Modified Homotopy Perturbation Research Articles

Series form solutions of time–space fractional Black–Scholes model via extended He-Aboodh algorithm

The objective of the current study is analyze linear and nonlinear time–space fractional Black–Scholes models via modified homotopy perturbation method (m-HPM). In current investigation, memory effects in financial markets are explored through fractional derivative in Caputo sense. The effectiveness of proposed methodology is checked numerically by finding residual errors and presented in tables. These tables also provide a benchmark for the comparison with already existing results in literature. Furthermore, solutions are graphically analyzed via 3D and contour plots across a range of parameters under varying market conditions. Analysis confirms the efficiency of m-HPM for predicting solutions of time–space fractional Black–Scholes models. The current study can contribute in understanding the applications of fractional calculus in finance, and can be a valuable computational tool for pricing financial derivatives in fractional environments.

Nonlinear oscillators dynamics using optimal and modified homotopy perturbation method

This study investigates the several strongly nonlinear oscillators (Van der Pol, Duffing and Rayleigh) for which we have applied the most powerful and advanced optimal semi-analytical technique: optimal and modified homotopy perturbation method (OM-HPM) for the convergent semi-analytical series solution. The numerical simulation demonstrates the high accuracy of the OM-HPM, which is straightforward, does not require any domain decomposition, special transformation, or pade approximations to get the convergent series solution. The key features for the high accuracy of the OM-HPM lies on the best optimal auxiliary linear operator. Therefore, OM-HPM offering a valuable tool for engineers and researcher to analyze the complex nonlinear oscillator.

Inspection of a Time-Delayed Excited Damping Duffing Oscillator

This paper examines a time delay in position and velocity to minimize the nonlinear vibration of an excited Duffing oscillator (DO). This model is highly beneficial for capturing the nonlinear characteristics of many different applications in engineering. To achieve an estimated uniform solution to the problem under consideration, a modified homotopy perturbation method (HPM) is utilized. This adaptation produces a more accurate precise approximation with a numerical solution (NS). This is obtained by employing Mathematica software 12 (MS) in comparison with the analytical solution (AS). The comparison signifies a good match between the two methodologies. The comparison is made with the aid of the NS. Consequently, the work allows for a qualitative assessment of the results of a representative analytical approximation approach. A promising stability analysis for the unforced system is performed. The time history of the accomplished results is illustrated in light of a diverse range of physical frequency and time-delay aspects. The outcomes are theoretically discussed and numerically explained with a set of graphs. The nonlinear structured prototype is examined via the multiple-scale procedure. It investigates how various controlling limits affect the organization of vibration performances. As a key assumption, according to cubic nonlinearity, two significant examples of resonance, sub-harmonic and super-harmonic, are explored. The obtained modulation equations, in these situations, are quantitatively investigated with regard to the influence of the applied backgrounds.

A Theoretical Approach to Understand the Nonlinear Processes in Enzymatic Electrochemical Biosensors.

The enzymatic biosensors' response can be monitored based on the results of nonlinear differential equations. The nonlinear reaction-diffusion equations proposed for this enzyme-based electrochemical biosensor include a nonlinear term associated with Michaelis-Menten kinetics. Herein, the system of nonlinear reaction-diffusion equations is solved using a modified homotopy perturbation method. For all values of the rate constants, the approximate analytical expressions for the concentration profiles, current, sensitivity, and gradient of biosensor have been determined. Performance factors of an enzymatic electrochemical biosensor, such as response time, sensitivity, accuracy, and resistance, are discussed. The analytical results and numerically simulated outcomes using Matlab software have been compared.

Mathematical analysis of intrahost spread and control of Dengue Virus: Unraveling the crucial role of antigenic immunity

Studying viral infections within the host is crucial for understanding healthy cell responses and immunity. This research employed a reaction-diffusion Caputo fractional order model and a modified homotopy perturbation method to explore how immune cell memory and mobility affect the dynamic interactions of cytotoxic T-lymphocyte cells (CTLs), B-cell antibodies, uninfected monocytes, infected monocytes, and Dengue Virus (DENV) particle concentration within the host. The model guaranteed a positive solution, with the existence and uniqueness of the solution assured by the Banach fixed point theorem and a general stability supported by the Hyers-Ulam theorem. Sensitivity analysis highlighted that the DENV invasion rate speeds the spread of infected cells, while simulation results and fractional order analysis revealed that cell-mediated memory from past exposure will enhance the activation of CTLs and B-cell antibodies to strengthen immunity against dengue virus.

