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This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model... more
This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model by using the extended ShGEEM, rational sine-cosine extended (ERSC), and sinh-cosh (ERSCh) methods. In addition, a numerical method for solving the KGE is described in this paper. We use a combination of two numerical techniques called fictitious time integration method and the group preserving scheme (GPS). Fictitious time integration method converts the main equation into a new problem then the GPS is used to gain the numerical solutions. Few experiments are provided to successfully demonstrate the correctness of the approach.
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This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model... more
This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model by using the extended ShGEEM, rational sine-cosine extended (ERSC), and sinh-cosh (ERSCh) methods. In addition, a numerical method for solving the KGE is described in this paper. We use a combination of two numerical techniques called fictitious time integration method and the group preserving scheme (GPS). Fictitious time integration method converts the main equation into a new problem then the GPS is used to gain the numerical solutions. Few experiments are provided to successfully demonstrate the correctness of the approach.
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Hyper-chaotic systems have useful applications in engineering applications due to their complex dynamics and high sensitivity. This research is supposed to introduce and analyze a new noninteger hyper-chaotic system. To design its... more
Hyper-chaotic systems have useful applications in engineering applications due to their complex dynamics and high sensitivity. This research is supposed to introduce and analyze a new noninteger hyper-chaotic system. To design its fractional model, we consider the Caputo fractional operator. To obtain the approximate solutions of the extracted system under the considered fractional-order derivative, we employ an accurate nonstandard finite difference (NSFD) algorithm. Moreover, the existence and uniqueness of the solutions are provided using the theory of fixed-point. Also, to see the performance of the utilized numerical scheme, we choose different values of fractional orders along with various amounts of the initial conditions (ICs). Graphs of solutions for each case are provided.
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During this paper, a specific type of fractal-fractional diffusion equation is presented by employing the fractal-fractional operator. We present a reliable and accurate operational matrix approach using shifted Chebyshev cardinal... more
During this paper, a specific type of fractal-fractional diffusion equation is presented by employing the fractal-fractional operator. We present a reliable and accurate operational matrix approach using shifted Chebyshev cardinal functions to solve the considered problem. Also, an operational matrix for the considered derivative is obtained from basic functions. To solve the introduced problem, we convert the main equation into an algebraic system by extracting the operational matrix methods. Graphs of exact and approximate solutions along with error graphs are presented. These figures show how the introduced approach is reliable and accurate. Also, tables are established to illustrate the values of solutions and errors. Finally, a comparison of the solutions at a specific time is given for each test problem.
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Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical... more
Numerical methods play an important role in modern mathematical research, especially studying the symmetry analysis and obtaining the numerical solutions of fractional differential equation. In the current work, we use two numerical schemes to deal with fractional differential equations. In the first case, a combination of the group preserving scheme and fictitious time integration method (FTIM) is considered to solve the problem. Firstly, we applied the FTIM role, and then the GPS came to integrate the obtained new system using initial conditions. Figure and tables containing the solutions are provided. The tabulated numerical simulations are compared with the reproducing kernel Hilbert space method (RKHSM) as well as the exact solution. The methodology of RKHSM mainly relies on the right choice of the reproducing kernel functions. The results confirm that the FTIM finds the true solution. Additionally, these numerical results indicate the effectiveness of the proposed methods.
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In this paper, we present a powerful numerical scheme based on energy boundary functions to get the approximate solutions of the time-fractional inverse Burger equation containing HH-derivative.This problem has never been investigated... more
In this paper, we present a powerful numerical scheme based on energy boundary functions to get the approximate solutions of the time-fractional inverse Burger equation containing HH-derivative.This problem has never been investigated earlier so, this is our motivation to work on this important problem. Some numerical examples are presented to verify the efficiency of the presented technique. Graphs of the exact and numerical solutions along with the plot of absolute error are provided for each example. Tables are given to see and compare the results point by point for each example.
The SARS-CoV-2 virus pandemic remains a pressing issue with its unpredictable nature, and it spreads worldwide through human interaction. Current research focuses on the investigation and analysis of fractional epidemic models that... more
The SARS-CoV-2 virus pandemic remains a pressing issue with its unpredictable nature, and it spreads worldwide through human interaction. Current research focuses on the investigation and analysis of fractional epidemic models that discuss the temporal dynamics of the SARS-CoV-2 virus in the community. In this work, we choose a fractional-order mathematical model to examine the transmissibility in the community of several symptoms of COVID-19 in the sense of the Caputo operator. Sensitivity analysis of $ R_{0} $ and disease-free local stability of the system are checked. Also, with the assistance of fixed point theory, we demonstrate the existence and uniqueness of the system. In addition, numerically we solve the fractional model and presented some simulation results via actual estimation parameters. Graphically we displayed the effects of numerous model parameters and memory indexes. The numerical outcomes show the reliability, validation, and accuracy of the scheme.
