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New Trends in Solving Partial Derivative Equations and Nonlinear Integral...
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New Trends in Solving Partial Derivative Equations and Nonlinear Integral Equations by Splitting Techniques and Nonlinear Iterative Methods

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 October 2020) | Viewed by 15099

Special Issue Editor

Prof. Dr. Eulalia Martínez Molada

E-Mail Website
Guest Editor
Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain
Interests: numerical analysis; iterative methods in Banach spaces; semilocal and local convergence; computational efficiency
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Large systems of ordinary, partial, or stochastic differential equations as well as integral equations can be treated by splitting techniques that decompose the problem, which involves dividing large operators into smaller sub-operators and then reducing the computational time and obtaining additional benefits.

After this process, if we have nonlinear parts, we deal with nonlinear solver schemes, such as iterative methods for nonlinear systems in Banach spaces, where the study of the local or semilocal convergence helps us to establish domains of existence and uniqueness for the solution of the problem. The dynamical study of the methods complements the study.

Topics for this Special Issue include but are not limited to the following:

  • Ordinary, partial, or stochastic differential equations, nonlinear integral equations;
    • Techniques to decompose a large problem into different parts:
      • Splitting techniques, e.g., AB-splitting, iterative splitting;
      • Time- or spatial decomposition techniques, e.g., Schwarz waveform relaxation, Picard iterations, Domain decomposition;
      • Functional- or exponential splitting methods;
      • Serial and parallel splitting techniques.
  • Iterative methods for nonlinear systems
    • Local or semilocal convergence;
    • Computational efficiency;
    • Dynamical study;
    • Steffensen-like methods;
    • Iterative methods with memory.

Prof. Dr. Eulalia Martínez Molada
Guest Editor

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Keywords

  • Operator splitting methods
  • Iterative splitting methods
  • Convergence and stability analysis
  • Parallel splitting methods
  • Time- and spatial splitting methods
  • Iterative methods for nonlinear equations.

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Published Papers (6 papers)

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Research

42 pages, 568 KiB  
Article
Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations
by Jürgen Geiser, Eulalia Martínez and Jose L. Hueso
Mathematics 2020, 8(11), 1950; https://doi.org/10.3390/math8111950 - 4 Nov 2020
Cited by 1 | Viewed by 2102
Abstract
The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting [...] Read more.
The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions. Full article
(This article belongs to the Special Issue New Trends in Solving Partial Derivative Equations and Nonlinear Integral Equations by Splitting Techniques and Nonlinear Iterative Methods)
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13 pages, 323 KiB  
Article
Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes
by José Manuel Gutiérrez, Miguel Ángel Hernández-Verón and Eulalia Martínez
Mathematics 2020, 8(10), 1747; https://doi.org/10.3390/math8101747 - 11 Oct 2020
Cited by 5 | Viewed by 2623
Abstract
This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor’s development. Then, we adapt an already known technique used for separable kernels to our case. First, [...] Read more.
This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor’s development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments. Full article
(This article belongs to the Special Issue New Trends in Solving Partial Derivative Equations and Nonlinear Integral Equations by Splitting Techniques and Nonlinear Iterative Methods)
17 pages, 838 KiB  
Article
Some High-Order Iterative Methods for Nonlinear Models Originating from Real Life Problems
by Malik Zaka Ullah, Ramandeep Behl and Ioannis K. Argyros
Mathematics 2020, 8(8), 1249; https://doi.org/10.3390/math8081249 - 31 Jul 2020
Cited by 2 | Viewed by 1724
Abstract
We develop a sixth order Steffensen-type method with one parameter in order to solve systems of equations. Our study’s novelty lies in the fact that two types of local convergence are established under weak conditions, including computable error bounds and uniqueness of the [...] Read more.
We develop a sixth order Steffensen-type method with one parameter in order to solve systems of equations. Our study’s novelty lies in the fact that two types of local convergence are established under weak conditions, including computable error bounds and uniqueness of the results. The performance of our methods is discussed and compared to other schemes using similar information. Finally, very large systems of equations (100×100 and 200×200) are solved in order to test the theoretical results and compare them favorably to earlier works. Full article
(This article belongs to the Special Issue New Trends in Solving Partial Derivative Equations and Nonlinear Integral Equations by Splitting Techniques and Nonlinear Iterative Methods)
28 pages, 713 KiB  
Article
Iterative and Noniterative Splitting Methods of the Stochastic Burgers’ Equation: Theory and Application
by Jürgen Geiser
Mathematics 2020, 8(8), 1243; https://doi.org/10.3390/math8081243 - 30 Jul 2020
Cited by 1 | Viewed by 2056
Abstract
In this paper, we discuss iterative and noniterative splitting methods, in theory and application, to solve stochastic Burgers’ equations in an inviscid form. We present the noniterative splitting methods, which are given as Lie–Trotter and Strang-splitting methods, and we then extend them to [...] Read more.
In this paper, we discuss iterative and noniterative splitting methods, in theory and application, to solve stochastic Burgers’ equations in an inviscid form. We present the noniterative splitting methods, which are given as Lie–Trotter and Strang-splitting methods, and we then extend them to deterministic–stochastic splitting approaches. We also discuss the iterative splitting methods, which are based on Picard’s iterative schemes in deterministic–stochastic versions. The numerical approaches are discussed with respect to decomping deterministic and stochastic behaviours, and we describe the underlying numerical analysis. We present numerical experiments based on the nonlinearity of Burgers’ equation, and we show the benefits of the iterative splitting approaches as efficient and accurate solver methods. Full article
(This article belongs to the Special Issue New Trends in Solving Partial Derivative Equations and Nonlinear Integral Equations by Splitting Techniques and Nonlinear Iterative Methods)
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22 pages, 483 KiB  
Article
Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations
by Jürgen Geiser, Jose L. Hueso and Eulalia Martínez
Mathematics 2020, 8(3), 302; https://doi.org/10.3390/math8030302 - 25 Feb 2020
Cited by 5 | Viewed by 3281
Abstract
This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection–diffusion–reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive [...] Read more.
This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection–diffusion–reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers’ and Maxwell–Stefan diffusion equations. Full article
(This article belongs to the Special Issue New Trends in Solving Partial Derivative Equations and Nonlinear Integral Equations by Splitting Techniques and Nonlinear Iterative Methods)
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16 pages, 5613 KiB  
Article
Impact on Stability by the Use of Memory in Traub-Type Schemes
by Francisco I. Chicharro, Alicia Cordero, Neus Garrido and Juan R. Torregrosa
Mathematics 2020, 8(2), 274; https://doi.org/10.3390/math8020274 - 18 Feb 2020
Cited by 11 | Viewed by 2541
Abstract
In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub’s method, they have been designed using linear approximations or the Newton’s interpolation polynomials. In both cases, the [...] Read more.
In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub’s method, they have been designed using linear approximations or the Newton’s interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub’s scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process. Full article
(This article belongs to the Special Issue New Trends in Solving Partial Derivative Equations and Nonlinear Integral Equations by Splitting Techniques and Nonlinear Iterative Methods)
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