra Karatas
In this article, we propose the integrator circuit model by the fractal-fractional operator in which fractional-order has taken in the Atangana-Baleanu sense. Through Schauder's fixed point theorem, we establish existence theory to ensure that the model posses at least one solution and via Banach fixed theorem, we guarantee that the proposed model has a unique solution. We derive the results for Ulam-Hyres stability by mean of non-linear functional analysis which shows that the proposed model is Ulam-Hyres stable under the new fractal-fractional derivative. We establish the numerical results of the model under consideration through Atanaga-Toufik method. We simulate the numerical results for different sets of fractional order and fractal dimension.(C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
A Quantitative Approach to nth -Order Nonlinear Fuzzy Integro-Differential Equation
(Springer, 2022) Ul Haq, Mansoor; Ullah, Aman; Ahmad, Shabir; Akgül, Ali
In recent decades, both the fuzzy differential and fuzzy integral equations have attracted the researcher because the fuzzy operators produce appropriate predictions of problems in an uncertain environment. In this paper, we use fuzzy concepts to study nth-order nonlinear integro-differential equations. For the proposed problem, through the modified fuzzy Laplace transform method, we derive the general procedure of the solution. To demonstrate the accuracy and appropriateness of the method, we present some numerical problems. We also provide graphical representation by the use of Matlab 2017 to compare the exact and approximate solution. We solve different problems having second-order, fifth-order, and a system of nonlinear fuzzy integro-differential equations through the developed scheme. We simulate the numerical results via 2D and 3D graphs for the different values of uncertainty. In the end, we provide the discussion and concluding remarks of the article. © 2022, The Author(s), under exclusive licence to Springer Nature India Private Limited.
A study of fractional order Ambartsumian equation involving exponential decay kernel
(Amer Inst Mathematical Sciences-Aims, 2021) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; De la Sen, Manuel
Recently, non-singular fractional operators have a significant role in the modeling of real-world problems. Specifically, the Caputo-Fabrizio operators are used to study better dynamics of memory processes. In this paper, under the non-singular fractional operator with exponential decay kernel, we analyze the Ambartsumian equation qualitatively and computationally. We deduce the result of the existence of at least one solution to the proposed equation through Krasnoselskii's fixed point theorem. Also, we utilize the Banach fixed point theorem to derive the result concerned with unique solution. We use the concept of functional analysis to show that the proposed equation is Ulam-Hyers and Ulam-Hyers-Rassias stable. We use an efficient analytical approach to compute a semi-analytical solution to the proposed problem. The convergence of the series solution to an exact solution is proved through non-linear analysis. Lastly, we present the solution for different fractional orders.
Analysis of a TB and HIV co-infection model under Mittag-Leffler fractal-fractional derivative
(Iop Publishing Ltd, 2022) Liu, Xuan; Ahmad, Shabir; Rahman, Mati Ur; Nadeem, Yasir; Akgul, Ali
In this paper, the nonlocal operator with the Mittag-Leffler kernel is used to analyze a TB-HIV co-infection model with recurrent TB and exogenous reinfection. The non-negative invariant region and basic reproduction number of the proposed model are demonstrated. By using the Krasnoselskii fixed result, we investigate that the TB-HIV co-infection model possesses at least one solution. We look at the existence of a unique solution using Banach's fixed point theorem. Functional analysis is used to demonstrate Ulam-Hyres stability. The numerical solution of the given model is obtained using the Adams-Bashforth technique. We illustrate the achieved results by studying the co-infection of TB and HIV for different fractional and fractal orders.
Analysis of fractal fractional Lorenz type and financial chaotic systems with exponential decay kernels
(Amer Inst Mathematical Sciences-Aims, 2022) Ul Haq, Ihtisham; Ahmad, Shabir; Saifullah, Sayed; Nonlaopon, Kamsing; Akgul, Ali
In this work, we formulate a fractal fractional chaotic system with cubic and quadratic nonlinearities. A fractal fractional chaotic Lorenz type and financial systems are studied using the Caputo Fabrizo (CF) fractal fractional derivative. This study focuses on the characterization of the chaotic nature, and the effects of the fractal fractional-order derivative in the CF sense on the evolution and behavior of each proposed systems. The stability of the equilibrium points for the both systems are investigated using the Routh-Hurwitz criterion. The numerical scheme, which includes the discretization of the CF fractal-fractional derivative, is used to depict the phase portraits of the fractal fractional chaotic Lorenz system and the fractal fractional-order financial system. The simulation results presented in both cases include the two- and three-dimensional phase portraits to evaluate the applications of the proposed operators.
