Research

15 pages, 434 KiB

Open AccessArticle

A Phase-Type Distribution for the Sum of Two Concatenated Markov Processes Application to the Analysis Survival in Bladder Cancer

by Belén García-Mora, Cristina Santamaría and Gregorio Rubio

Mathematics 2020, 8(12), 2099; https://doi.org/10.3390/math8122099 - 24 Nov 2020

Cited by 1 | Viewed by 1229

Abstract

Stochastic processes are useful and important for modeling the evolution of processes that take different states over time, a situation frequently found in fields such as medical research and engineering. In a previous paper and within this framework, we developed the sum of [...] Read more.

Stochastic processes are useful and important for modeling the evolution of processes that take different states over time, a situation frequently found in fields such as medical research and engineering. In a previous paper and within this framework, we developed the sum of two independent phase-type (PH)-distributed variables, each of them being associated with a Markovian process of one absorbing state. In that analysis, we computed the distribution function, and its associated survival function, of the sum of both variables, also PH-distributed. In this work, in one more step, we have developed a first approximation of that distribution function in order to avoid the calculation of an inverse matrix for the possibility of a bad conditioning of the matrix, involved in the expression of the distribution function in the previous paper. Next, in a second step, we improve this result, giving a second, more accurate approximation. Two numerical applications, one with simulated data and the other one with bladder cancer data, are used to illustrate the two proposed approaches to the distribution function. We compare and argue the accuracy and precision of each one of them by means of their error bound and the application to real data of bladder cancer. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

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17 pages, 354 KiB

Open AccessArticle

Modified King’s Family for Multiple Zeros of Scalar Nonlinear Functions

by Ramandeep Behl, Munish Kansal and Mehdi Salimi

Mathematics 2020, 8(5), 827; https://doi.org/10.3390/math8050827 - 19 May 2020

Cited by 4 | Viewed by 2258

Abstract

There is no doubt that there is plethora of optimal fourth-order iterative approaches available to estimate the simple zeros of nonlinear functions. We can extend these method/methods for multiple zeros but the main issue is to preserve the same convergence order. Therefore, numerous [...] Read more.

There is no doubt that there is plethora of optimal fourth-order iterative approaches available to estimate the simple zeros of nonlinear functions. We can extend these method/methods for multiple zeros but the main issue is to preserve the same convergence order. Therefore, numerous optimal and non-optimal modifications have been introduced in the literature to preserve the order of convergence. Such count of methods that can estimate the multiple zeros are limited in the scientific literature. With this point, a new optimal fourth-order scheme is presented for multiple zeros with known multiplicity. The proposed scheme is based on the weight function strategy involving functions in ratio. Moreover, the scheme is optimal as it satisfies the hypothesis of Kung–Traub conjecture. An exhaustive study of the convergence is shown to determine the fourth order of the methods under certain conditions. To demonstrate the validity and appropriateness for the proposed family, several numerical experiments have been performed. The numerical comparison highlights the effectiveness of scheme in terms of accuracy, stability, and CPU time. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

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28 pages, 843 KiB

Open AccessArticle

Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models

by Habib ur Rehman, Poom Kumam, Ioannis K. Argyros, Meshal Shutaywi and Zahir Shah

Mathematics 2020, 8(5), 822; https://doi.org/10.3390/math8050822 - 19 May 2020

Cited by 21 | Viewed by 3060

Abstract

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second [...] Read more.

In this paper, we propose two modified two-step proximal methods that are formed through the proximal-like mapping and inertial effect for solving two classes of equilibrium problems. A weak convergence theorem for the first method and the strong convergence result of the second method are well established based on the mild condition on a bifunction. Such methods have the advantage of not involving any line search procedure or any knowledge of the Lipschitz-type constants of the bifunction. One practical reason is that the stepsize involving in these methods is updated based on some previous iterations or uses a stepsize sequence that is non-summable. We consider the well-known Nash–Cournot equilibrium models to support our well-established convergence results and see the advantage of the proposed methods over other well-known methods. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

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10 pages, 746 KiB

Open AccessArticle

Oscillation Theorems for Advanced Differential Equations with p-Laplacian Like Operators

by Omar Bazighifan and Poom Kumam

Mathematics 2020, 8(5), 821; https://doi.org/10.3390/math8050821 - 19 May 2020

Cited by 35 | Viewed by 1939

Abstract

The main objective of this paper is to establish new oscillation results of solutions to a class of even-order advanced differential equations with a p-Laplacian like operator. The key idea of our approach is to use the Riccati transformation and the theory [...] Read more.

The main objective of this paper is to establish new oscillation results of solutions to a class of even-order advanced differential equations with a p-Laplacian like operator. The key idea of our approach is to use the Riccati transformation and the theory of comparison with first and second-order delay equations. Some examples are provided to illustrate the main results. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

21 pages, 5159 KiB

Open AccessArticle

Malmquist Productivity Analysis of Top Global Automobile Manufacturers

by Chia-Nan Wang, Hector Tibo and Hong Anh Nguyen

Mathematics 2020, 8(4), 580; https://doi.org/10.3390/math8040580 - 14 Apr 2020

Cited by 10 | Viewed by 3920

Abstract

The automobile industry is one of the largest economies in the world, by revenue. Being one of the industries with higher employment output, this has become a major determinant of economic growth. In view of the declining automobile production after a period of [...] Read more.

The automobile industry is one of the largest economies in the world, by revenue. Being one of the industries with higher employment output, this has become a major determinant of economic growth. In view of the declining automobile production after a period of continuous growth in the 2008 global auto crisis, the re-evaluation of automobile manufacturing is necessary. This study applies the Malmquist productivity index (MPI), one of the many models in the Data Envelopment Analysis (DEA), to analyze the performance of the world’s top 20 automakers over the period of 2015–2018. The researchers assessed the technical efficiency, technological progress, and the total factor productivity of global automobile manufacturers, using a variety of input and output variables which are considered to be essential financial indicators, such as total assets, shareholder’s equity, cost of revenue, operating expenses, revenue, and net income. The results show that the most productive automaker on average is Volkswagen, followed by Honda, BAIC, General Motors, and Suzuki. On the contrary, Mitsubishi and Tata Motors were the worst-performing automakers during the studied period. This study provides a general overview of the global automobile industry. This paper can be a valuable reference for car managers, policymakers, and investors, to aid their decision-making on automobile management, investment, and development. This research is also a contribution to organizational performance measurement, using the DEA Malmquist model. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

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14 pages, 802 KiB

Open AccessArticle

Local Convergence for Multi-Step High Order Solvers under Weak Conditions

by Ramandeep Behl and Ioannis K. Argyros

Mathematics 2020, 8(2), 179; https://doi.org/10.3390/math8020179 - 02 Feb 2020

Cited by 1 | Viewed by 1496

Abstract

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the [...] Read more.

Our aim in this article is to suggest an extended local convergence study for a class of multi-step solvers for nonlinear equations valued in a Banach space. In comparison to previous studies, where they adopt hypotheses up to 7th Fŕechet-derivative, we restrict the hypotheses to only first-order derivative of considered operators and Lipschitz constants. Hence, we enlarge the suitability region of these solvers along with computable radii of convergence. In the end of this study, we choose a variety of numerical problems which illustrate that our works are applicable but not earlier to solve nonlinear problems. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

12 pages, 332 KiB

Open AccessArticle

New Improvement of the Domain of Parameters for Newton’s Method

by Cristina Amorós, Ioannis K. Argyros, Daniel González, Ángel Alberto Magreñán, Samundra Regmi and Íñigo Sarría

Mathematics 2020, 8(1), 103; https://doi.org/10.3390/math8010103 - 08 Jan 2020

Cited by 6 | Viewed by 2261

Abstract

There is a need to extend the convergence domain of iterative methods for computing a locally unique solution of Banach space valued operator equations. This is because the domain is small in general, limiting the applicability of the methods. The new idea involves [...] Read more.

There is a need to extend the convergence domain of iterative methods for computing a locally unique solution of Banach space valued operator equations. This is because the domain is small in general, limiting the applicability of the methods. The new idea involves the construction of a tighter set than the ones used before also containing the iterates leading to at least as tight Lipschitz parameters and consequently a finer local as well as a semi-local convergence analysis. We used Newton’s method to demonstrate our technique. However, our technique can be used to extend the applicability of other methods too in an analogous manner. In particular, the new information related to the location of the solution improves the one in previous studies. This work also includes numerical examples that validate the proven results. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

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14 pages, 1015 KiB

Open AccessArticle

Semi-Local Analysis and Real Life Applications of Higher-Order Iterative Schemes for Nonlinear Systems

by Ramandeep Behl, Sonia Bhalla, Ioannis K. Argyros and Sanjeev Kumar

Mathematics 2020, 8(1), 92; https://doi.org/10.3390/math8010092 - 06 Jan 2020

Cited by 2 | Viewed by 1890

Abstract

Our aim is to improve the applicability of the family suggested by Bhalla et al. (Computational and Applied Mathematics, 2018) for the approximation of solutions of nonlinear systems. Semi-local convergence relies on conditions with first order derivatives and Lipschitz constants in contrast to [...] Read more.

Our aim is to improve the applicability of the family suggested by Bhalla et al. (Computational and Applied Mathematics, 2018) for the approximation of solutions of nonlinear systems. Semi-local convergence relies on conditions with first order derivatives and Lipschitz constants in contrast to other works requiring higher order derivatives not appearing in these schemes. Hence, the usage of these schemes is improved. Moreover, a variety of real world problems, namely, Bratu’s 1D, Bratu’s 2D and Fisher’s problems, are applied in order to inspect the utilization of the family and to test the theoretical results by adopting variable precision arithmetics in Mathematica 10. On account of these examples, it is concluded that the family is more efficient and shows better performance as compared to the existing one. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

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9 pages, 240 KiB

Open AccessArticle

A Comparison of Methods for Determining the Time Step When Propagating with the Lanczos Algorithm

by N. Mohankumar and Tucker Carrington

Mathematics 2019, 7(11), 1109; https://doi.org/10.3390/math7111109 - 15 Nov 2019

Cited by 2 | Viewed by 1786

Abstract

To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 [...] Read more.

To use the short iterative Lanczos algorithm to solve the time-dependent Schroedinger equation, one must choose, for a given Lanczos space size, a time step. We compare the derivation of the well-known Lubich and Hochbruck time step from SIAM J. Numer. Anal. 34 (1997) 1911 with the a priori time step we proposed in Mohankumar and Carrington (MC) Comput. Phys. Commun., 181 (2010) 1859 and demonstrate that the MC time step is somewhat larger, i.e., that the MC error bound is tighter. In addition, we use the MC approach to derive an error bound and time step for imaginary time propagation. The error bound we derive is much tighter than the error bound of Stewart and Leyk. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications 2020)

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