A space-time Caputo fractional order and modified homotopy perturbation method for evaluating the pathological response of tumor-immune cells

Tumors result from genetic mutations or environmental factors that prompt cells to divide uncontrollably. This study aims to examine the behavior of tumor-immune cell growth in the presence of chemotherapy drug diffusion at a Caputo fractional order. To accomplish this, we employed the modified homotopy perturbation method to solve a proposed system of nonlinear differential equations. We obtained the analytical solutions to study the spatiotemporal pathological response of tumor-immune cell growth. Our analysis also considered the impact of the Caputo-fractional order on the system's dynamics and compared the results with the classical integer-order scenario. Our findings demonstrated that the proposed method is an effective and precise technique for understanding the intricate interactions of tumor-immune cell growth. Additionally, we revealed that the Caputo-fractional order plays a significant role in the system's behavior and should not be overlooked in future analyses of such systems. In conclusion, this study holds important implications for cancer research by providing insights into the behavior of tumor-immune cell growth in the presence of time-fractional administration of chemotherapy drugs.

Analytical Solution of One-Dimensional Keller-Segel Equations via New Homotopy Perturbation Method

For solving a system of nonlinear partial differential equations (PDE) emerging in an attractor one-dimensional chemotaxis model, we used a relatively new analytical method called the new modified homotopy perturbation method (NMHPM). We use NMHPM for solving one-dimensional Keller-Segel models for different types. Some properties show biologically acceptable dependency on parameter values, and numerical solutions are provided. NMHPM’s stability and reduced computing time provide it with a broader range of applications. The algorithm provides analytical approximations for different types of Keller-Segel equations. Some numerical illustrations are given to show the efficiency of the algorithm.

Analysis and estimation of the COVID-19 pandemic by modified homotopy perturbation method

The Bernoulli equation is useful to assess the motility and recovery rate with respect to time in order to measure the COVID-19 outbreak. The homotopy perturbation method was applied in the current article to compute the Bernoulli equation. For the existence and uniqueness of solutions, we also used the Caputo–Fabrizio Integral and differential operators. Additionally, we conducted a corresponding investigation for derivatives of integer and fractional orders on the estimated motility and recovery rate.

The modified homotopy perturbation method and its application to the dynamics of price evolution in Caputo-fractional order Black Scholes model

BackgroundFollowing a financial loss in trades due to lack of risk management in previous models from market practitioners, Fisher Black and Myron Scholes visited the academic setting and were able to mathematically develop an option pricing equation named the Black–Scholes model. In this study, we address the solution of a Caputo fractional-order Black–Scholes model using an analytic method named the modified initial guess homotopy perturbation method.MethodologyForemost, the classical Black Scholes model relaxed for European option style is generalized to be of Caputo derivative. The introduced method is established by coupling a power series function of arbitrary order with the renown He’s homotopy perturbation method. The convergence of the method is demonstrated using the fixed point theorem, and its methodology is illustrated by solving a generalized theoretical form of the fractional order Black Scholes model. The applicability of the method is proven by solving three different fractional order Black–Scholes equations derived from different market scenarios and its effectiveness is confirmed as feasible series of arbitrary orders that accelerate fast to the exact solution at an integer order were obtained. The computation of these results was carried out using Mathematica 12 software. Subsequently, the obtained outcomes were utilized in Maple 18 software to conduct a series of numerical simulations. These simulations aimed to analyze the influence of the fractional order on the dynamics of payoff functions regarding the share value as the option approached its expiration date under varying market constraints. In all three scenarios, the results showed that option values decrease as the expiration date approaches the integer order. Furthermore, the comparative outcomes reveal that Caputo fractional order derivatives control the flexibility of the classical Black–Scholes model because its payoff curve exhibits more sensitivity to changes associated with market characteristic parameters, such as volatility and interest rates.RecommendationsWe propose that the results of this work should be examined and implemented by mathematicians and economists to better comprehend the influence of Caputo-fractional order derivatives in understanding the dynamics of option price evolution of financial assets.

Dynamic pull-down theory for the Toda oscillator

The pull-down instability is a newly found phenomenon for oscillators with even nonlinearities. It is a seemingly periodic motion, which will suddenly pull down after some cycles. The phenomenon is similar to the pull-in instability in the micro-electromechanical systems. This paper restudies the well-known Toda oscillator, which is considered to be in periodic motion. Its pseudo-periodic property is studied and the pull-down instability is found for the first time ever. To this end, this paper establishes a variational principle for the Toda oscillator, and the criterion for prediction of the pull-down motion is obtained. A simple formulation to calculate the pull-down instability is proved and verified through examples. Furthermore, an approximate frequency formula is obtained by Li–He modified homotopy perturbation method.

