Search Results (162)

The present study addresses the European option pricing problem based on the Black–Scholes (B-S) model using a hybrid analytical approach known as the Sawi homotopy perturbation transform scheme (SHPTS). We formulate fractional derivatives in the Caputo sense to effectively capture the memory effects inherent in financial models. The competency and reliability of the SHPTS are demonstrated through two illustrative examples. This method produces a closed-form series solution that converges to the precise solution. We perform convergence and visual analyses to demonstrate the competency and reliability of the proposed scheme. The numerical findings further reveal that the strategy is straightforward to apply and very successful in resolving the fractional form of the B-S problem. Full article

This study investigates the solution structure of the nonlinear Rayleigh oscillator equation through two widely used semi-analytical techniques: the Laplace–Adomian Decomposition Method (LADM) and the Homotopy Perturbation Method (HPM). The Rayleigh oscillator exhibits inherent asymmetry in its nonlinear damping term, which disrupts the time-reversal symmetry present in linear oscillatory systems. Applying the LADM and HPM, we derive approximate solutions for the Rayleigh oscillator. Due to the absence of exact analytical solutions in the literature, these approximations are benchmarked against high-precision numerical results obtained using Mathematica’s NDSolve function. We perform a detailed error analysis across different damping parameter values ε and time intervals. Our results reveal how the asymmetric damping influences the accuracy and convergence behavior of each method. This study highlights the role of nonlinear asymmetry in shaping the solution dynamics and provides insight into the suitability of the LADM and HPM under varying conditions. Full article

The electroosmotic flow (EOF) of non-Newtonian fluids plays a significant role in microfluidic systems. The EOF of Powell–Eyring fluid within a parallel-plate microchannel, under the influence of both electric field and pressure gradient, is investigated. Navier’s boundary condition is adopted. The velocity distribution’s approximate solution is derived via the homotopy perturbation technique (HPM). Optimized initial guesses enable accurate second-order approximations, dramatically lowering computational complexity. The numerical solution is acquired via the modified spectral local linearization method (SLLM), exhibiting both high accuracy and computational efficiency. Visualizations reveal how the pressure gradient/electric field, the electric double layer (EDL) width, and slip length affect velocity. The ratio of pressure gradient to electric field exhibits a nonlinear modulating effect on the velocity. The EDL is a nanoscale charge layer at solid–liquid interfaces. A thinner EDL thickness diminishes the slip flow phenomenon. The shear-thinning characteristics of the Powell–Eyring fluid are particularly pronounced in the central region under high pressure gradients and in the boundary layer region when wall slip is present. These findings establish a theoretical base for the development of microfluidic devices and the improvement of pharmaceutical carrier strategies. Full article

This study analyzes the family of one of the most essential fractional differential equations due to its wide applications in physics and engineering: the multidimensional fractional linear and nonlinear diffusion equations. The Caputo fractional derivative operator is used to treat the time-fractional derivative. To complete the analysis and generate more stable and highly accurate approximations of the proposed models, three extremely effective techniques, known as the direct Tantawy technique, the new iterative transform technique (NITM), and the homotopy perturbation transform method (HPTM), which combine the Elzaki transform (ET) with the new iterative method (NIM), and the homotopy perturbation method (HPM), are employed. These reliable approaches produce more stable and highly accurate analytical approximations in series form, which converge to the exact solutions after a few iterations. As the number of terms/iterations in the problems series solution rises, it is found that the derived approximations are closely related to each problem’s exact solutions. The two- and three-dimensional graphical representations are considered to understand the mechanism and dynamics of the nonlinear phenomena described by the derived approximations. Moreover, both the absolute and residual errors for all generated approximations are estimated to demonstrate the high accuracy of all derived approximations. The obtained results are encouraging and appropriate for investigating diffusion problems. The primary benefit lies in the fact that our proposed plan does not necessitate any presumptions or limitations on variables that might affect the real problems. One of the most essential features of the proposed methods is the low computational cost and fast computations, especially for the Tantawy technique. The findings of the present study will be valuable as a tool for handling fractional partial differential equation solutions. These approaches are essential in solving the problem and moving beyond the restrictions on variables that could make modeling the problem challenging. Full article

This work reveals an advanced numerical scheme for obtaining approximate solutions to nonlinear fractional Kuramoto–Sivashinsky (K-S) equations involving Caputo derivatives. We introduce the Sumudu transform (ST), which converts the fractional derivatives into their classical counterparts to produce a nonlinear recurrence equation. By using the homotopy perturbation method (HPM), we construct a homotopy with an embedding parameter to solve this recurrence relation. Our proposed technique is known as the Sumudu homotopy transform method (SHTM), which delivers results after fewer iterations and achieves precise outcomes with minimal computational effort. The proposed technique effectively eliminates the necessity for complex discretization or linearization, making it highly suitable for nonlinear problems. We showcase two numerical cases, along with two- and three-dimensional visualizations, to validate the accuracy and effectiveness of this technique. It also produces rapidly converging series solutions that closely align with the precise results. Full article