Analysis of fractal-fractional model of tumor-immune interaction
(Elsevier, 2021) Ahmad, Shabir; Ullah, Aman; Abdeljawad, Thabet; Akgul, Ali; Mlaiki, Nabil
Recently, Atangana proposed new operators by combining the fractional and fractal calculus. These recently proposed operators, referred to as fractal-fractional operators, have been widely used to study the complex dynamics of a problem. Cancer is a prevalent disease today and is difficult to cure. The immune system tends to fight it as cancer sets up in the body. In this manuscript, the novel operators have been used to analyze the relationship between the immune system and cancer cells. The tumor-immune model has been studied qualitatively and quantitatively via Atangana-Baleanu fractal-fractional operator. The existence and uniqueness results of the model under Atangana-Baleanu fractal-fractional operator have proved through fixed point theorems. The Ulam-Hyres stability for the model has derived through non-linear analysis. Numerical results have developed through Lagrangian-piece wise interpolation for the different fractal-fractional operators. To visualize the relationship between immune cells and cancers cells under novel operators in a various sense, we simulate the numerical results for the different sets of fractional and fractal orders.
ANALYSIS OF HIDDEN ATTRACTORS OF NON-EQUILIBRIUM FRACTAL-FRACTIONAL CHAOTIC SYSTEM WITH ONE SIGNUM FUNCTION
(World Scientific Publ Co Pte Ltd, 2022) Zhang, Lei; Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Akgul, Esra Karatas
In 2017, Atangana proposed more generalized operators depending on two parameters: one is fractional order (FO) and other is fractal dimension. The novel operators are defined with three different kernels. These operators produced excellent dynamics of the chaotic systems. In this paper, the Caputo fractal-fractional operator is used to explore a chaotic system which contains only one signum function. The existence theory is developed by using the fixed-point result of Leray-Schauder to prove that the considered chaotic system possesses at least one solution. The proposed chaotic system has a unique solution, according to Banach's fixed-point theorem. We demonstrate that the suggested chaotic system is Ulam-Hyres (UH) stable under the novel operator of power law kernel by employing nonlinear functional analysis. The Adams-Bashforth technique is used to evaluate the numerical outcomes of the considered model. We show the complex structure of numerical solutions for different FO and fractal dimension values.
Analysis of Kink Behaviour of KdV-mKdV Equation under Caputo Fractional Operator with Non-Singular Kernel
(Mdpi, 2022) Ali, Sajjad; Ullah, Aman; Ahmad, Shabir; Nonlaopon, Kamsing; Akgul, Ali
The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, which is a modified and generalized form of the ordinary KdV equation. We use the fractional operator in the Caputo sense to analyze the equation. We examine some theoretical results concerned with the solution's existence, uniqueness, and stability. We employ a modified Laplace method to extract the numerical results of the considered equation. We use MATLAB-2020 to simulate the results in a few fractional orders. We report the effects of the fractional order on the wave dynamics of the proposed equation.
Analysis of the fractional tumour-immune-vitamins model with Mittag-Leffler kernel
(Elsevier, 2020) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Baleanu, Dumitru
Recently, Atangana-Baleanu fractional derivative has got much attention of the researchers due to its nonlocality and non-singularity. This operator contains an accurate kernel that describes the better dynamics of systems with a memory effect. In this paper, we investigate the fractional-order tumour-immune-vitamin model (TIVM) under Mittag-Leffler derivative. The existence of at least one solution and a unique solution has discussed through fixed point results. We established the Hyres-Ulam stability of the proposed model under the Mittag-Leffler derivative. The fractional Adams-Bashforth method has used to achieve numerical results. Finally, we simulate the obtained numerical results for different fractional orders to show the effect of vitamin intervention on decreased tumour cell growth and cancer risk. At the end of the paper, the conclusion has provided.
