Research

17 pages, 2186 KiB

Open AccessArticle

An Efficient Optimal Derivative-Free Fourth-Order Method and Its Memory Variant for Non-Linear Models and Their Dynamics

by Himani Sharma, Munish Kansal and Ramandeep Behl

Math. Comput. Appl. 2023, 28(2), 48; https://doi.org/10.3390/mca28020048 - 22 Mar 2023

Viewed by 1102

Abstract

We propose a new optimal iterative scheme without memory free from derivatives for solving non-linear equations. There are many iterative schemes existing in the literature which either diverge or fail to work when

f^{'} (x) = 0

. However, our [...] Read more.

We propose a new optimal iterative scheme without memory free from derivatives for solving non-linear equations. There are many iterative schemes existing in the literature which either diverge or fail to work when

f^{'} (x) = 0

. However, our proposed scheme works even in these cases. In addition, we extended the same idea for iterative methods with memory with the help of self-accelerating parameters estimated from the current and previous approximations. As a result, the order of convergence increased from four to seven without the addition of any further functional evaluation. To confirm the theoretical results, numerical examples and comparisons with some of the existing methods are included which reveal that our scheme is more efficient than the existing schemes. Furthermore, basins of attraction are also included to describe a clear picture of the convergence of the proposed method as well as some of the existing methods. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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20 pages, 352 KiB

Open AccessArticle

On Some Fixed Point Iterative Schemes with Strong Convergence and Their Applications

by Anku, Mona Narang and Vinay Kanwar

Math. Comput. Appl. 2023, 28(2), 45; https://doi.org/10.3390/mca28020045 - 20 Mar 2023

Viewed by 1378

Abstract

In this paper, a new one-parameter class of fixed point iterative method is proposed to approximate the fixed points of contractive type mappings. The presence of an arbitrary parameter in the proposed family increases its interval of convergence. Further, we also propose new [...] Read more.

In this paper, a new one-parameter class of fixed point iterative method is proposed to approximate the fixed points of contractive type mappings. The presence of an arbitrary parameter in the proposed family increases its interval of convergence. Further, we also propose new two-step and three-step fixed point iterative schemes. We also discuss the stability, strong convergence and fastness of the proposed methods. Furthermore, numerical experiments are performed to check the applicability of the new methods, and these have been compared with well-known similar existing methods in the literature. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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11 pages, 2204 KiB

Open AccessArticle

A Multiplicative Calculus Approach to Solve Applied Nonlinear Models

by Gurjeet Singh, Sonia Bhalla and Ramandeep Behl

Math. Comput. Appl. 2023, 28(2), 28; https://doi.org/10.3390/mca28020028 - 21 Feb 2023

Cited by 2 | Viewed by 1259

Abstract

Problems such as population growth, continuous stirred tank reactor (CSTR), and ideal gas have been studied over the last four decades in the fields of medical science, engineering, and applied science, respectively. Some of the main motivations were to understand the pattern of [...] Read more.

Problems such as population growth, continuous stirred tank reactor (CSTR), and ideal gas have been studied over the last four decades in the fields of medical science, engineering, and applied science, respectively. Some of the main motivations were to understand the pattern of such issues and how to obtain the solution to them. With the help of applied mathematics, these problems can be converted or modeled by nonlinear expressions with similar properties. Then, the required solution can be obtained by means of iterative techniques. In this manuscript, we propose a new iterative scheme for computing multiple roots (without prior knowledge of multiplicity m) based on multiplicative calculus rather than standard calculus. The structure of our scheme stands on the well-known Schröder method and also retains the same convergence order. Some numerical examples are tested to find the roots of nonlinear equations, and results are found to be competent compared with ordinary derivative methods. Finally, the new scheme is also analyzed by the basin of attractions that also supports the theoretical aspects. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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15 pages, 3974 KiB

Open AccessArticle

Higher-Order Multiplicative Derivative Iterative Scheme to Solve the Nonlinear Problems

by Gurjeet Singh, Sonia Bhalla and Ramandeep Behl

Math. Comput. Appl. 2023, 28(1), 23; https://doi.org/10.3390/mca28010023 - 09 Feb 2023

Cited by 1 | Viewed by 1310

Abstract

Grossman and Katz (five decades ago) suggested a new definition of differential and integral calculus which utilizes the multiplicative and division operator as compared to addition and subtraction. Multiplicative calculus is a vital part of applied mathematics because of its application in the [...] Read more.