In this study, we model the fractal-fractional system of the Computer virus problem using the Atangana–Baleanu operator. Moreover, we have presented the existence and the uniqueness of the results under applying the Schauder fixed point... more
In this study, we model the fractal-fractional system of the Computer virus problem using the Atangana–Baleanu operator. Moreover, we have presented the existence and the uniqueness of the results under applying the Schauder fixed point and Banach fixed theorems. We have used the Atangana–Toufik technique to obtain the approximate solutions by choosing various values of orders. Different values of fractal-fractional orders along with different amounts of initial conditions are selected to examine the performance of the suggested numerical method on the new fractal-fractional system. Also, graphs in different dimensions are presented to exhibit the solutions, clearly.
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This study develops a fractional model using the Caputo–Fabrizio derivative with order α for platelet-poor plasma arising in a blood coagulation system. The existence of solutions ensures that there are solutions to the considered system... more
This study develops a fractional model using the Caputo–Fabrizio derivative with order α for platelet-poor plasma arising in a blood coagulation system. The existence of solutions ensures that there are solutions to the considered system of equations. Approximate solutions to the recommended model are presented by selecting different numbers of fractional orders and initial conditions (ICs). For each case, graphs of solutions are supplied through different dimensions.
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The long head of the biceps tendon is widely recognized as an important pain generator, especially in anterior shoulder pain and dysfunction with athletes and working individuals. The purpose of this review is to provide a current... more
The long head of the biceps tendon is widely recognized as an important pain generator, especially in anterior shoulder pain and dysfunction with athletes and working individuals. The purpose of this review is to provide a current understanding of the long head of the biceps tendon anatomy and its surrounding structures, function, and relevant clinical information such as evaluation, treatment options, and complications in hopes of helping orthopaedic surgeons counsel their patients. An understanding of the long head of the biceps tendon anatomy and its surrounding structures is helpful to determine normal function as well as pathologic injuries that stem proximally. The biceps-labral complex has been identified and broken down into different regions that can further enhance a physician’s knowledge of common anterior shoulder pain etiologies. Although various physical examination maneuvers exist meant to localize the anterior shoulder pain, the lack of specificity requires orthopaed...
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The fractal–fractional derivative with the Mittag–Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover,... more
The fractal–fractional derivative with the Mittag–Leffler kernel is employed to design the fractional-order model of the new circumscribed self-excited spherical attractor, which is not investigated yet by fractional operators. Moreover, the theorems of Schauder’s fixed point and Banach fixed existence theory are used to guarantee that there are solutions to the model. Approximate solutions to the problem are presented by an effective method. To prove the efficiency of the given technique, different values of fractal and fractional orders as well as initial conditions are selected. Figures of the approximate solutions are provided for each case in different dimensions.
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In this work, we established some exact solutions for the (1 + 1)-dimensional and (2 + 1)-dimensional fifth-order integrable equations ((1+1)D and (2+1)D FOIEs) which is considered based on the improved tanh(ϕ(ξ)/2)-expansion method... more
In this work, we established some exact solutions for the (1 + 1)-dimensional and (2 + 1)-dimensional fifth-order integrable equations ((1+1)D and (2+1)D FOIEs) which is considered based on the improved tanh(ϕ(ξ)/2)-expansion method (IThEM), by utilizing Maple software. We obtained new periodic solitary wave solutions. The obtained solutions include soliton, periodic, kink, kink-singular wave solutions. Comparing our new results with Wazwaz results, namely, Hereman-Nuseri method [2, 3] show that our results give the further solutions. Many other such types of nonlinear equations arising in uid dynamics, plasma physics and nonlinear physics.
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Abstract 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... more
Abstract 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.
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In this paper, a computational approach is suggested to obtain unknown space–time-dependent source term of the fractional diffusion equations. We assign a time-dependent source term and a linear space with the zero components which... more
In this paper, a computational approach is suggested to obtain unknown space–time-dependent source term of the fractional diffusion equations. We assign a time-dependent source term and a linear space with the zero components which represent a series of boundary functions. In linear space, an energy border functional equation is obtained. After that, some numerical examples are provided to verify the accuracy of the method. Also, two tables are presented to display the values of solutions.
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Abstract A simple and powerful numerical method is proposed for solving the time-fractional Burger-Huxley equation (TFBH) equation within Caputo type fractional derivative. A fictitious coordinate θ is applied into the equation in order... more
Abstract A simple and powerful numerical method is proposed for solving the time-fractional Burger-Huxley equation (TFBH) equation within Caputo type fractional derivative. A fictitious coordinate θ is applied into the equation in order to convert the dependent variable u ( x , t ) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of the method of line and group preserving scheme (GPS) is introduced to find the approximate solutions. The reliability and accuracy of this scheme has been shown through some examples of the TFBH equation.