Analysis of Time Fractional Diffusion Equation Arising in Ocean Pollution with Different Kernels
(Springer, 2023) Ullah, Inayat; Ullah, Aman; Ahmad, Shabir; Ikramullah; Akgül, Ali
The objective of this paper is to look the solution of the ocean oil equations under the three different fractional operators. We analyze the fractional ocean oil equation in one dimension using the Caputo fractional derivative. Then, using the Caputo–Fabrizio derivative, we investigate the same ocean oil equation. Finally, the Atangana–Baleanu derivative is applied to the same problem. In comparison to other analytical approaches, the Laplace transform (LT) is an easy and efficient method which has a good convergence rate for the precise solution. As a result, we employ the LT to achieve the suggested equation’s series solution. To explore the efficiency and validity of the suggested method, we present two examples of the provided equation. The error analysis of is carried out through computationally and graphically. The comparison between different Caputo, CF and ABC ocean oil equation is provided through numeric data and graphs. Finally, we offer a conclusion as well as a physical explanation of the figures. © 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited.
Bifurcation behavior and PDcontrol mechanism of a fractional delayed genetic regulatory model
(Pergamon-Elsevier Science Ltd, 2023) Li, Peiluan; Gao, Rong; Xu, Changjin; Ahmad, Shabir; Li, Ying; Akgul, Ali
It is crucial for us to build genetic regulatory models to reveal the relationship between genes and protein efficaciously. In this work, a novel fractional delayed genetic regulatory model is built. For one thing, we explore the peculiarity of the solution of the fractional delayed genetic regulatory model. Some sufficient conditions on the existence and uniqueness, non-negativeness and boundedness of the solution to the fractional delayed genetic regulatory model are acquired. Secondly, the stability trait and the appearance of bifurcation have been discussed at length. Thirdly, by virtue of PD ?????? controller, we availably dominate the time of appearance of bifurcation of the fractional delayed genetic regulatory model. Finally, simulation plots are displayed to sustain the acquired crucial results. The acquired conclusions in this study are entirely innovative and own great potential value in dominating the balance of genes and protein in genetic regulatory models.
Bifurcations, stability analysis and complex dynamics of Caputo fractal-fractional cancer model
(Pergamon-Elsevier Science Ltd, 2022) Xuan, Liu; Ahmad, Shabir; Ullah, Aman; Saifullah, Sayed; Akguel, Ali; Qu, Haidong
The association of cancer and immune cells has complex nature and produces chaotic behavior when it is simulated. The newly introduced operators which combine the fractal and fractional operators produce excellent and profound hidden attractors in a chaotic system which is sometimes not possible to get hidden attractors using integer order operators. The cancer model is considered under fractal fractional operator in Caputo sense. Linear stability of different equilibrium points is analyzed. The primary objective of the current paper is to analyze different bifurcations like pitch-fork, quasi, and inverse period-doubling bifurcations. Another important objective of this article is to study hidden limit cycle type chaotic structures of the cancer model via Caputo fractalfractional operator. The existence and uniqueness of the solution and Ulam-Hyres (UH) stability are studied through the concepts of nonlinear analysis. The numerical solution is derived through the predictor-corrector method. The obtained results were presented and validated through numerical simulations. The lyapunov spectra of the state variables are presented through graphical illustration and table. Sensitivity of the state variables to the initial conditions are simulated for initial conditions 0.1 and 0.11. For various values of fractal dimensions and fractional orders, the time series oscillations and hidden limit cycles type chaotic attractors are graphically presented through MATLAB-17.
Bright Soliton Behaviours of Fractal Fractional Nonlinear Good Boussinesq Equation with Nonsingular Kernels
(Mdpi, 2022) Sadiq, Gulaly; Ali, Amir; Ahmad, Shabir; Nonlaopon, Kamsing; Akgul, Ali
In this manuscript, we investigate the nonlinear Boussinesq equation (BEQ) under fractal-fractional derivatives in the sense of the Caputo-Fabrizio and Atangana-Baleanu operators. We use the double modified Laplace transform (LT) method to determine the general series solution of the Boussinesq equation. We study the convergence, existence, uniqueness, boundedness, and stability of the solution of the considered good BEQ under the aforementioned derivatives. The obtained solutions are presented with numerical illustrations considering a particular example by two cases based on both derivatives with suitable initial conditions. The results are illustrated graphically where good agreements are obtained. Our results show that fractal-fractional derivatives are a very effective tool for studying nonlinear systems. Furthermore, when t increases, the solitary waves of the system oscillate. As the fractional order a or fractal dimension beta increases, the soliton solutions become coherently close to the exact solution. For compactness, an error analysis is performed. The absolute error reveals an approximate linear evolution in the soliton solutions as time increases and that the system does not blow up nonlinearly.
