Maple Software Research Articles

Application of the New Mapping Method to Complex Three Coupled Maccari’s System Possessing M-Fractional Derivative

In this academic investigation, an innovative mapping approach is applied to complex three coupled Maccari’s system to unveil novel soliton solutions. This is achieved through the utilization of M-Truncated fractional derivative with employing the new mapping method and computer algebraic syatem (CAS) such as Maple. The derived solutions in the form of hyperbolic and trigonometric functions. Our study elucidates a variety of soliton solutions such as periodic, singular, dark, kink, bright, dark-bright solitons solutions. To facilitate comprehension, with certain solutions being visually depicted through 2-dimensional, contour, 3-dimensional, and phase plots depicting bifurcation characteristics utilizing Maple software. Furthermore, the incorporation of M-Truncated derivative enables a more extensive exploration of solution patterns. Our study establishes a connection between computer science and soliton physics, emphasizing the pivotal role of soliton phenomena in advancing simulations and computational modeling. Analytical solutions are subsequently generated through the application of the new mapping method. Following this, a thorough examination of the dynamic nature of the equation is conducted from various perspectives. In essence, understanding the dynamic characteristics of systems is of great importance for predicting outcomes and advancing new technologies. This research significantly contributes to the convergence of theoretical mathematics and applied computer science, emphasizing the crucial role of solitons in scientific disciplines.

Magnetic williamson hybrid nanofluid flow around an inclined stretching cylinder with joule heating in a porous medium

The study of an inclined stretching cylinder in a porous medium holds significant implications for understanding complex fluid dynamics and heat transfer phenomena, mainly when influenced by a hybrid nanofluid containing Cu and Al2O3, Joule heating, and a magnetic field in the presence of Williamson fluid. The nonlinear ordinary differential equations are derived by applying pertinent similarity transformations to partial differential equations. Subsequently, these non-dimensional ordinary differential equations are transformed into a system of first-order ODEs and numerically solved using Maplesoft's symbolic computation software MAPLE 2023. The influence of governing parameters on dimensionless velocity, temperature, concentration, skin friction, heat, and mass transfer rates is investigated. This investigation is crucial for various practical applications, including geothermal reservoirs, enhanced oil recovery, and environmental processes involving porous media. It is demonstrated that Biot and Eckert numbers and nanofluid parameters contribute to increased surface temperature, while the Lewis number leads to a decrease in dimensionless concentration. Moreover, the curvature parameter is associated with an increase in dimensionless concentration, and skin friction demonstrates a direct relationship with the magnetic parameter while inversely correlating with the cylinder's curvature.These results contribute to the scientific understanding of these complex interactions and provide valuable insights for engineering applications, such as designing efficient heat exchangers, optimizing cooling systems, and advancing technologies involving fluid flow and magnetic fields.

DERIVATION OF EXPRESSION FOR PHOTOCURRENT DENSITY FOR NON-DESTRUCTIVE TESTING OF 3D PRINTING FILAMENT BY MEANS OF TERAHERTZ SPECTROSCOPY

This report presents a revised expression for the photocurrent density in terahertz spectroscopy, which is a non-destructive testing technique of particular interest to the authors in the context of 3D printed parts. 3D printing, also known as additive manufacturing, involves creating three-dimensional objects based on computer-aided design (CAD) models. The process entails depositing, joining, or solidifying material under computer control, layer by layer.
 
 Defects in 3D printing, such as weak infill, gaps in thin walls, inconsistent extrusion, layer separation, and bed drop, can lead to low printing quality and render some printed parts unfit and unsafe for use. Moreover, the ability to tamper with internal layers without altering the exterior could result in the production of maliciously defective parts without detection. Therefore, it is crucial to test 3D printed details and filaments at each stage of processing using non-destructive methods.
 
 A comprehensive review of the relevant literature indicates the potential for enhancing measurement accuracy through various improvements in terahertz spectrometer models. The mathematical model for the photocurrent involves a convolution integral of the current density and the laser radiation pulse that irradiates the surface of the material under study. The expression within the integral incorporates parameters such as the duration of the optical pulse, carrier lifetime, and momentum relaxation time. By evaluating the integral, the result can be obtained as two terms, each being a product of an exponent and a complementary error function with the same parameters mentioned earlier.
 
 The calculation involves several steps, including a change of variables during integration. Verification using Maple software demonstrates agreement with analytical calculations and suggests a pathway for further refinement of the expression for the photocurrent density. The Maple program influenced the results by means of repeating same calculation with aid of computer and allowing to compare if analytical results are same and true, also it could be use for simulation and example calculation, for results graphical representation. 
 