This work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated techniques employed to derive analytical solutions. The proposed methods are novel and remarkable hybrid integral transform schemes that effectively incorporate the Adomian decomposition method, homotopy perturbation method, and Yang transform method. They efficiently yield rapidly convergent series-type solutions through an iterative process that requires fewer computations. The Caputo operator, used to express the fractional derivatives in the equations, provides a robust framework for analyzing the behavior of non-integer-order systems. To validate the accuracy and reliability of the obtained solutions, numerical simulations and graphical representations are presented. Furthermore, the results are compared with exact solutions using various tables and graphs, illustrating the effectiveness and ease of implementation of the proposed approaches for various fractional partial differential equations. The influence of the non-integer parameter on the solutions behavior is specifically examined, highlighting its function in regulating wave propagation and diffusion. In addition, a comparison with the natural transform iterative method and optimal auxiliary function method demonstrates that the proposed methods are more accurate than these alternative approaches. The results highlight the potential of YTDM and HPTM as reliable tools for solving nonlinear fractional differential equations and affirm their relevance in wave mechanics, fluid dynamics, and other fields where fractional-order models are applied. Full article

The time-fractional generalized Burger–Fisher equation (TF-GBFE) is utilized in many physical applications and applied sciences, including nonlinear phenomena in plasma physics, gas dynamics, ocean engineering, fluid mechanics, and the simulation of financial mathematics. This mathematical expression explains the idea of dissipation and shows how advection and reaction systems can work together. We compare the homotopy perturbation transform method and the new iterative method in the current study. The suggested approaches are evaluated on nonlinear TF-GBFE. Two-dimensional (2D) and three-dimensional (3D) figures are displayed to show the dynamics and physical properties of some of the derived solutions. A comparison was made between the approximate and accurate solutions of the TF-GBFE. Simple tables are also given to compare the integer-order and fractional-order findings. It has been verified that the solution generated by the techniques given converges to the precise solution at an appropriate rate. In terms of absolute errors, the results obtained have been compared with those of alternative methods, including the Haar wavelet, OHAM, and q-HATM. The fundamental benefit of the offered approaches is the minimal amount of calculations required. In this research, we focus on managing the recurrence relation that yields the series solutions after a limited number of repetitions. The comparison table shows how well the methods work for different fractional orders, with results getting closer to precision as the fractional-order numbers get closer to integer values. The accuracy of the suggested techniques is greatly increased by obtaining numerical results in the form of a fast-convergent series. Maple is used to derive the approximate series solution’s behavior, which is graphically displayed for a number of fractional orders. The computational stability and versatility of the suggested approaches for examining a variety of phenomena in a broad range of physical science and engineering fields are highlighted in this work. Full article

This work explores modern mathematical avenues as part of fractional calculus research. We apply fractional dispersion relations to the fractional wave equation to numerically examine various formulations of the generalized fractional wave equation. The research explores Drinfeld–Sokolov–Wilson and shallow water equations as fundamental differential equations forming the basis of wave theory studies. This work presents effective methods to obtain the numerical solution of the fractional-order FDSW and FSW coupled system equations. The analysis employs Caputo fractional derivatives during studies of fractional orders. This study develops the new iterative transform technique (NITM) and homotopy perturbation transform method (HPTM) using Elzaki transform (ET) with a new iteration method and a homotopy perturbation method. The proposed techniques generate approximation solutions that adopt an infinite fractional series with fractional order solutions converging towards analytic integer solutions. The proposed method demonstrates its precision through tabular simulations of computed approximations and their absolute error values while representing results with 2D and 3D graphics. The paper presents the physical analysis of solution dynamics across diverse $ϵ$ ranges during a suitable time frame. The developed computational techniques yield numerical and graphical output, which are compared to analytic results to verify the solution convergence. The computational algorithms have proven their high accuracy, flexibility, effectiveness, and simplicity in evaluating fractional-order mathematical models. Full article

This paper proposes an innovative method that combines the homotopy analysis method with the Jafari transform, applying it for the first time to solve systems of fractional-order linear and nonlinear differential equations. The method constructs approximate solutions in the form of a series and validates its feasibility through comparison with known exact solutions. The proposed approach introduces a convergence parameter ℏ, which plays a crucial role in adjusting the convergence range of the series solution. By appropriately selecting initial terms, the convergence speed and computational accuracy can be significantly improved. The Jafari transform can be regarded as a generalization of classical transforms such as the Laplace and Elzaki transforms, enhancing the flexibility of the method. Numerical results demonstrate that the proposed technique is computationally efficient and easy to implement. Additionally, when the convergence parameter $\hslash =-1$, both the homotopy perturbation method and the Adomian decomposition method emerge as special cases of the proposed method. The knowledge gained in this study will be important for model solving in the fields of mathematical economics, analysis of biological population dynamics, engineering optimization, and signal processing. Full article

This article presents a computational analysis of approximate solutions for the time-fractional nonlinear Kawahara problem (KP) and the modified Kawahara problem (modified KP). This study utilizes the natural homotopy transform scheme (NHTS), which integrates the natural transform (NT) with the homotopy perturbation scheme (HPS). We derive the algebraic expression of nonlinear terms through the implementation of HPS. The fractional derivatives are considered in the Caputo form. Numerical results and visualizations present the practical interest and effectiveness of the fractional derivatives. The accuracy of the approximate results, coupled with their precise outcomes, emphasizes the reliability of the method. These findings demonstrate that NHTS is a robust and effective approach for solving time-fractional problems through series expansions. Full article

In this work, we introduce an innovative analytical–numerical approach to solving nonlinear fractional differential equations by integrating the homotopy perturbation method with the new integral transform. The Kawahara equation and its modified form, which is significant in fluid dynamics and wave propagation, serve as test cases for the proposed methodology. Additionally, we apply the fractional new integral transform–homotopy perturbation method (FNIT-HPM) to a nonlinear system of coupled Burgers’ equations, further demonstrating its versatility. All calculations and simulations are performed using Mathematica 12 software, ensuring precision and efficiency in computations. The FNIT-HPM framework effectively transforms complex fractional differential equations into more manageable forms, enabling rapid convergence and high accuracy without linearization or discretization. By evaluating multiple case studies, we demonstrate the efficiency and adaptability of this approach in handling nonlinear systems. The results highlight the superior accuracy of the FNIT-HPM compared to traditional methods, making it a powerful tool for addressing complex mathematical models in engineering and physics. Full article

The aim of this article is to introduce analytical and approximate techniques to obtain the solution of time-fractional Navier–Stokes equations. This proposed technique consists is coupling the homotopy perturbation method (HPM) and Laplace transform (LT). The time-fractional derivative used is the Caputo–Hadamard fractional derivative (CHFD). The effectiveness of this method is demonstrated and validated through two test problems. The results show that the proposed method is robust, efficient, and easy to implement for both linear and nonlinear problems in science and engineering. Additionally, its computational efficiency requires less computation compared to other schemes. Full article

We investigate the topological degree for generalized monotone operators of class ${\left(S\right)}_{+}$ with compact set-valued perturbations. It is assumed that perturbations can be represented as the composition of a continuous single-valued mapping and an upper semicontinuous set-valued mapping with aspheric values. This allows us to extend the standard degree theory for convex-valued operators to set-valued mappings whose values can have complex geometry. Several theoretical aspects concerning the definition and main properties of the topological degree for such set-valued mappings are discussed. In particular, it is shown that the introduced degree has the homotopy invariance property and can be used as a convenient tool in checking the existence of solutions to corresponding operator inclusions. To illustrate the applicability of our approach to studying models of real processes, we consider an optimal feedback control problem for the steady-state internal flow of a generalized Newtonian fluid in a 3D (or 2D) bounded domain with a Lipschitz boundary. By using the proposed topological degree method, we prove the solvability of this problem in the weak formulation. Full article

The present study dealt with a comprehensive mathematical model to explore the nonlinear forced vibration of a magnetostrictive laminated beam. This system was subjected to mechanical impact, a nonlinear Winkler–Pasternak foundation, and an electromagnetic actuator considering the thickness effect. The expressions of the nonlinear differential equations were obtained for the pinned–pinned boundary conditions with the help of the Galerkin–Bubnov procedure and Hamiltonian approach. The nonlinear differential equations were studied using an original, explicit, and very efficient technique, namely the optimal auxiliary functions method (OAFM). It should be emphasized that our procedure assures a rapid convergence of the approximate analytical solutions after only one iteration, without the presence of a small parameter in the governing equations or boundary conditions. Detailed results are presented on the effects of some parameters, among them being analyzed were the damping, frequency, electromagnetic, and nonlinear elastic foundation coefficients. The local stability of the equilibrium points was performed by introducing two variable expansion method, the homotopy perturbation method, and then applying the Routh–Hurwitz criteria and eigenvalues of the Jacobian matrix. Full article

In this article, we examine a deterministic Zika virus model that takes into account the vector and sexual transmission route, in the absence of disease-induced deaths, symmetrically observing the impact of human knowledge and vector control. In order to construct the model, we suppose that the Zika virus is first spread to humans through mosquito bites, and then to their sexual partner. In this article, we conduct analytical studies which often begin by proving the existence and uniqueness of solutions for the Zika virus model using the fractional derivative from the Caputo–Fabrizio derivative. Then, the uniqueness of the solution is investigated. After that, we also identify under which circumstances and symmetry the model provides a unique solution. Full article