A novel approach for an approximate solution of a nonlinear equation of charged damped oscillator with one degree of freedom

A novel technique is proposed for finding an approximate solution of the strongly nonlinear ordinary differential equation for the charged damped pendulum with one degree of freedom. The method relies on a transformation of the governing nonlinear differential equation that keeps unchanged the order of the highest derivative, in conjunction with a modified homotopy perturbation technique (MHPM). Only quadratic damping is considered for the numerical computations. To validate the used technique, the obtained results are compared to those arising from a numerical solution by Runge–Kutta of the fourth order (RK4) and by finite differences (FD). Good agreement between the two solutions is reached when quadratic damping is suppressed. In the presence of damping, agreement takes place only for a rather limited range of times. Plots of the analytical solutions are provided for both cases. The proposed method may be used to analyze a wide class of nonlinear differential equations.

Gohar Fractional Derivative: Theory and Applications

The local fractional derivatives marked the beginning of a new era in fractional calculus. Due to their that have never been observed before in the field, they are able to fill in the gaps left by the nonlocal fractional derivatives and substantially increase the field’s theoretical and applied potential. In this article, we introduce a new local fractional derivative that possesses some classical properties of the integer-order calculus, such as the product rule, the quotient rule, the linearity, and the chain rule. It meets the fractional extensions of Rolle’s theorem and the mean value theorem and has more properties beyond those of previously defined local fractional derivatives. We reveal its geometric interpretation and physical meaning. We prove that a function can be differentiable in its sense without being classically differentiable. Moreover, we apply it to solve the Riccati fractional differential equations to demonstrate that it provides more accurate results with less error in comparison with the previously defined local fractional derivatives when applied to solve fractional differential equations. The numerical results obtained in this work by our local fractional derivative are shown to be in excellent agreement with those produced by other analytical and numerical methods such as the enhanced homotopy perturbation method (EHPM), the improved Adams-Bashforth-Moulton method(IABMM), the modified homotopy perturbation method (MHPM), the Bernstein polynomial method (BPM), the fractional Taylor basis method (FTBM), and the reproducing kernel method (RKM).

An optimal and modified homotopy perturbation method for strongly nonlinear differential equations

An optimal and modified homotopy perturbation method for strongly nonlinear differential equations

Dynamical analysis of an inverted pendulum with positive position feedback controller approximate uniform solution

The inverted pendulum is controlled in this article by using the nonlinear control theory. From classical analytical mechanics, its substructure equation of motion is derived. Because of the inclusion of the restoring forces, the Taylor expansion is employed to facilitate the analysis. An estimated satisfactory periodic solution is obtained with the aid of the modified Homotopy perturbation method. A numerical technique based on the fourth-order Runge–Kutta method is employed to justify the previous solution. On the other hand, a positive position feedback control is developed to dampen the vibrations of an IP system subjected to multi-excitation forces. The multiple time scale perturbation technique of the second order is introduced as a mathematical method to solve a two-degree-of-freedom system that simulates the IP with the PPF at primary and 1:1 internal resonance. The stability of these solutions is checked with the aid of the Routh–Hurwitz criterion. A set of graphs, based on the frequency response equations resulting from the MSPT method, is incorporated. Additionally, a numerical simulation is set up with RK-4 to confirm the overall controlled performance of the studied model. The quality of the solution is confirmed by the match between the approximate solution and the numerical simulation. Numerous other nonlinear systems can be controlled using the provided control method. Illustrations are offered that pertain to implications in design and pedagogy. The linearized stability of IP near the fixed points as well as the phase portraits is depicted for the autonomous and non-autonomous cases. Because of the static stability of the IP, it is found that its instability can be suppressed by the increase of both the generalized force as well as the torsional constant stiffness of the spring. Additionally, the presence of the magnetic field enhances the stability of IP.

On the solvability of a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator

In this paper, utilizing the technique of generalized Darbo's fixed‐point theorem associated with measure of noncompactness in Banach space, we analyze the existence of solution for a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator. The existing result was obtained to strengthen the ones mentioned previously in the literature. An example for a class of nonlinear functional integral equations is also presented to validate our main result. Finally, we propose the numerical method formed by the modified homotopy perturbation approach to resolving the problem with acceptable accuracy.

Analytical and numerical study of a vibrating magnetic inverted pendulum

The current study investigates the stability structure of the base periodic motion of an inverted pendulum (IP). A uniform magnetic field affects the motion in the direction of the plane configuration. Furthermore, a non-conservative force as one that dampens air is considered. Its underlying equation of motion is derived from traditional analytical mechanics. The mathematical analysis is made simpler by substituting the Taylor theory in order to expand the restoring forces. The modified Homotopy perturbation method (HPM) is employed to achieve a roughly adequate regular result. To support the prior result, a numerical method based on the fourth-order Runge-Kutta method (RK4) is employed. The graphs for both the analytic and numerical solutions are highly consistent with one another, which indicates that the perturbation strategy is accurate. The solution time history curve exhibits a decaying performance and indicates that it is steady and without chaos. The resonance and non-resonance cases are found through the stability study by using the time scale method. In all perturbation approaches, the methodology of multiple time scales is actually regarded as a further standard approach. The time history is used to create a collection of graphs. Some graphical representations are used to illustrate how the typical physical values affect the behavior of the discovered solution. It has been discovered that the statically unstable IP can have its instability reduced by raising the spring torsional constant stiffness as well as the damped coefficient. Moreover, the magnetic field has a significant role in the stability configuration, which explains that at higher values of this field, the decaying waves take much more time than the smaller values of this field. Accordingly, it can be employed in various engineering devices that need a certain period of time to be more stable.

