Search Results (108)

The paper consists of various types of wave solutions for the truncated M-fractional Bateman–Burgers equation, a significant mathematical physics equation. This model describes the nonlinear waves and solitons in different physical fields such as optical fibers, plasma physics, fluid dynamics, traffic flow, etc. Through the application of the ${exp}_a$ function method and the modified simplest equation method, we are able to obtain exact series of soliton solutions. The results differ from the current solutions of the Bateman–Burgers model because of the fractional derivative. The achieved results could be helpful in various engineering and scientific domains. The Mathematica software is used to assist in obtaining and verifying the exact solutions and to obtain contour plots of the solutions in two and three dimensions. To ensure that the model in question is stable, a stability analysis is also carried out using the modulation instability method. Future research on the system in question and related systems will benefit from the findings. The methods used are simple and effective. Full article

The Pennes bioheat equation is the most widely used model for describing heat transfer in living tissue during thermal exposure. It is derived from the classical Fourier law of heat conduction and assumes energy exchange between blood vessels and surrounding tissues. The literature presents various numerical methods for solving the bioheat equation, with exact solutions developed for different boundary conditions and geometries. However, analytical models based on this framework are rarely reported. This study aims to develop an analytical three-dimensional model using MATHEMATICA software, with subsequent mathematical validation performed through COMSOL simulations, to characterize heat transfer in biological tissues induced by laser irradiation under various therapeutic conditions. The objective is to refine the conventional bioheat equation by introducing three key improvements: (a) incorporating a non-Fourier framework for the Pennes equation, thereby accounting for the relaxation time in thermal response; (b) integrating Dirac functions and the telegraph equation into the bioheat model to simulate localized point heating of diseased tissue; and (c) deriving a closed-form analytical solution for the Pennes equation in both its classical (Fourier-based) and improved (non-Fourier-based) formulations. This paper investigates the nuanced relationship between the relaxation time parameter in the telegraph equation and the thermal relaxation time employed in the bioheat transfer equation. Considering all these aspects, the optimal thermal relaxation time determined for these simulations was 1.16 s, while the investigated thermal exposure time ranged from 0.01 s to 120 s. This study introduces a generalized version of the model, providing a more realistic representation of heat exchange between biological tissue and blood flow by accounting for non-uniform temperature distribution. It is important to note that a reasonable agreement was observed between the two modeling approaches: analytical (MATHEMATICA) and numerical (COMSOL) simulations. As a result, this research paves the way for advancements in laser-based medical treatments and thermal therapies, ultimately contributing to more optimized therapeutic outcomes. Full article

Reducing setup costs and improving product quality are critical objectives in a sustainable production processes. The significance of these goals lies in their direct impact on efficiency. It affects competitiveness and customer satisfaction. Businesses can reduce setup costs to maximize resource usage. It can reduce downtime between production runs and improve overall operational agility. Sustained performance and expansion in contemporary manufacturing environments focus on setup cost reduction and product quality improvement. The present paper discusses a production inventory model for the product, which produces by-products as secondary products from the same manufacturing process. Setup cost is reduced for the setup of production and refining processes. A production process may change from being under control to an uncontrolled one. As a result of this, imperfect products are formed. This paper considers product quality improvement for both produced and processed items. The outcome shows that dealing with by-products helps make the system more profitable. Sensitivity analysis is performed for various costs and parameters. Mathematica 11 software was used for calculation and graphical work. Full article

For a significant fluid model and the truncated M-fractional (1 + 1)-dimensional nonlinear generalized water wave equation, distinct types of truncated M-fractional wave solitons are obtained. Ocean waves, tidal waves, weather simulations, river and irrigation flows, tsunami predictions, and more are all explained by this model. We use the improved $(G^{\prime }/G)$ expansion technique and a modified extended direct algebraic technique to obtain these solutions. Results for trigonometry, hyperbolic, and rational functions are obtained. The impact of the fractional-order derivative is also covered. We use Mathematica software to verify our findings. Furthermore, we use contour graphs in two and three dimensions to illustrate some wave solitons that are obtained. The results obtained have applications in ocean engineering, fluid dynamics, and other fields. The stability analysis of the considered equation is also performed. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. Furthermore, the used methods are useful for other nonlinear fractional partial differential equations in different areas of applied science and engineering. Full article

Web-tapered beams with I-sections, which are aesthetic and structurally efficient, have been widely used in steel structures. Web-tapered I-section beams bent about the strong axis may undergo out-of-plane buckling through lateral deflection and twisting. This primary stability failure mode in slender beams is known as lateral-torsional buckling (LTB). Unlike prismatic I-beams, the complex mode shape of web-tapered I-section beams makes it challenging or even impossible to derive a closed-form expression for the LTB load under certain transverse loading conditions. Therefore, the LTB assessment of web-tapered I-section beams is primarily performed using finite element analysis (FEA). However, this method involves multiple steps, requires specialized expertise, and demands significant computational resources, making it impractical in certain cases. This study proposes an analytical approach based on the Ritz method to evaluate the LTB of simply supported web-tapered beams with doubly or mono-symmetric I-sections. The proposed analytical method accounts for web tapering, I-section mono-symmetry, types and positions of transverse loads, and beam slenderness. The method was implemented in Mathematica to allow the rapid evaluation of the LTB capacity of web-tapered I-beams. The study validates the LTB loads computed using the developed Mathematica package against results from shell-based FEA. An excellent agreement was observed between the analytically and numerically calculated LTB loads. Full article