Grossman and Katz (five decades ago) suggested a new definition of differential and integral calculus which utilizes the multiplicative and division operator as compared to addition and subtraction. Multiplicative calculus is a vital part of applied mathematics because of its application in the areas of biology, science and finance, biomedical, economic, etc. Therefore, we used a multiplicative calculus approach to develop a new fourth-order iterative scheme for multiple roots based on the well-known King’s method. In addition, we also propose a detailed convergence analysis of our scheme with the help of a multiplicative calculus approach rather than the normal one. Different kinds of numerical comparisons have been suggested and analyzed. The obtained results (from line graphs, bar graphs and tables) are very impressive compared to the earlier iterative methods of the same order with the ordinary derivative. Finally, the convergence of our technique is also analyzed by the basin of attractions, which also supports the theoretical aspects. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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20 pages, 4615 KiB

Open AccessArticle

Chemical MHD Hiemenz Flow over a Nonlinear Stretching Sheet and Brownian Motion Effects of Nanoparticles through a Porous Medium with Radiation Effect

by Faisal Salah, Abdelmgid O. M. Sidahmed and K. K. Viswanathan

Math. Comput. Appl. 2023, 28(1), 21; https://doi.org/10.3390/mca28010021 - 07 Feb 2023

Cited by 3 | Viewed by 1413

Abstract

In this paper, the numerical solutions for magneto-hydrodynamic Hiemenz fluid over a nonlinear stretching sheet and the Brownian motion effects of nanoparticles through a porous medium with chemical reaction and radiation are studied. The repercussions of thermophoresis and mass transfer at the stagnation [...] Read more.

In this paper, the numerical solutions for magneto-hydrodynamic Hiemenz fluid over a nonlinear stretching sheet and the Brownian motion effects of nanoparticles through a porous medium with chemical reaction and radiation are studied. The repercussions of thermophoresis and mass transfer at the stagnation point flow are discussed. The plate progresses in the contrary direction or in the free stream orientation. The underlying PDEs are reshaped into a set of ordinary differential equations employing precise transformation. They are addressed numerically using the successive linearization method, which is an efficient systematic process. The main goal of this study is to compare the solutions obtained using the successive linearization method to solve the velocity and temperature equations in the presence of

m

changes, thereby demonstrating its accuracy and suitability for solving nonlinear differential equations. For comparison, tables containing the results are presented. This contrast is significant because it demonstrates the accuracy with which a set of nonlinear differential equations can be solved using the successive linearization method. The resulting solution is examined and discussed with respect to a number of engineering parameters. Graphs exemplify the simulation of distinct parameters that govern the motion factors. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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15 pages, 5282 KiB

Open AccessArticle

Entropy Generation of Cu–Al₂O₃/Water Flow with Convective Boundary Conditions through a Porous Stretching Sheet with Slip Effect, Joule Heating and Chemical Reaction

by Maria Immaculate Joyce, Jagan Kandasamy and Sivasankaran Sivanandam

Math. Comput. Appl. 2023, 28(1), 18; https://doi.org/10.3390/mca28010018 - 02 Feb 2023

Cited by 6 | Viewed by 1693

Abstract

Currently, the efficiency of heat exchange is not only determined by enhancements in the rate of heat transfer but also by economic and accompanying considerations. Responding to this demand, many scientists have been involved in improving heat transfer performance, which is referred to [...] Read more.