 The connection between the obtained mathematical expression and its relation to 3D printing (additive manufacturing) exists. The explanation is in that the 3D printer uses filament, filament has defects, defectoscopy of filament in the terahertz domain have models and methods. The research of defectoscopy models and methods is helpful to increase accuracy of measurement of filament defect parameters and account on it and improve the quality of 3D printed details.

Soliton solutions to a nonlinear wave equation via modern methods

AbstractIn this pioneering study, we have systematically derived traveling wave solutions for the highly intricate Zoomeron equation, employing well-established mathematical frameworks, notably the modified (G′/G)-expansion technique. Twenty distinct mathematical solutions have been revealed, each distinguished by distinguishable characteristics in the domains of hyperbolic, trigonometric, and irrational expressions. Furthermore, we have used the formidable computational capabilities of Maple software to construct depictions of these solutions, both in two-dimensional and three-dimensional visualizations. The visual representations vividly capture the essence of our findings, showcasing a diverse spectrum of wave profiles, including the kink-type shape, soliton solutions, bell-shaped waveforms, and periodic traveling wave profiles, all of which are clarified with careful precision.

Triple-diffusive magneto-nanofluid flow over a nonlinearly stretching sheet

Magnetic nanofluid constitutes a special class of nanofluids that exhibit both magnetic and fluid properties. Due to its extensive applications, researchers used the magneto-nanofluid to investigate the thermos-diffusion effects on boundary layer flows. The impact on the thermos-diffusion of magneto-nanofluid over a linear stretching sheet was first introduced by Awad et al., (boundary value problems, 1–13, 2013), then Goyal et al., (microfluidics and nano fluidics, 17(3), 591–604) examined the effects of Brownian motion, thermophoresis, and cross-diffusion on the enhancement of convection features of the power-law stretching sheet. In the present study, we extend Goyal’s finding of the thermophoresis effect by including a magnetic field. Consequently, now five partial differential equations with the magnetic field parameter need to be solved, which are converted to ordinary differential equations with similarity transformations. Then, the resulting nonlinear coupled differential equations are solved analytically using the method of Directly Defining the inverse Mapping with the help of Maple software. This is the first time this novel method is used to solve a system with four non-linear coupled differential equations. The analytical results of the study are compared with the available numerical results for a special case and found good agreement. This leads to the conclusion that this novel method is accurate and can be applied to solve not only higher order but also a system of several non-linear differential equations. Further, the skin friction is analyzed for various values of the magnetic parameter. It is observed that, the velocity profile decreases as the magnetic parameter increases. However, quite the opposite is true on the skin friction.

The dynamical behavior analysis of the fractional perturbed Gerdjikov–Ivanov equation

In this article, the fractional perturbed Gerdjikov–Ivanov equation is investigated. Firstly, the fractional perturbed Gerdjikov–Ivanov equation is transformed into an ordinary differential equation through traveling wave transformation. Secondly, using the trial method of rank homogeneous equation polynomials and the principle of homogeneous equilibrium, a two-dimensional planar dynamic system is presented and its bifurcation behavior is studied. Then, its two-dimensional phase portraits are drawn by using Maple software. Finally, disturbance factors are introduced into the planar dynamical system to study its chaotic behavior, and some two-dimensional phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis graphs of the perturbed system are plotted by using Maple software. The novelty lies in studying the dynamic behavior of the objective equation without the need for solving.

Implementation of Problem Based Learning (PBL) Model Using Maple Software in Solving a Trigonometry Material Problem

This study aims to investigate the implementation and effectiveness of the PBL learning model with Maple Software in solving Trigonometry problems during mathematics lessons. It is descriptive research with a quantitative approach. The participants were 20 tenth-grade students from an vocational school in the second semester of the 2022/2023 academic year. Data were collected through student questionnaires and written tests. The questionnaires assessed student understanding in solving Trigonometry problems with Maple Software, while the tests included 5 questions to evaluate problem-solving skills using the software. The findings indicate that: (1) The PBL model with Maple Software enhances student engagement and effectively solves Trigonometry problems. (2) The PBL model using Maple Software is effective in improving students' problem-solving abilities in Trigonometry.

Dynamical exploration of optical soliton solutions for M-fractional Paraxial wave equation.