Quaternions are used in various applications, especially in those where it is necessary to model and represent rotational movements, both in the plane and in space, such as in the modeling of the movements of robots and mechanisms. In this article, a methodology to model the rigid rotations of coupled bodies by means of unit quaternions is presented. Two parallel robots were modeled: a planar RRR robot and a spatial motion PRRS robot using the proposed methodology. Inverse kinematic problems were formulated for both models. The planar RRR robot model generated a system of 21 nonlinear equations and 18 unknowns and a system of 36 nonlinear equations and 33 unknowns for the case of space robot PRRS; both systems of equations were of the polynomial algebraic type. The systems of equations were solved using the Broyden–Fletcher–Goldfarb–Shanno nonlinear programming algorithm and Mathematica V12 symbolic computation software. The modeling methodology and the algebra of unitary quaternions allowed the systematic study of the movements of both robots and the generation of mathematical models clearly and functionally. Full article

The study offers a comprehensive investigation of periodic solutions in highly nonlinear oscillator systems, employing advanced analytical and numerical techniques. The motivation stems from the urgent need to understand complex dynamical behaviors in physics and engineering, where traditional linear approximations fall short. This work precisely applies He’s Frequency Formula (HFF) to provide theoretical insights into certain classes of strongly nonlinear oscillators, as illustrated through five broad examples drawn from various scientific and engineering disciplines. Additionally, the novelty of the present work lies in reducing the required time compared to the classical perturbation techniques that are widely employed in this field. The proposed non-perturbative approach (NPA) effectively converts nonlinear ordinary differential equations (ODEs) into linear ones, equivalent to simple harmonic motion. This method yields a new frequency approximation that aligns closely with the numerical results, often outperforming existing approximation techniques in terms of accuracy. To aid readers, the NPA is thoroughly explained, and its theoretical predictions are validated through numerical simulations using Mathematica Software (MS). An excellent agreement between the theoretical and numerical responses highlights the robustness of this method. Furthermore, the NPA enables a detailed stability analysis, an area where traditional methods frequently underperform. Due to its flexibility and effectiveness, the NPA presents a powerful and efficient tool for analyzing highly nonlinear oscillators across various fields of engineering and applied science. Full article

This study examines the flow dynamics of synovial fluid within a lubricated knee joint during movement, incorporating the effect of inertia and linear re-absorption at the synovial membrane. The fluid behavior is modeled using a couple-stress fluid framework, which accounts for mechanical phenomena and employs a lubricated membrane. synovial membrane plays a crucial role in reducing drag and enhancing joint lubrication for the formation of a uniform lubrication layer over the cartilage surfaces. The mathematical model of synovial fluid flow through the knee joint presents a set of non-linear partial differential equations solved by a recursive approach and inverse method through the software Mathematica 11. The results indicate that synovial fluid flow generates high pressure and shear stress away from the entry point due to the combined effects of inertial forces, linear re-absorption, and micro-rotation within the couple-stress fluid. Axial flow intensifies at the center of the knee joint during activity in the presence of linear re-absorption and molecular rotation, while transverse flow increases away from the center and near to synovium due to its permeability. These findings provide critical insights for biomedical engineers to quantify pressure and stress distributions in synovial fluid to design artificial joints. Full article

The aim of this work is to model the nonlinear dynamics of conservative oscillators, with restoring force originating from even-order potentials. In particular, we extend our previous findings on inverting the time-integral equation that arises in the solution of such dynamical systems, a task that is almost always intractable in exact form. This is faced and solved by approximating the restoring force with its Chebyshev series truncated to order five; such a quintication approach yields a quinticate oscillator, whose associated time-integral can be inverted in closed form. Our solution procedure is based on the quinticate oscillator coefficients, upon which a second-order polynomial is constructed, which appears in the time-integrand of the quinticate problem, and whose roots determine the expression of the closed-form solution, as well as that of its period. The presented algorithm is implemented in the Mathematica software and validated on some conservative nonlinear oscillators taken from the relevant literature. 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 almost every production firm is to gain maximum profit along with customer satisfaction. The formation of imperfect products is an obvious process in a production system, which is not a good thing from a business point of view. This paper considers an inventory model for an imperfect production system. All the imperfect products are assumed to be reworkable. An investment occurs for in-process inspection to reduce the rate of formation of imperfect items. A comparison is performed with a production system without in-process inspection to demonstrate the effectiveness of the model. The study shows that the implementation of in-process inspection significantly reduces the total cost of the system as compared to a production system without in-process inspection. The results obtained show that the use of in-process inspection can reduce the total cost by up to 9.3%. Moreover, reducing the formation of defective items saves energy as well as resources. In addition, to reduce carbon emissions, a penalty is implemented on carbon emissions caused by manufacturing, reworking, disposal, and indirect emissions caused by the transportation of disposed items to the treatment facility. As everyone should now be concerned about the environment, green technology is implemented to reduce the amount of carbon emissions to some extent. A classical optimization technique is used to achieve decision variables, i.e., optimal production quantity (Q), fraction of profit invested in in-process inspection ($P_f$), and green technology investment (G), such that the total cost of the system is minimized. A sensitivity analysis is performed to determine the effects of various parameters on the decision variables and total cost. Maple 18 and Mathematica 11 software are used for mathematical work and graphical representation. Full article

Retailing strategy can be considered as the most critical factor for the success of industries. Managing deteriorating products in retail demands a strategic approach aimed at mitigating losses while maximizing profitability. This entails a proactive stance towards identifying products nearing expiration, becoming obsolete or showing signs of deterioration. Offering discounts or promotions can stimulate consumer interest and clear out inventory. The promotion of products within the context of retail management involves a multifaceted approach aimed at increasing awareness, generating interest, and ultimately driving sales. Sustainability helps retailers to develop social as well as economic consistency. Every country and their respective governments are currently working towards sustainable development. New technologies in this direction have been introduced. The present paper introduces a retailing model considering green technology as it is becoming popular to lower environmental risks. The items considered in this study are perishable in nature. As product prices and the promotion of products highly influence demand, a demand pattern dependent on price and promotion is therefore considered. This paper presents a sustainable retail-based inventory model that considers preservation technology to lower the rate of deterioration and increase product shelf life. As carbon emissions is currently the biggest threat to the environment, enforcing a penalty may lower its emissions. Carbon emissions costs due to storage, transportation, and preservation are considered herein. This model studies the effect of various cost parameters on the model. A numerical analysis is performed to validate the result. The results of this study show that the implementation of preservation technology not only increases cycle time but also significantly reduces total cost, hence increasing profit. Sensitivity analysis is performed to show the behaviors of different cost parameters on total cost and decision variables. Mathematica 11 and Maple 18 software are used for graphical representation. Full article

We determine the radius of $\alpha$-spirallikeness of order $\cos \left(\alpha \right)/2$ for entire functions represented as infinite products of their positive zeros. The discussion includes several examples featuring special functions such as Gamma functions, Bessel functions, Struve functions, Wright functions, Ramanujan-type entire functions, and q-Bessel functions. We also consider combinations of classical Bessel functions, including both first-order and second-order derivatives. Additionally, several other special functions that can be incorporated into the established classes are described. We utilize Mathematica 12 software to compute the numerical values of the radius for some functions. Full article

In engineering, physics, and other fields, implicit ordinary differential equations are essential to simulate complex systems. However, because of their intrinsic nonlinearity and difficulty separating higher-order derivatives, implicit ordinary differential equations pose substantial challenges. When applied to these types of equations, traditional numerical methods frequently have problems with convergence or require a significant amount of computing power. In this work, we present the multi-stage differential transform method, a novel semi-analytical approach for effectively solving first- and second-order implicit ordinary differential systems, in conjunction with Adomian polynomials. The main contribution of this method is that it simplifies the solution procedure and lowers processing costs by enabling the differential transform method to be applied directly to implicit systems without transforming them into explicit or quasi-linear forms. We obtain straightforward and effective algorithms that build solutions incrementally utilizing the characteristics of Adomian polynomials, providing benefits in theory and practice. By solving several implicit ODE systems that are difficult for traditional software programs such as Maple 2024, Mathematica 14, or Matlab 24.1, we validate our approach. The multi-stage differential transform method’s contribution includes expanded convergence intervals for numerical results, more accurate approximate solutions for wider domains, and the efficient calculation of exact solutions as a convergent power series. Because of its ease of implementation in educational computational tools and substantial advantages in terms of simplicity and efficiency, our method is suitable for researchers and practitioners working with complex implicit differential equations. Full article

This study presents an innovative iterative method designed to approximate common fixed points of generalized contractive mappings. We provide theorems that confirm the convergence and stability of the proposed iteration scheme, further illustrated through examples and visual demonstrations. Moreover, we apply s-convexity to the iteration procedure to construct orbits under convexity conditions, and we present a theorem that determines the condition when a sequence diverges to infinity, known as the escape criterion, for the transcendental sine function $sin\left(u^m\right)-\alpha u+\beta$, where $u,\alpha ,$$\beta \in \mathbb{C}$ and $m\ge 2$. Additionally, we generate chaotic fractals for this orbit, governed by escape criteria, with numerical examples implemented using MATHEMATICA software. Visual representations are included to demonstrate how various parameters influence the coloration and dynamics of the fractals. Furthermore, we observe that enlarging the Mandelbrot set near its petal edges reveals the Julia set, indicating that every point in the Mandelbrot set contains substantial data corresponding to the Julia set’s structure. Full article