Currently, the efficiency of heat exchange is not only determined by enhancements in the rate of heat transfer but also by economic and accompanying considerations. Responding to this demand, many scientists have been involved in improving heat transfer performance, which is referred to as heat transfer enhancement, augmentation, or intensification. This study deals with the influence on hybrid Cu–Al₂CO₃/water nanofluidic flows on a porous stretched sheet of velocity slip, convective boundary conditions, Joule heating, and chemical reactions using an adapted Tiwari–Das model. Nonlinear fundamental equations such as continuity, momentum, energy, and concentration are transmuted into a non-dimensional ordinary nonlinear differential equation by similarity transformations. Numerical calculations are performed using HAM and the outcomes are traced on graphs such as velocity, temperature, and concentration. Temperature and concentration profiles are elevated as porosity is increased, whereas velocity is decreased. The Biot number increases the temperature profile. The rate of entropy is enhanced as the Brinkman number is raised. A decrease in the velocity is seen as the slip increases. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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15 pages, 590 KiB

Open AccessArticle

A Model for the Generalised Dispersion of Synovial Fluids on Nutritional Transport with Joint Impacts of Electric and Magnetic Field

by B. Rushi Kumar, R. Vijayakumar and A. Jancy Rani

Math. Comput. Appl. 2023, 28(1), 3; https://doi.org/10.3390/mca28010003 - 27 Dec 2022

Cited by 1 | Viewed by 1335

Abstract

This work analyses the effect of electromagnetic fields on cartilaginous cells in human joints and the nutrients that flow from the synovial fluid to the cartilage. The perturbation approach and the generalised dispersion model is used to solve the governing equation of momentum [...] Read more.

This work analyses the effect of electromagnetic fields on cartilaginous cells in human joints and the nutrients that flow from the synovial fluid to the cartilage. The perturbation approach and the generalised dispersion model is used to solve the governing equation of momentum and mass transfer. The dispersion coefficient increases with dimensionless time. It aids in grasping the level of nutritional transport to the synovial joint. Low-molecular-weight solutes have a lower concentration distribution at the same depth in articular cartilage than high-molecular-weight solutes. Thus, diffusion dominates nutrition transport for low-molecular-weight solutes, whereas a mechanical pumping action dominates nutrition transport for high-molecular-weight solutes. The report says that the cells in the centre of the cartilage surface receive more nutrients during imbibition and exudation than the cells on the periphery, and the earliest indications of cartilage degradation emerge in the uninflected regions. As a result, cartilage nutrition is considered necessary to joint mobility. It also predicts that, as the viscoelastic parameter increases, the concentration in the articular cartilage diminishes, resulting in the cartilage cells receiving less nutrition, which might lead to harmful effects. The dispersion coefficient and mean concentration for distinct factors, such as the Hartmann number, porous parameter, and viscoelastic parameters of gel formation, have been computed and illustrated through graphics. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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12 pages, 2664 KiB

Open AccessArticle

Impacts of Casson Model on Hybrid Nanofluid Flow over a Moving Thin Needle with Dufour and Soret and Thermal Radiation Effects

by Vinodh Srinivasa Reddy, Jagan Kandasamy and Sivasankaran Sivanandam

Math. Comput. Appl. 2023, 28(1), 2; https://doi.org/10.3390/mca28010002 - 27 Dec 2022

Cited by 7 | Viewed by 1391

Abstract

The current study used a novel Casson model to investigate hybrid Al₂O₃-Cu/Ethylene glycol nanofluid flow over a moving thin needle under MHD, Dufour–Soret effects, and thermal radiation. By utilizing the appropriate transformations, the governing partial differential equations are transformed [...] Read more.

The current study used a novel Casson model to investigate hybrid Al₂O₃-Cu/Ethylene glycol nanofluid flow over a moving thin needle under MHD, Dufour–Soret effects, and thermal radiation. By utilizing the appropriate transformations, the governing partial differential equations are transformed into ordinary differential equations. The transformed ordinary differential equations are solved analytically using HAM. Furthermore, we discuss velocity profiles, temperature profiles, and concentration profiles for various values of governing parameters. Skin friction coefficient increases by upto 45% as the Casson parameter raised upto 20%, and the heat transfer rate also increases with the inclusion of nanoparticles. Additionally, local skin friction, a local Nusselt number, and a local Sherwood number for many parameters are entangled in this article. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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13 pages, 4359 KiB