This work explores diverse novel soliton solutions due to fractional derivative, dispersive, and nonlinearity effects for the nonlinear time M-fractional paraxial wave equation. The advanced exp [-φ(ξ)] expansion method integrates the nonlinear M-fractional Paraxial wave equation for achieving creative solitonic and traveling wave envelopes to reconnoiter such dynamics. As a result, trigonometric and hyperbolic solutions have been found via the proposed method. Under the conditions of the constraint, fruitful solutions are gained and verified with the use of the symbolic software Maple 18. For any chosen set of the allowed parameters 3D, 2D and density plots illustrate, this inquisition achieved kink shape, the collision of kink type and rogue wave, periodic rogue wave, some distinct singular periodic soliton waves for time M-fractional Paraxial wave equation. As certain nonlinear effects cancel out dispersion effects, optical solitons typically can travel great distances without dissipating. We have constructed reasonable soliton solutions and managed the actual meaning of the acquired solutions of action by characterizing the particular advantages of the summarized parameters by the portrayal of figures and by interpreting the physical occurrences. New precise voyaging wave configurations are obtained using symbolic computation and the previously described methodologies. However, the movement role of the waves is explored, and the modulation instability analysis is used to describe the stability of waves in a dispersive fashion of the obtained solutions, confirming that all created solutions are precise and stable.

Comparative heat transfer analysis of Fe3O4/Graphene-water hybrid nanofluid in unsteady hydrodynamic Lapwood-Brinkmann Forchheimer medium over a curved elongating region

In the current work, MHD flow across an elongating surface is investigated using water based (Fe3O4 and Graphene) hybrid nanoparticles with an unsteady state. A mathematical model is elucidated for curvilinear coordinates in partial differential equations using appropriate boundary conditions subject to a magnetic field, Forchheimer and thermal radiation from the flow's geometry. The Newton Rapshon built-in shooting procedure was adopted with Maple software to numerically analyse transformed non-dimensional systems of governing equations. Graphs have been plotted to further illustrate the influence of emerging factors on the fluid velocity and temperature profiles. It is discovered that the Forchhemier and magnetic parameters both increase the friction drag, also an expansion in the radiation parameter and high-volume proportion of nanoparticles both raise the heat transfer rate, respectively. Furthermore, growing values of the curvature parameter cause the heat transfer rate to decrease and the friction drag to increase. Heat transmission in fluids governs both the heating and cooling systems of engineering applications, so it is observed from the study that hybrid nanofluids have a greater heat transfer rate. Also, our results are compared with existing literature for a particular case, and found to be in good agreement.

A Mathematical Model to Assess the Impact of Tourism Activities on the Economy of Vendors at Bukit Maras, Terengganu, Malaysia

Bukit Maras is one of the recreational spots that has been identified as having great potential to be promoted as a tourist destination in Terengganu. In this study, it is intriguing to see the whether the tourism activities that occurred at Bukit Maras have a positive impact on the economic activity of the local vendors from the point of view of mathematical modelling. The objective of this paper is to formulate a mathematical model in the form of ordinary differential equations (ODEs), inspired by the concept of symbiosis interaction in ecology. Further, some mathematical analyses of this model such as finding the steady states and their stability, were also proved. In addition, some numerical simulations are performed using Maple software to produce the plots of time series as well as phase portraits. To prove the positive impact of the presence of tourists on the economics of vendors, a parameter called the spending rate by the tourists was varied. The results showed that as the spending rate increased, it was expected that the number of vendors would increase too. This indicated that the economics of the vendors have improved. Moreover, it is also predicted that the number of tourists will continuously increase in the future, provided that the attractiveness factors of the Bukit Maras are well preserved. These results are significant for the local government to provide more funding for the development of Bukit Maras so that both tourists and vendors can be sustained in the future.

Analysis of New 4D Hyperchaotic Systems MACM and Implementation Using Runge-kutta

In general, hyperchaotic system is defined as a chaotic system with more than one positive Lyapunov exponent, this implies that its chaotic dynamics extend in several different directions simultaneously. The discovery of reliable and effective methods for solving hyperchaotic systems is carried out extensively. The aim of this research is to implement nonlinear hyperchaotic using runge-kutta. The NHS system is built by inspecting, modifying and adding one state to the MACM chaotic system. In this study, we use the same initial conditions x_0=y_0=z_0=w_0=1 and the parameters a = 2,b = 2,c = 0.5, and d = 14.5. At first we experiments modified approaches, namely Classic Runge-Kutta Method 3 Order, Runge-Kutta Method Based On Arithmetic Mean, Metode Runge-Kutta Method Based On Geometric Mean, Modified Runge-Kutta Geometric Mean Based Method-1, RKLCM Approaches Instead of RKGM using MAPLE software as a calculation tool. In this work, we observe that the Runge-Kutta Method Based on Arithmetic Mean has the smallest error value with an average variable x of 0.000606953, variable y of 0.009280181, variable z of 0.000378639 and variable w of 0.60781585. Then also in RKLCM Approaches Instead of RKGM it has the largest error value with an average variable x of 0.483789793, variable y of 0.620803849, variable z of 0.92097139, and variable w of 0.596137549.