Fractional Modeling of Non-Newtonian Casson Fluid between Two Parallel Plates

In this manuscript, fractional modeling of non-Newtonian Casson fluid squeezed between two parallel plates is performed under the influence of magneto-hydro-dynamic and Darcian effects. The Casson fluid model is fractionally transformed through mixed similarity transformations. As a result, partial differential equations (PDEs) are transformed to a fractional ordinary differential equation (FODE). In the current modeling, the continuity equation is satisfied while the momentum equation of the integral order Casson fluid is recovered when the fractional parameter is taken as α = 1 . A modified homotopy perturbation algorithm is used for the solution and analysis of highly nonlinear and fully fractional ordinary differential equations. Obtained solutions and errors are compared with existing integral order results from the literature. Graphical analysis is also performed at normal and radial velocity components for different fluid and fractional parameters. Analysis reveals that a few parameters are showing different behavior in a fractional environment as compared to existing integer-order cases from the literature. These findings affirm the importance of fractional calculus in terms of more generalized analysis of physical phenomena.

A heuristic approach to the prediction of a periodic solution for a damping nonlinear oscillator with the non-perturbative technique

The present work attracts attention to obtaining a new result of the periodic solution of a damped nonlinear Duffing oscillator and a damped Klein–Gordon equation. It is known that the frequency response equation in the Duffing equation can be derived from the homotopy analysis method only in the absence of the damping force. We suggest a suitable new scheme successfully to produce a periodic solution without losing the damping coefficient. The novel strategy is centered on establishing an alternate equation apart from any difficulty in handling the influence of the linear damped term. This alternative equation was obtained with the rank upgrading technique. The periodic solution of the problem is presented using the non-perturbative method and validated by the modified homotopy perturbation technique. This technique is successful in obtaining new results toward a periodic solution, frequency equation, and the corresponding stability conditions. This methodology yields a more effective outcome of the damped nonlinear oscillators. With the help of this procedure, one can analyze many problems in the domain of physical engineering that involve oscillators and a linear damping influence. Moreover, this method can help all interested plasma authors for modeling different nonlinear acoustic oscillations in plasma.

New modied Atangana-Baleanu fractional derivative applied to solve nonlinear fractional dierential equations

The main goal of this work is to present a new modified version of the Atangana-Baleanu fractional derivative with Mittag-Leffler non-singular kernel and strong memory. This proposal presents important advantages when specific initial conditions are impossed. The new modified version of the Atangana-Baleanu fractional derivative with Mittag-Leffler non-singular kernel has been constructed considering the fulfillment of the initial conditions with special interest because they are decisive in the obtaintion of analytical and numerical solutions of the fractional differential equations. The advantage of this new fractional derivative in the fulfilling of initial conditions plays a central role for the implementation of different perturbative analytical methods, such as the homotopy perturbation method and the modified homotopy perturbation method. These methods will be applied to solve nonlinear fractional differential equations. This novel modified derivative can be applied in the future in different mathematical modeling areas where satisfy the initial conditions is of great relevance to get more accurate description of real-world problems.

FRACTIONAL-ORDER SINE-GORDON EQUATION INVOLVING NONSINGULAR DERIVATIVE

The sine-Gordon equation has received attention since 1970s due to the existence of soliton solutions. The aforesaid equation has significant applications in the quantum field theory. The aforementioned problem has been treated by using various numerical and analytical techniques under the ordinary as well as fractional-order derivatives. The mentioned equation has been investigated under the usual Caputo fractional-order derivative. Since in some cases the nonsingular-type derivatives produce more significant results in the mathematical modelings of real-world nonlinear problems, therefore, the proposed problem is considered in this paper under the fractional-order case in the context of Atangana–Baleanu–Caputo (ABC) derivative for the analytical and approximate results. This fractional derivative has some useful properties involving Mittag-Leffler-type kernel that is nonlocal and nonsingular. Furthermore, Modified Homotopy Perturbation Method (MHPM) is utilized for the required approximate solution. We give appropriate examples depicting the sine-Gordon model. Also, we present our results for the approximate solution graphically to support all the results.