Open AccessArticle

Go-MoS₂/Water Flow over a Shrinking Cylinder with Stefan Blowing, Joule Heating, and Thermal Radiation

by Manoj Kumar Narayanaswamy, Jagan Kandasamy and Sivasankaran Sivanandam

Math. Comput. Appl. 2022, 27(6), 110; https://doi.org/10.3390/mca27060110 - 14 Dec 2022

Cited by 6 | Viewed by 1276

Abstract

The impacts of Stefan blowing along with slip and Joule heating on hybrid nanofluid (HNF) flow past a shrinking cylinder are investigated in the presence of thermal radiation. Using the suitable transformations, the governing equations are converted into ODEs, and the MATLAB tool [...] Read more.

The impacts of Stefan blowing along with slip and Joule heating on hybrid nanofluid (HNF) flow past a shrinking cylinder are investigated in the presence of thermal radiation. Using the suitable transformations, the governing equations are converted into ODEs, and the MATLAB tool bvp4c is used to solve the resulting equations. As Stefan blowing increases, temperature and concentration profiles are accelerated but the velocity profile diminishes and also the heat transfer rate improves up to 25% as thermal radiation upsurges. The mass transfer rate diminishes as increasing Stefan blowing. The Sherwood number, the Nusselt number, and the skin friction coefficient are numerically tabulated and graphs are also plotted. The outcomes are conscientiously and thoroughly discussed. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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18 pages, 8351 KiB

Open AccessArticle

Role of Nanoparticles and Heat Source/Sink on MHD Flow of Cu-H₂O Nanofluid Flow Past a Vertical Plate with Soret and Dufour Effects

by Ramesh Kune, Hari Singh Naik, Borra Shashidar Reddy and Christophe Chesneau

Math. Comput. Appl. 2022, 27(6), 102; https://doi.org/10.3390/mca27060102 - 28 Nov 2022

Cited by 5 | Viewed by 1511

Abstract

The study is devoted to investigating the effect of an unsteady non-Newtonian Casson fluid over a vertical plate. A mathematical analysis is presented for a Casson fluid by taking into consideration Soret and Dufour effects, heat generation, heat radiation, and chemical reaction. The [...] Read more.

The study is devoted to investigating the effect of an unsteady non-Newtonian Casson fluid over a vertical plate. A mathematical analysis is presented for a Casson fluid by taking into consideration Soret and Dufour effects, heat generation, heat radiation, and chemical reaction. The novelty of the problem is the physical interpretation of Casson fluid before and after adding copper water-based nanoparticles to the governing flow. It is found that velocity was decreased and the temperature profile was enhanced. A similarity transformation is used to convert the linked partial differential equations that control flow into non-linear coupled ordinary differential equations. The momentum, energy, and concentration formulations are cracked by means of the finite element method. The thermal and solute layer thickness growth is due to the nanoparticles’ thermo-diffusion. The effects of relevant parameters such as the Casson fluid parameter, radiation, Soret and Dufour effects, chemical reaction, and Prandtl number are discussed. A correlation of the average Nusselt number and Sherwood number corresponding to active parameters is presented. It can be noticed that increasing the Dufour number leads to an uplift in heat transfer. Fluid velocity increases with Grashof number and decreases with magnetic effect. The impact of heat sources and radiation is to increase the thermal conductivity. Concentration decreases with the Schmidt number. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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14 pages, 583 KiB

Open AccessArticle

Three-Dimensional Non-Linearly Thermally Radiated Flow of Jeffrey Nanoliquid towards a Stretchy Surface with Convective Boundary and Cattaneo–Christov Flux

by Kandasamy Jagan and Sivanandam Sivasankaran

Math. Comput. Appl. 2022, 27(6), 98; https://doi.org/10.3390/mca27060098 - 19 Nov 2022

Cited by 7 | Viewed by 1051

Abstract

The objective of this paper is to investigate the 3D non-linearly thermally radiated flow of a Jeffrey nanofluid towards a stretchy surface with the Cattaneo–Christov heat flux (CCHF) model in the presence of a convective boundary condition.The Homotopy Analysis Method (HAM) is used [...] Read more.

The objective of this paper is to investigate the 3D non-linearly thermally radiated flow of a Jeffrey nanofluid towards a stretchy surface with the Cattaneo–Christov heat flux (CCHF) model in the presence of a convective boundary condition.The Homotopy Analysis Method (HAM) is used to solve the ordinary differential equation that is obtained by reforming the governing equation using suitable transformations. The equations obtained from HAM are plotted graphically for different parameters. In addition, the skin-friction coefficient, local Nusselt number, and Sherwood number for various parameters are calculated and discussed. The velocity profile along the x- and y-directions decrease with a raise in the ratio of relaxation to retardation times. The concentration and temperature profile rises while magnifying the ratio of relaxation to retardation times. While raising the ratio parameter, the x-direction velocity, temperature, and concentration profile diminishes, whereas the y-direction velocity profile magnifies. Magnifying the Deborah number results in a rise in the velocity profile along the x- and y-directions, and a decline in the temperature and concentration profile. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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18 pages, 1290 KiB

Open AccessArticle

An Efficient Two-Step Iterative Family Adaptive with Memory for Solving Nonlinear Equations and Their Applications

by Himani Sharma, Munish Kansal and Ramandeep Behl

Math. Comput. Appl. 2022, 27(6), 97; https://doi.org/10.3390/mca27060097 - 18 Nov 2022

Cited by 1 | Viewed by 1319

Abstract

We propose a new iterative scheme without memory for solving nonlinear equations. The proposed scheme is based on a cubically convergent Hansen–Patrick-type method. The beauty of our techniques is that they work even though the derivative is very small in the vicinity of [...] Read more.

We propose a new iterative scheme without memory for solving nonlinear equations. The proposed scheme is based on a cubically convergent Hansen–Patrick-type method. The beauty of our techniques is that they work even though the derivative is very small in the vicinity of the required root or

f^{'} (x) = 0

. On the contrary, the previous modifications either diverge or fail to work. In addition, we also extended the same idea for an iterative method with memory. Numerical examples and comparisons with some of the existing methods are included to confirm the theoretical results. Furthermore, basins of attraction are included to describe a clear picture of the convergence of the proposed method as well as that of some of the existing methods. Numerical experiments are performed on engineering problems, such as fractional conversion in a chemical reactor, Planck’s radiation law problem, Van der Waal’s problem, trajectory of an electron in between two parallel plates. The numerical results reveal that the proposed schemes are of utmost importance to be applied on various real–life problems. Basins of attraction also support this aspect. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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14 pages, 10766 KiB

Open AccessArticle

Impacts of Stefan Blowing on Hybrid Nanofluid Flow over a Stretching Cylinder with Thermal Radiation and Dufour and Soret Effect

by Manoj Kumar Narayanaswamy, Jagan Kandasamy and Sivasankaran Sivanandam

Math. Comput. Appl. 2022, 27(6), 91; https://doi.org/10.3390/mca27060091 - 02 Nov 2022

Cited by 5 | Viewed by 1618

Abstract

The focal interest in this article is to investigate the Stefan blowing and Dufour and Soret effects on hybrid nanofluid (HNF) flow towards a stretching cylinder with thermal radiation. The governing equations are converted into ODE by using suitable transformations. The boundary value [...] Read more.

The focal interest in this article is to investigate the Stefan blowing and Dufour and Soret effects on hybrid nanofluid (HNF) flow towards a stretching cylinder with thermal radiation. The governing equations are converted into ODE by using suitable transformations. The boundary value problem solver (bvp4c), which is a package in the MATLAB, is used to solve the resulting ODE equations. Results show that rise in the Stefan blowing enhances velocity, temperature, and concentration profiles. Heat transfer rate increases by up to 10% in the presence of 4% nanoparticle/HNF but mass transfer rate diminishes. Additionally, skin friction coefficient, Nusselt number and Sherwood number are examined for many parameters entangled in this article. Additionally, results are deliberatively discussed in detail. Full article

(This article belongs to the Special Issue Mathematical and Computational Modeling on Fluid Flow and Heat Transfer)

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