Mathematical and Computational Applications

13 pages, 270 KiB

Open AccessArticle

New Model for Hill’s Problem in the Framework of Continuation Fractional Potential

by Elbaz I. Abouelmagd

Math. Comput. Appl. 2024, 29(3), 34; https://doi.org/10.3390/mca29030034 - 02 May 2024

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In this work, we derived a new type model for spatial Hill’s system considering the created perturbation by the parameter effect of the continuation fractional potential. The new model is considered a reduced system from the restricted three-body problem under the same effect [...] Read more.

In this work, we derived a new type model for spatial Hill’s system considering the created perturbation by the parameter effect of the continuation fractional potential. The new model is considered a reduced system from the restricted three-body problem under the same effect for describing Hill’s problem. We identified the associated Lagrangian and Hamiltonian functions of the new system, and used them to verify the existence of the new equations of motion. We also proved that the new model has different six valid solutions under different six symmetries transformations as well as the original solution, where the new model is an invariant under these transformations. The several symmetries of Hill’s model can extremely simplify the calculation and analysis of preparatory studies for the dynamical behavior of the system. Finally, we confirm that these symmetries also authorize us to explore the similarities and differences among many classes of paths that otherwise differ from the obtained trajectories by restricted three-body problem. Full article

17 pages, 10189 KiB

Open AccessArticle

Evaluation of Aortic Valve Pressure Gradients for Increasing Severities of Rheumatic and Calcific Stenosis Using Empirical and Numerical Approaches

by Lindi Grobler, Ryno Laubscher, Johan van der Merwe and Philip G. Herbst

Math. Comput. Appl. 2024, 29(3), 33; https://doi.org/10.3390/mca29030033 - 28 Apr 2024

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Abstract

The evaluation and accurate diagnosis of the type and severity of aortic stenosis relies on the precision of medical imaging technology and clinical correlations and the expertise of medical professionals. The application of the clinical correlation to different aortic stenosis morphologies and severities [...] Read more.

The evaluation and accurate diagnosis of the type and severity of aortic stenosis relies on the precision of medical imaging technology and clinical correlations and the expertise of medical professionals. The application of the clinical correlation to different aortic stenosis morphologies and severities is investigated. The manner in which numerical techniques can be used to simulate the blood flow through pathological aortic valves was analysed and compared to the ground-truth CFD model. Larger pressure gradients are estimated in all severities of rheumatic aortic valves compared to calcific aortic valves. The zero-dimensional morphology-insensitive model underpredicted the transvalvular pressure gradient with the greatest error. The 1D model underestimated the pressure gradient in rheumatic cases and overestimated the pressure gradient in calcific cases. The pressure gradients estimated by the clinical approach depends on the location of the flow vena contracta and is sensitive to the severity and type of valve lesion. Through the analysis of entropy generation within the flow domain, the dominant parameters and regions driving adverse pressure gradients were identified. It is concluded that sudden expansion is the dominant parameter leading to higher pressure gradients in rheumatic heart valves compared to calcific ones. Full article

(This article belongs to the Special Issue Current Problems and Advances in Computational and Applied Mechanics (AfriComp6))

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15 pages, 512 KiB

Open AccessArticle

Recognizable Languages of k-Forcing Automata

by Marzieh Shamsizadeh, Mohammad Mehdi Zahedi, Khadijeh Abolpour and Manuel De la Sen

Math. Comput. Appl. 2024, 29(3), 32; https://doi.org/10.3390/mca29030032 - 25 Apr 2024

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Abstract

In this study, we show that automata theory is also a suitable tool for analyzing a more complex type of the k-forcing process. First, the definition of k-forcing automata is presented according to the definition of k-forcing for graphs. Moreover, we study and [...] Read more.

In this study, we show that automata theory is also a suitable tool for analyzing a more complex type of the k-forcing process. First, the definition of k-forcing automata is presented according to the definition of k-forcing for graphs. Moreover, we study and discuss the language of k-forcing automata for particular graphs. Also, for some graphs with different k-forcing sets, we study the languages of their k-forcing automata. In addition, for some given recognizable languages, we study the structure of graphs. After that, we show that k-forcing automata arising from isomorph graphs are also isomorph. Also, we present the style of words that can be recognized with k-forcing automata. Moreover, we introduce the structure of graphs the k-forcing automata arising from which recognize some particular languages. To clarify the notions and the results obtained in this study, some examples are submitted as well. Full article

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15 pages, 313 KiB

Open AccessArticle

A New Generalized Definition of Fractal–Fractional Derivative with Some Applications

by Francisco Martínez and Mohammed K. A. Kaabar

Math. Comput. Appl. 2024, 29(3), 31; https://doi.org/10.3390/mca29030031 - 25 Apr 2024

Viewed by 242

Abstract

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical [...] Read more.

In this study, a new generalized fractal–fractional (FF) derivative is proposed. By applying this definition to some elementary functions, we show its compatibility with the results of the FF derivative in the Caputo sense with the power law. The main elements of classical differential calculus are introduced in terms of this new derivative. Thus, we establish and demonstrate the basic operations with derivatives, chain rule, mean value theorems with their immediate applications and inverse function’s derivative. We complete the theory of generalized FF calculus by proposing a notion of integration and presenting two important results of integral calculus: the fundamental theorem and Barrow’s rule. Finally, we analytically solve interesting FF ordinary differential equations by applying our proposed definition. Full article

16 pages, 4439 KiB

Open AccessArticle

Simulation of Temperature Field in Steel Billets during Reheating in Pusher-Type Furnace by Meshless Method

by Qingguo Liu, Umut Hanoglu, Zlatko Rek and Božidar Šarler

Math. Comput. Appl. 2024, 29(3), 30; https://doi.org/10.3390/mca29030030 - 24 Apr 2024

Viewed by 330

Abstract

Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. [...] Read more.

Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. Inside each of the billets, the heat diffusion equation is solved on a two-dimensional central slice of the billet. The diffusion equation is solved in a strong form by the Local Radial Basis Function Collocation Method (LRBFCM) with explicit time-stepping. The ray tracing procedure solves the radiation, where the view factors are computed with the Monte Carlo method. The changing number of billets in the furnace at the start and the end of the loading and unloading of the furnace is considered. A sensitivity study on billets’ temperature evolution is performed as a function of a different number of rays used in the Monte Carlo method, different stopping times of the billets in the furnace, and different spacing between the billets. The temperature field simulation is also essential for automatically optimizing the furnace’s productivity, energy consumption, and the billet’s quality. For the first time, the LRBFCM is successfully demonstrated for solving such a complex industrial problem. Full article

(This article belongs to the Special Issue Radial Basis Functions)

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Math. Comput. Appl., Volume 29, Issue 3 (June 2024) – 5 articles

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