Unveiling optical soliton solutions and bifurcation analysis in the space–time fractional Fokas–Lenells equation via SSE approach

The space–time fractional Fokas–Lenells (STFFL) equation serves as a fundamental mathematical model employed in telecommunications and transmission technology, elucidating the intricate dynamics of nonlinear pulse propagation in optical fibers. This study employs the Sardar sub-equation (SSE) approach within the STFFL equation framework to explore uncharted territories, uncovering a myriad of optical soliton solutions (OSSs) and conducting a thorough analysis of their bifurcations. The discovered OSSs encompass a diverse array, including bright-dark, periodic, multiple bright-dark solitons, and various other types, forming a captivating spectrum. These solutions reveal an intricate interplay among bright-dark solitons, complex periodic sequences, rhythmic breathers, coexistence of multiple bright-dark solitons, alongside intriguing phenomena like kinks, anti-kinks, and dark-bell solitons. This exploration, built upon meticulous literature review, unveils previously undiscovered wave patterns within the dynamic framework of the STFFL equation, significantly expanding the theoretical understanding and paving the way for innovative applications. Utilizing 2D, contour, and 3D diagrams, we illustrate the influence of fractional and temporal parameters on these solutions. Furthermore, comprehensive 2D, 3D, contour, and bifurcation analysis diagrams scrutinize the nonlinear effects inherent in the STFFL equation. Employing a Hamiltonian function (HF) enables detailed phase-plane dynamics analysis, complemented by simulations conducted using Python and MAPLE software. The practical implications of the discovered OSS solutions extend to real-world physical events, underlining the efficacy and applicability of the SSE scheme in solving time–space nonlinear fractional differential equations (TSNLFDEs). Hence, it is crucial to acknowledge the SSE technique as a direct, efficient, and reliable numerical tool, illuminating precise outcomes in nonlinear comparisons.

Mathematical Model of Celebrity Worship Tendency Among K-Pop Fans in South Sulawesi

Celebrity worship is a form of almost obsessional participation in which people idolize and even start to “worship” their favorite celebrities. Twitter reveals that in 2021, Indonesia was ranked fourth as the country with the most number of Korean Pop (K-Pop) fans. In this study, we constructed a SEIRS mathematical model of celebrity worship tendency among K-Pop fans (CWKF) because there have not been specific mathematical models built on the problem of celebrity worship tendency among K-Pop fans. The CWKF free equilibrium point (E0), endemic equilibrium point (E1), and using the Next Generation Matrix for the Basic Reproduction Number (R0) are calculated through analysis. Using the Routh-Hurwitz criteria for equilibrium analysis, we found that E0 is locally asymptotically stable if R0 < 1, and E1 is locally asymptotically stable if R0 > 1. Secondly, we conduct numerical simulations using Maple software, and research data consisting of 204 individuals from various regions in South Sulawesi serves as the basis for this mathematical analysis. The numerical simulation obtained supports the dynamic analysis results that the number of individuals who are K-Pop fans will decrease in the population and disappear within a certain period, even without any special treatment.

Modulation Instability and Dynamical Analysis of New Abundant Closed‐Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M‐Fractional Derivative

In this work, we study the modulation instability (MI) and closed‐form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV‐ZK) equation with a truncated M‐fractional derivative. The mKdV‐ZK equation can be used to describe the behavior of ion‐acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved F‐expansion scheme and unified solver techniques for the mKdV‐ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three‐dimensional (3D) plots with density plots and two‐dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed‐form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved F‐expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell‐shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena.

A new study on the Newell-Whitehead-Segel equation with Caputo-Fabrizio fractional derivative

<p>In this research, we propose a new numerical method that combines with the Caputo-Fabrizio Elzaki transform and the q-homotopy analysis transform method. This work aims to analyze the Caputo-Fabrizio fractional Newell-Whitehead-Segel (NWS) equation utilizing the Caputo-Fabrizio q-Elzaki homotopy analysis transform method. The Newell-Whitehead-Segel equation is a partial differential equation employed for modeling the dynamics of reaction-diffusion systems, specifically in the realm of pattern generation in biological and chemical systems. A convergence analysis of the proposed method was performed. Two-dimensional and three-dimensional graphs of the solutions have been drawn with the Maple software. It is seen that the resulting proposed method is more powerful and effective than the Aboodh transform homotopy perturbation method and conformable Laplace decomposition method in the results.</p>