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Abstract In this work, a powerful approach is presented to solve the nonlinear wave equation (NWE): utt (x, t) − u(x, t)uxx (x, t) = g(x, t), where c is a positive constant. Firstly, a discretization is implemented on the original... more
Abstract In this work, a powerful approach is presented to solve the nonlinear wave equation (NWE): utt (x, t) − u(x, t)uxx (x, t) = g(x, t), where c is a positive constant. Firstly, a discretization is implemented on the original equation. Then, a geometric technique is applied to obtain the approximate solutions. Some examples are displayed to prove the power and capability of the offered approach to work out the wave problem.
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A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): A B C D 0 + , t β u ( x , t ) = ζ u x x ( x , t ) − κ u x ( x , t ) + $^{ABC}\\mathcal{D}_{0^+,t}^{\\beta}u(x,t) =\\zeta... more
A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): A B C D 0 + , t β u ( x , t ) = ζ u x x ( x , t ) − κ u x ( x , t ) + $^{ABC}\\mathcal{D}_{0^+,t}^{\\beta}u(x,t) =\\zeta u_{xx}(x,t)- \\kappa u_x(x,t)+$ F(x, t), 0 < β ≤ 1. The time-fractional derivative A B C D 0 + , t β u ( x , t ) $^{ABC}\\mathcal{D}_{0^+,t}^{\\beta}u(x,t)$ is described in the Atangana-Baleanu Caputo concept. The basis of our approach is transforming the original equation into a new equation by imposing a transformation involving a fictitious coordinate. Then, a geometric scheme namely the group preserving scheme (GPS) is implemented to solve the new equation by taking an initial guess. Moreover, in order to present the power of the presented approach some examples are solved, successfully.
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Abstract.In this work we offer a robust numerical algorithm based on the Lie group to solve the time-fractional diffusion-wave (TFDW) equation. Firstly, we use a fictitious time variable $ \xi$ξ to convert the related variable u(x, t)... more
Abstract.In this work we offer a robust numerical algorithm based on the Lie group to solve the time-fractional diffusion-wave (TFDW) equation. Firstly, we use a fictitious time variable $ \xi$ξ to convert the related variable u(x, t) into a new space with one extra dimension. Then by using a composition of the group preserving scheme (GPS) and a semi-discretization of new variable, we approximate the solutions of the problem. Finally, various numerical experiments are performed to illustrate the power and accuracy of the given method.
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In this work a poweful approach is presented to solve the time-fractional gas dynamics equation. In fact, we use a fictitious time variable y to convert the dependent variable w(x, t) into a new one with one more dimension. Then by taking... more
In this work a poweful approach is presented to solve the time-fractional gas dynamics equation. In fact, we use a fictitious time variable y to convert the dependent variable w(x, t) into a new one with one more dimension. Then by taking a initial guess and implementing the group preserving scheme we solve the problem. Finally four examples are solved to illustrate the power of the offered method.
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Abstract In this work, we investigate the SEIR and Blood Coagulation systems using a specific type of fractional derivative. SEIR epidemic model which outlines the close communication of contagious disease is estimated to dominate the... more
Abstract In this work, we investigate the SEIR and Blood Coagulation systems using a specific type of fractional derivative. SEIR epidemic model which outlines the close communication of contagious disease is estimated to dominate the measles epidemic for infected groups. Moreover, Blood coagulation is a protective tool that restricts the loss of blood upon the rupture of endothelial tissues. This process is a complicated one that is managed by various mechanical and biochemical mechanisms. Indeed, the fractional Atangana-Baleau-Caputo derivative operator is exercised to achieve the new models of fractional equations of the SEIR epidemic and Blood Coagulation. Moreover, the existence and uniqueness of the considered systems are checked. Also, simulations are provided under selecting different amounts of fractional orders using Atangana-Toufik method. Additionally, chaotic behaviors of the proposed models by adopting different values of orders are presented, clearly to show the robustness and reliability of the recommended scheme. During graphs of simulations which are obtained under applying various values of orders, show that the used algorithm is highly effective to solve such fractional systems employing various initial conditions(ICs)compared to the other methods.
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HIV-1 infection is a dangerous diseases like Cancer, AIDS, etc. Many mathematical models have been introduced in the literature, which are investigated with different approaches. In this article, we generalize the HIV-1 model through... more
HIV-1 infection is a dangerous diseases like Cancer, AIDS, etc. Many mathematical models have been introduced in the literature, which are investigated with different approaches. In this article, we generalize the HIV-1 model through nonsingular fractional operator. The non-integer mathematical model of HIV-1 infection under the Caputo-Fabrizio derivative is presented in this paper. The concept of Picard-Lindelof and fixed-point theory are used to address the existence of a unique solution to the HIV-1 model under the suggested operator. Also, the stability of the suggested model is proved through the Picard iteration and fixed point theory approach. The model's approximate solution is constructed through three steps Adams-Bashforth numerical method. Numerical simulations are provided for different values of fractional-order to study the complex dynamics of the model. Lastly, we provide the oscillatory and chaotic behavior of the proposed model for various fractional orders.