Bright, dark and hybrid multistrip optical soliton solutions of a non-linear Schrodinger equation using modified extended tanh technique with new Riccati solutions
(Springer, 2023) Ahmad, Shafiq; Salman; Ullah, Aman; Ahmad, Shabir; Akgul, Ali
This paper focus on new optical soliton solutions of a nonlinear Schrodinger equation with self-frequency shift , and kerr nonlinearity terms. We utilize the extended version of modified tanh expansion technique associated with new Riccati solutions. We extract optical soliton solutions such as dark and bright optical solitons, breathers type, periodic and hybrid type optical soliton solutions. We display all the solutions with help of Mathematica in 2D and 3D graphs.
CHAOTIC BEHAVIOR OF BHALEKAR-GEJJI DYNAMICAL SYSTEM UNDER ATANGANA-BALEANU FRACTAL FRACTIONAL OPERATOR
(World Scientific Publ Co Pte Ltd, 2022) Ahmad, Shabir; Ullah, Aman; Akgul, Ali; Abdeljawad, Thabet
In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal-fractional derivatives in which fractional derivative is taken in Atangana-Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder's fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam-Hyers stable under the new fractal-fractional derivative. We establish the numerical results of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we study the chaos behavior of the proposed model via simulation at 2D and 3D phase. We show the effect of fractal dimension on integer and fractional order through simulations.
Complex dynamics of multi strain TB model under nonlocal and nonsingular fractal fractional operator
(Elsevier, 2021) Adnan; Ahmad, Shabir; Ullah, Aman; Riaz, Muhammad Bilal; Ali, Amir; Akgul, Ali; Partohaghighi, Mohammad
Researchers have recently begun to use fractal fractional operators in the Atangana-Baleanu sense to analyze complicated dynamics of various models in applied sciences, as the Atangana-Baleanu operator generalizes the integer and fractional order operators. To analyze the complex dynamics of the multi-strain TB model, we use the AB-fractal fractional operator. We use the Banach fixed point theorem to ensure that at most one solution exists to the model. Further, the Ulam-Hyers type stability of the model is investigated with the help of functional analysis. The Adams-Bashforth approach is used to get numerical results for the proposed model. The analysis of the chaotic behavior of the proposed TB model was missing in the literature. Therefore, for different values of fractional and fractal order, we study the nonlinear dynamics and chaotic behavior of the obtained results of the proposed model.
Computational analysis of fuzzy fractional order non-dimensional Fisher equation
(Iop Publishing Ltd, 2021) Ahmad, Shabir; Ullah, Aman; Ullah, Abd; Akgul, Ali; Abdeljawad, Thabet
In recent decades, fuzzy differential equations of integer and arbitrary order are extensively used for analyzing the dynamics of a mathematical model of the physical process because crisp operators of integer and arbitrary order are not able to study the model being studied when there is uncertainty in values used in modeling. In this article, we have considered the time-fractional Fisher equation in a fuzzy environment. The basic aim of this article is to deduce a semi-analytical solution to the fuzzy fractional-order non-dimensional model of the Fisher equation. Since the Laplace-Adomian method has a good convergence rate. We use the Laplace- Adomian decomposition method (LADM) to determine a solution under a fuzzy concept in parametric form. We discuss the convergence and error analysis of the proposed method. For the validity of the proposed scheme, we provide few examples with detailed solutions. We provide comparisons between exact and approximate solutions through graphs. In the end, the conclusion of the paper is provided.
Computational analysis of the third order dispersive fractional PDE under exponential-decay and Mittag-Leffler type kernels
(Wiley, 2023) Ahmad, Shabir; Ullah, Aman; Shah, Kamal; Akgul, Ali
This article aims to investigate the fractional dispersive partial differential equations (FPDEs) under non-singular and non-local kernels. First, we study the fractional dispersive equations under the Caputo-Fabrizio fractional derivative in one and higher dimension. Second, we investigate the same equations under the Atangana-Baleanu derivative. The Laplace transform has an excellent convergence rate for the exact solution as compared to the other analytical methods. Therefore, we use Laplace transform to obtain the series solution of the proposed equations. We provide two examples of each equation to confirm the validity of the proposed scheme. The results and simulations of examples show higher convergence of the fractional-order solution to the integer-order solution. In the end, we provide the conclusion and physical interpretation of the figures.

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