Optimizing Bank Profitability Model Considering the Role of Modern Banking Technologies

The aim of this study is to present an optimized model for bank profitability by considering the role of modern banking technologies. This research, being exploratory in nature due to its model development, is also considered applied because its results are utilized by beneficiaries. A quantitative method is used in this study to optimize bank profitability while considering the role of modern banking technologies. In the first step, influential components are identified through library techniques and document review, followed by mathematical programming for modeling. The proposed model is optimized using the epsilon-constraint algorithm in the MAPLE software and further optimized using MATLAB software. The research population, as a case study, includes documents and articles related to the banking industry. Initially, a single-component bank, considered as part of the system, was examined. Assume the bank operates in a seven-hour shift, six days a week. The maintenance activity time and fintech interventions were set to 7 time units, and the service delivery and corrective actions were set to 12 time units. With a 10% and 20% increase in the δ_E and δ_(M/C) data, significant changes in the objective function and decision variables were observed. However, changes in other data did not result in noticeable alterations in the model, indicating the importance of preventing the process from shifting to an out-of-control state and the significance of changes in the mean and standard deviation of the targeted quality characteristic. The proposed model determines the optimal value of each of the four decision variables: sample size (n), sampling interval (h), control limit coefficient (k), and intervals for preventive fintech actions (t_PM), which minimizes the total expected cost from integration per time unit. A numerical example demonstrated the economic impact of cost parameters on the integrated policy of profit control and preventive fintech actions. Additionally, in this study, the integrated model was compared with the independent model through a numerical example, and the significant difference in total cost highlighted the superiority of the integrated model over the independent model.

Analysis of Random Difference Equations Using the Differential Transformation Method

The differential transformation method (DTM) is one of the best methods easily applied to linear and nonlinear difference equations with random coefficients. In this study, we apply the theorems related to the DTM to the given examples and investigate the behaviour of the approximate analytical solutions. The expected value, variance, coefficient of variation, and confidence intervals of the solutions of random difference equations obtained from discrete probability distributions such as uniform, geometric, Poisson, and binomial distributions will be calculated. Maple and MATLAB software packages are used to plot the solution graphs and also to interpret the solution behaviour.

The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel

<abstract> <p>The orthogonal polynomials approach with Gegenbauer polynomials is an effective tool for analyzing mixed integral equations (MIEs) due to their orthogonality qualities. This article reviewed recent breakthroughs in the use of Gegenbauer polynomials to solve mixed integral problems. Previous authors studied the problem with a continuous kernel that combined both Volterra (V) and Fredholm (F) components; however, in this paper, we focused on a singular Carleman kernel. The kernel of FI was measured with respect to position in the space <inline-formula><tex-math id="M1">\begin{document}$ {L}_{2}[-\mathrm{1, 1}], $\end{document}</tex-math></inline-formula> while the kernel of Ⅵ was considered as a function of time in the space <inline-formula><tex-math id="M2">\begin{document}$ C[0, T], T < 1 $\end{document}</tex-math></inline-formula>. The existence of a unique solution was discussed in <inline-formula><tex-math id="M3">\begin{document}$ {L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right] $\end{document}</tex-math></inline-formula> space. The solution and its error stability were both investigated and commented on. Finally, numerical examples were reviewed, and their estimated errors were assessed using Maple (2022) software.</p> </abstract>

The novel numerical solutions for time-fractional Fornberg-Whitham equation by using fractional natural transform decomposition method

<p>The time-fractional partial differential equations were solved by the fractional natural transform decomposition method. Fractional derivatives are Caputo sense. The Fornberg-Whitham equation is a generalization of the Korteweg-de Vries (KdV) equation, which describes the propagation of long waves in shallow water. It includes higher-order dispersion terms, making it applicable to a wider range of dispersive media the fractional natural transform decomposition method (FNTDM) was also used to examine applications, and the solutions obtained by this method have been compared to those obtained by the variational iteration method, fractional variational iteration method, and homotopy perturbation method. In addition, the MAPLE package drew graphs of the solutions of nonlinear time-fractional partial differential equations, taking into account physics. The method described in this study exhibited a notable degree of computational precision and straightforwardness when used to the analysis and resolution of intricate phenomena pertaining to fractional nonlinear partial differential equations within the domains of science and technology.</p>

Symmetry Analysis and Wave Solutions of Time Fractional Kupershmidt Equation

This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem.