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We define the neutrosophic normed space and the statistical convergence in neutrosophic normed space. We give the statistically Cauchy sequence in neutrosophic normed space and present the statistically completeness in connection with a neutrosophic normed space.

The generalized regularized long wave (GRLW) equation has been developed to model a variety of physical phenomena such as ion-acoustic and magnetohydrodynamic waves in plasma, nonlinear transverse waves in shallow water and phonon packets in nonlinear crystals. This paper aims to develop and analyze a powerful numerical scheme for the nonlinear generalized regularized long wave (GRLW) equation by Petrov--Galerkin method in which the element shape functions are cubic and weight functions are quadratic B-splines. The suggested method is performed to three test problems involving propagation of the single solitary wave, interaction of two solitary waves and evolution of solitons with the Maxwellian initial condition. The variational formulation and semi-discrete Galerkin scheme of the equation are firstly constituted. We estimate accuracy of such a spatial approximation. Then Fourier stability analysis of the linearized scheme shows that it is unconditionally stable. To verify practicality and robustness of the new scheme error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated. The obtained numerical results are compared with other published results and shown to be precise and effective., 28 pages

In this article, we consider a generalized equal width wave (GEW) equation which is a significant nonlinear wave equation as it can be used to model many problems occurring in applied sciences. Here we study a Petrov–Galerkin method for the model problem, in which element shape functions are quadratic and weight functions are linear B-splines. We investigate the existence and uniqueness of solutions of the weak form of the equation. Then, we establish the theoretical bound of the error in the semi-discrete spatial scheme as well as of a full discrete scheme at t = t n. Furthermore, a powerful Fourier analysis has been applied to show that the proposed scheme is unconditionally stable. Finally, propagation of solitary waves and evolution of solitons are analyzed to demonstrate the efficiency and applicability of the proposed scheme. The three invariants (I1, I2 and I3) of motion have been commented to verify the conservation features of the proposed algorithms. Our proposed numerical scheme has been compared with other published schemes and demonstrated to be valid, effective and it outperforms the others.

In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element approximation have been studied respectively. In addition to that a powerful Fourier series analysis has been performed and indicated that our method is unconditionally stable. Finally, proficiency and practicality of the method have been demonstrated by illustrating it on two important problems of the GEW equation including propagation of single solitons and collision of double solitary waves. The performance of the numerical algorithm has been demonstrated for the motion of single soliton by computing L ∞ and L 2 norms and for the other problem computing three invariant quantities I 1 , I 2 and I 3 . The presented numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid.

Countries aiming for sustainability in economic growth and development ensure the reliability of energy supplies. For countries to provide their energy needs uninterruptedly, it is important for domestic and renewable energy sources to be utilised. For this reason, the supply of reliable and sustainable energy has become an important issue that concerns and occupies mankind. Of the renewable energy sources, wind energy is a clean, reliable and inexhaustible source of energy with low operating costs. Turkey is a rich nation in terms of wind energy potential. Forecasting of investment efficiency is an important issue before and during the investment period in wind energy investment process because of high investment costs. It is aimed to forecast the wind energy products monthly with multilayer neural network approach in this study. For this aim a feed forward back propagation neural network model has been established. As a set of data, wind speed values 48 months (January 2012-December 2015) have been used. The training data set occurs from 36 monthly wind speed values (January 2012-December 2014) and the test data set occurs from other values (January-December 2015). Analysis findings show that the trained Artificial Neural Networks (ANNs) have the ability of accurate prediction for the samples that are not used at training phase. The prediction errors for the wind energy plantation values are ranged between 0.00494-0.015035. Also the overall mean prediction error for this prediction is calculated as 0.004818 (0.48%). In general, we can say that ANNs be able to estimate the aspect of wind energy plant productions.

In this article, a collocation fi nite element method based on septic B-splines as a tool has been carried out to obtain the numerical solutions of the nonlinear generalized Rosenau-RLW equation. One of the advantages of this method is that when the bases are chosen at a high degree, better numerical solutions are obtained. Effectiveness of the method is demonstrated by solving the equation with various initial and boundary conditions. Also, in order to detect the performance of the method we have computed L2 and L1 error norms and two lowest invariants IM and IE: The obtained numerical results have been compared with some of those in the literature for similar parameters. This comparison clearly shows that the obtained results are better than and found in good conformity with the some earlier results. Stability analysis denotes that our algorithm, based on a Crank Nicolson approximation in time, is unconditionally stable.

WOS: 000435964300009, In this study, some new order relations on family of sets are introduced by using Minkowski difference. The relations between these orders and the ordering cone of the vector space are obtained. It is shown that depending on the corresponding cone, these order relations are partial orders on the family of nonempty bounded sets. Some relationships between these order relations and upper and lower set less order relations are investigated. Also, two scalarizing functions are introduced in order to replace set optimization problems with respect to these partial order relations with scalar optimization problems. Moreover, necessary and sufficient optimality conditions are presented.

I n this paper, we show that the system of difference equations 1 11 0 , , , N , 111 n n nn nn n nn n n n n n n xy yz zx xyz n xy yz zx + ++ + ++ = = = ∈ +++ where the initial values xyz , , are real numbers, are solvable in explicit form via some changes of variables and tricks. Also, we determine the forbidden set of the initial values xyz , , for the above mentioned system and investigate asymptotic behavior of the well-defined solutions by using these explicit formulas

In this article, numerical solutions of the modified Korteweg-de Vries (MKdV) equation have been obtained by a numerical technique attributed on collocation method using quintic B-spline finite elements. The suggested numerical scheme is controlled by applying three test problems involving single solitary wave, interaction of two and three solitary waves. To check the performance of the newly applied method, the error norms, L2 and L∞, as well as the three lowest invariants, I1, I2 and I3, have been calculated. The acquired numerical results are compared with some of those available in the literature. Linear stability analysis of the algorithm is also examined.

In this paper, using the concept of ideal convergence, which extends the idea of ordinary convergence and statistical convergence, we are concerned with the I-uniform convergence and the I-pointwise convergence of sequences of functions defined on a set of real numbers D. We present the Arzelà–Ascoli theorem by means of ideal convergence and also the relationship between I-equicontinuity and I-continuity for a family of functions.

In this work, we develop a formulation for the approximate-analytical solution of fractional partial differential equations (PDEs) by using conformable fractional derivative. Firstly, we redefine the conformable fractional Adomian decomposition method (CFADM) and conformable fractional modified homotopy perturbation method (CFMHPM). Then, we solve some initial boundary value problems (IBVP) by using the proposed methods, which can analytically solve the fractional partial differential equations (FPDE). In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of the IBVP. Also, we have found out that the proposed models are very efficient and powerful techniques in finding approximate solutions for the IBVP of fractional order in the conformable sense.

The purpose of the study is to perform a less-dimensional visualization process for the purpose of determining the images of the students on the concept of number. The Ward clustering analysis combined with the self-organizing map (SOM) was used for this purpose. The conceptual understanding tool, which consisted of the open-ended question “write the first ten things you remember when the term number is mentioned” to the study group, which consisted of 212 fifth grade students. The analysis results showed that students mostly explained the concept of number by associating with mathematics and other sciences with the terms like “addition, subtraction, division, multiplication, fraction, share, equation, cluster, angle, square, rectangle, plus, minus, shape, operation, step, graphics, value, equal, equals”, which are related with “operations-calculations” and “mathematical terms”. The students mostly established relations with mathematical side of the number. At the end of the study, the dataset obtained from the conceptual understanding tool was used for training the SOM, and a visualization approach that revealed the images of the students on number concept was recommended.

A symmetric C 0 finite element method for the biharmonic problem is constructed and analyzed. In our approach, we introduce one-sided discrete second-order derivatives and Hessian matrices to formulate our scheme. We show that the method is stable and converge with optimal order in a variety of norms. A distinctive feature of the method is that the results hold without extrinsic penalization of the gradient across interelement boundaries. Numerical experiments are given that support the theoretical results, and the extension to Kirchhoff plates is also discussed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1254–1278, 2014

This study obtains a fuzzy solution to the mathematical model of a cancer tumor under Caputo-generalized Hukuhara partial differentiability by using fuzzy integral transforms. In order to solve the fuzzy partial fractional differential equations, the two-variable fuzzy Laplace—Carson transforms and the two-variable fuzzy Fourier transform under the generalized Hukuhara partial differentiability and Caputo-generalized Hukuhara partial differentiability are investigated. Following this, an algorithm for the proposed method is presented. Finally, as a practical model of fuzzy fractional diffusion equations, the fuzzy mathematical model of the net killing rate of cancer cells in tumors is investigated. This technique is powerful and essential for the development of a fuzzy analytical method for solving fuzzy partial fractional differential equations. 2-s2.0-85119302470

In this paper, we propose the new fuzzy arithmetic operations based on transmission average on pseudo-hexagonal fuzzy numbers from [F. Abbasi, T. Allahviranloo and S. Abbasbandy, A new attitude coupled with fuzzy thinking to fuzzy rings and fields, Journal of Intelligent and Fuzzy Systems 29 (2015) 851–861]. The properties of these propose operations and their fundamental qualities are discussed in detail. Several illustrative examples were given to show the accomplishment and ability of the proposed method. Finally, as a applied rudimentary case study an electric network model of a football stadium is considered in fuzzy environment. The reliability of components for the proposed model is considered as pseudo-hexagonal fuzzy number.

In this paper, we are going to utilize newly developed Higher Order Haar wavelet method (HOHWM) and classical Haar wavelet method (HWM) to numerically solve the Regularized Long Wave (RLW) equation. Spatial variable of the RLW equation is treated with HOHWM and HWM separately. On the other hand temporal variable is discretized by finite differences combined with Strang splitting approach. The presented methods applied to three different test problems and the obtained results are given in tables as well as depicted in figures. The obtained results are compared with analytical results wherever they exist. The error norms L-2 and L-infinity and invariants I-1, I-2 and I-3 are used to show the accuracy of the methods when comparing the present results with those in the literature. (C)& nbsp;2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Let S denote the class of functions f(z) = z + a2z2+... analytic and univalent in the open unit disc D = {z ? C z| 0 in , and prove that every close-to-convex function is univalent. The normalized class of close-to-convex functions denoted by K. These classesare related by the proper inclusions C ? S* ? K ? S. In this paper, we generalize the close-to-convex functions and denote K(?) the class of such functions. Various properties of this class of functions is alos studied.©2013 Melike Aydog?an et al.

In this dissertation I develop a general theoretical framework for designing healthcare strategies for controlling infectious diseases. The dissertation employs a variety of mathematical methods, such as mathematical modelling, adaptive dynamics, and optimisation. The design of strategies for controlling infectious diseases by means of healthcare interventions (such as vaccination, quarantine, and individual isolation and treatment) is a complicated affair for a number of reasons. Firstly, pathogens tend to adapt to the current state of healthcare interventions (e.g., by becoming more virulent or infectious), which renders present interventions less effective. For a control strategy to be successful in the long term, such evolutionary responses of the pathogen need to be anticipated and taken into account. Secondly, the optimal strategy will depend on the particular objective (such as the minimisation of virulence or infectiousness, the total eradication of the disease, or others). The choice of an objective is political, but in order to be able to make a rational choice, the relation between the objective and the corresponding optimal control strategy has to be known first. Thirdly, which strategies can be practically implemented depends on constraints (such as the available budget, knowledge, technology, infrastructure, and others) that also affect the design of the control strategies. Fourthly, pathogen-specific characteristics (such as cross-immunity, superinfection, and co-infection) play a significant role in the infection process. They can complicate the infection process and make infection control more difficult. In order to develop highly effective control strategies, these pathogen characteristics need to be incorporated into research. The dissertation focuses on three sub-projects, corresponding to the three articles listed below. The first sub-project focuses on the study of the feedback loop between healthcare interventions and virulence evolution, with virulence being defined as the disease-induced death rate. The second sub-project is concerned with the study of optimal strategies for various healthcare interventions in terms of multiple objectives and budget levels. The third sub-project aims to study the effects of healthcare interventions on pathogen evolution (in particular, the evolution of the within-host growth rate) in the context of cross-immunity acting between different strains. Tässä väitöskirjassa kehitän yleisen teoreettisen viitekehyksen infektiotauteja torjuvien terveydenhuollon strategioiden suunnittelulle. Väitöskirja käyttää monenlaisia matemaattisia menetelmiä, kuten matemaattista mallintamista, sopeutuvaa dynamiikkaa ja optimointia. Tarttuvien tautien hallintastrategioiden suunnittelu terveysinterventioiden avulla (kuten rokotus, karanteeni ja yksilöllinen eristäminen ja hoito) on monimutkainen asia useista syistä. Ensinnäkin patogeenit taipuvat sopeutumaan nykyisiin terveysinterventioihin (esim. muuttumalla virulentimmiksi tai tarttuvammiksi), mikä tekee nykyisistä interventioista vähemmän tehokkaita. Jotta hallintastrategia olisi pitkällä aikavälillä menestyksekäs, patogeenin evolutiivisiin vastauksiin on varauduttava ja otettava ne huomioon. Toiseksi optimaalinen strategia riippuu erityisestä tavoitteesta (kuten virulenssin tai tarttuvuuden minimoinnista, taudin kokonaisvaltaisesta hävittämisestä tai muista). Tavoitteen valinta on poliittinen, mutta järkevän valinnan tekemiseksi tavoitteen ja vastaavan optimaalisen hallintastrategian suhde on ensin tunnettava. Kolmanneksi se, mitkä strategiat voidaan käytännössä toteuttaa, riippuu rajoituksista (kuten käytettävissä olevasta budjetista, tiedosta, teknologiasta, infrastruktuurista ja muista), jotka vaikuttavat myös hallintastrategioiden suunnitteluun. Neljänneksi patogeenikohtaiset ominaisuudet (kuten risti-immuniteetti, superinfektio ja yhteisinfektio) ovat merkittävä rooli infektioprosessissa. Ne voivat monimutkaistaa infektioprosessia ja tehdä infektioiden torjunnasta vaikeampaa. Jotta voitaisiin kehittää erittäin tehokkaita hallintastrategioita, nämä patogeenin ominaisuudet on otettava huomioon tutkimuksessa. Väitöskirja keskittyy kolmeen alaprojektiin, jotka vastaavat alla lueteltuja kolmea artikkelia. Ensimmäinen alaprojekti keskittyy terveydenhuollon interventioiden ja virulenssin evoluution väliseen palautelooppiin, jossa virulenssi määritellään tautiin liittyvän kuolleisuusasteena. Toinen alaprojekti käsittelee optimaalisten strategioiden tutkimusta eri terveydenhuollon interventioihin useiden tavoitteiden ja budjettitasojen näkökulmasta. Kolmas alaprojekti pyrkii tutkimaan terveydenhuollon interventioiden vaikutuksia patogeenien evoluutioon (erityisesti isäntäsolun kasvunopeuden evoluutioon) eri kantojen välisessä ristisuojautumisessa.

3rd International Conference on Applied Mathematics and Approximation Theory, AMAT 2015, 28 May 2015 through 31 May 2015, In this chapter, using power series method we study some Korovkin type approximation theorems which deal with the problem of approximating a function by means of a sequence of linear operators acting on weighted spaces. © Springer International Publishing Switzerland 2016.

WOS: 000382960700003, In this study, some relationships between quasidifferentiability and weakly subdifferentiability of a function are investigated, and some results on calculating the quasidifferential in terms of weak exhausters are presented. Moreover, under some assumptions weak subdifferentiability of a quasidifferentiable function is proved and the weak subdifferential is obtained by using the quasidifferential. Under some assumptions, it is proved that a positively homogeneous and lower semicontinuous function is quasidifferentiable. It is also shown that directional differentiability and weak subdifferentiability imply quasidifferentiability. Furthermore, a method to evaluate the quasidifferential is given by using reduced weak lower exhausters., Commission of Scientific Research Projects of Anadolu University [1201F018], This research was supported by Commission of Scientific Research Projects of Anadolu University [1201F018].

The purpose of this paper is twofold. First is the study of the nonexistence of positive solutions of the parabolic problem {∂u∂t=Δpu+V(x)up-1+λuqinΩ×(0,T),u(x,0)=u0(x)≥0inΩ,|∇u|p-2∂u∂ν=β|u|p-2uon∂Ω×(0,T),where Ω is a bounded domain in RN with smooth boundary ∂Ω, Δpu= div (| ∇ u| p-2∇ u) is the p-Laplacian of u, V∈Lloc1(Ω), β∈Lloc1(∂Ω), λ∈ R, the exponents p and q satisfy 1 < p< 2 , and q> 0. Then, we present some sharp Hardy and Leray type inequalities with remainder terms that provide us concrete potentials to use in the partial differential equation we are interested in.

The main purpose of this paper is to introduce fuzzy Adams–Bashforth (A–B) and fuzzy Adams–Moulton (A–M) methods based on the generalized Hukuhara (gH)-differentiability and employ them as the predictor and corrector, respectively. The local truncation error, stability and convergence of these methods are discussed in the sequel. Finally, some fuzzy linear and nonlinear initial value problems (IVPs) are solved. The numerical results obtained here show that our methods provide a suitable approximation for the exact solution. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.

There may be cases where experts do not have in-depth knowledge of the problem to be solved in decision-making problems. In such cases, experts may fail to express their views on certain aspects of the problem, resulting in incomplete preferences, in which some preference values are not provided or are missing. In this article, we present a new model for group decision-making (GDM) methods in which experts’ preferences can be expressed as incomplete Fermatean fuzzy preference relations. This model is guided by the additive-consistency property and only uses the preference values the expert provides. An additive consistency definition characterized by a Fermatean fuzzy priority vector has been given. The additive consistency property is also used to measure the level of consistency of the information provided by the experts. The proposed additive consistency definition’s property is presented, as well as a model for obtaining missing judgments in incomplete Fermatean fuzzy preference relations. We present a method for adjusting the inconsistency for Fermatean fuzzy preference relations, a model for obtaining the priority vector, and a method for increasing the consensus degrees of Fermatean fuzzy preference relations. In addition, we present a GDM method in environments with incomplete Fermatean fuzzy preference relations. To show that our method outperforms existing GDM methods in incomplete Fermatean fuzzy preference relations environments, we have provided an example and compared it with some methods. It has been seen that our proposed GDM method is beneficial for GDM in deficient Fermatean fuzzy preference relation environments and produces meaningful results for us.

WOS: 000360382200001, In this work, the notion of weak superdifferential is presented. Some calculation rules are given to evaluate weak subdifferential and weak superdifferential of some classes of functions represented by support functions. Moreover, some methods are obtained to calculate weak subdifferential of convex functions. In addition, the concept of weak lower and weak upper exhausters of positively homogeneous functions are introduced by using weak subdifferential and weak superdifferential, respectively. In terms of weak exhausters, some optimality conditions are given to find local or global minimizers/maximizers of some classes of functions., Anadolu University Scientific Research Projects Commission [1201F018], This study was supported by Anadolu University Scientific Research Projects Commission [grant number 1201F018].

WOS: 000353814300001, In this work, it is proved that weak exhausters of a positively homogeneous function can be reduced if weak subdifferential/superdifferential can be represented as the sum of a subset of it and weak subdifferential of zero function. It is also shown that weak exhausters can be reduced if this subset is the set of minimal elements of weak subdifferential/superdifferential with respect to weak subdifferential of zero function. At the end, some optimality conditions are given using reduced weak exhausters., Anadolu University Scientific Research Projects Commission [1201F018], The authors would like to thank the editor and anonymous referees for their kind and helpful remarks and comments. This study was supported by Anadolu University Scientific Research Projects Commission under the grant no 1201F018.

In the present paper we investigate a class of harmonic mappings for which the second dilatation is a close-to-convex function of complex order b, b is an element of C/{0} (Lashin in Indian J. Pure Appl. Math. 34(7):1101-1108, 2003). Publisher's Version

This study presents a new analytical method to extract the fuzzy solution of the fuzzy initial value problem (FIVP) of fourth-order fuzzy ordinary differential equations (FODEs) using the Laplace operator under the strongly generalized Hukuhara differentiability (SGH-differentiability). To this end, firstly the fourth-order derivative of the fuzzy valued function (FVF) according to the type of the SGH-differentiability is introduced, and then the relationships between the fourth-order derivative of the FVF and its Laplace transform are established. Furthermore, considering the types of differentiabilities and switching points, some fundamental theorems related to the Laplace transform of the fourth-order derivative of the FVF are stated and proved in detail and a method to solve FIVP by the fuzzy Laplace transform is presented in detail. An application of our proposed method in Resistance-Inductance circuit (RL circuit) is presented. Finally, FIVP's solution is graphically analyzed to visualize and support theoretical results.

In this paper, the concept of the full Z-linear programming problem (FZLP) is first introduced. It is worth to mention that in these type of problems, all parameters, including the coefficients of variables in the objective functions, the coefficients of the variables in the constraints, the right-hand side of the constraints, as well as the decision variables, are valued as Z-numbers. Clearly, it is critical to provide a method to solve these types of problems, and to find their optimal solutions. Therefore, this paper develops a novel and practical method to find the solution of the problem using the concept of Z-numbers. To prove the applicability of the concept, two illustrative examples are provided to demonstrate the precision and effectiveness of this method. WOS:000718216500001 Q1

In this paper, considering fuzzy measure theory and matrix-valued fuzzy norm spaces, we study a differential system of non-autonomous cellular neural networks with mixed delays (NA-CNN-MD). Our main goal is to investigate the existence of a unique solution for the presented system of NA-CNN-MD in the matrix-valued fuzzy spaces and using the fuzzy measure. After investigating the existence of a unique solution of NA-CNNs-MD which are rho-pseudo almost periodically, we prove the Mittag-Leffler stability and Mittag-Leffler attractiveness for rho-pseudo almost periodic functions. Finally, two examples are given to illustrate the theoretical results. WOS:000876308000001 Q1

A septic B-spline collocation method is implemented to find the numerical solution of the modified regularized long wave (MRLW) equation. Three test problems including the single soliton and interaction of two and three solitons are studied to validate the proposed method by calculating the error normsL2andL∞and the invariantsI1,I2, andI3. Also, we have studied the Maxwellian initial condition pulse. The numerical results obtained by the method show that the present method is accurate and efficient. Results are compared with some earlier results given in the literature. A linear stability analysis of the method is also investigated.

A numerical solution of the modified Burgers’ equation (MBE) is obtained by using quartic B-spline subdomain finite element method (SFEM) over which the nonlinear term is locally linearized and using quartic B-spline differential quadrature (QBDQM) method. The accuracy and efficiency of the methods are discussed by computingL2andL∞error norms. Comparisons are made with those of some earlier papers. The obtained numerical results show that the methods are effective numerical schemes to solve the MBE. A linear stability analysis, based on the von Neumann scheme, shows the SFEM is unconditionally stable. A rate of convergence analysis is also given for the DQM.

In this article, we present an efficient method for solving Abel’s integral equations. This important equation is consisting of an integral equation that is modeling many problems in literature. Our proposed method is based on first taking the truncated Taylor expansions of the solution function and fractional derivatives, then substituting their matrix forms into the equation. The main character behind this technique’s approach is that it reduces such problems to solving a system of algebraic equations, thus greatly simplifying the problem. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. Figures and tables are demonstrated to solutions impress. Also, all numerical examples are solved with the aid of Maple.

In this study, one-dimensional (1D) and two-dimensional (2D) coupled Schrodinger-Boussinesq (SBq) equations are examined numerically. A local meshless method based on radial basis function-finite difference (RBF-FD) method for spatial approximation is devised. We use polyharmonic splines as radial basis function along with augmented polynomials. By using polyharmonic splines we avoid to choose optimal shape parameter which requires special algorithms in meshless methods. For temporal discretization, low-storage ten-stage fourth-order explicit strong stability preserving Runge Kutta method is used which gives more flexibility on temporal step width. L ∞ and L 2 error norms are calculated to show accuracy of the proposed method. Further, conserved quantities are monitoried during numerical simulations to see how good the proposed method preserves them. Stability of the proposed method is dicussed numerically. Some codes are developed in Julia programming language to achieve more speed up in numerical simulations. Obtained results and their comparison with some studies such as wavelet, difference schemes and Fourier spectral methods available in literature verify the efficiency and reliability of the proposed method.

WOS: 000323567500001, In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of Gamma (P-M). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics., Research Project Offices (BAP) of SelcukSelcuk University [13701071]; UludagUludag University [2012-12, 2012-19]; Faculty Research Fund, Sungkyunkwan University; Sungkyunkwan University BK21 ProjectMinistry of Education & Human Resources Development (MOEHRD), Republic of Korea; BK21 Math Modelling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea, The third and fourth authors are partially supported by Research Project Offices (BAP) of Selcuk (with Project Number 13701071) and Uludag (with Project Numbers 2012-12 and 2012-19) Universities, respectively. The second author is supported by the Faculty Research Fund, Sungkyunkwan University, 2012 and Sungkyunkwan University BK21 Project, BK21 Math Modelling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea.

WOS: 000326449800001, In this paper, we determine necessary and sufficient conditions for Bruck-Reilly and generalized Bruck-Reilly *-extensions of arbitrary monoids to be regular, coregular and strongly pi-inverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and p-adic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and Bruck-Reilly extensions contain a mixture fixed-point results of algebra, topology and geometry within the purposes of this journal., Research Project Offices (BAP) of Selcuk UniversitySelcuk University [13701071]; Research Project Offices (BAP) of Uludag UniversityUludag University [2012-15, 2012-19], The second and fourth authors are partially supported by Research Project Offices (BAP) of Selcuk (with Project No. 13701071) and Uludag (with Project No. 2012-15 and 2012-19) Universities, respectively.

Baar and Braha [1], introduced the sequence spaces l(infinity), C and C-0 of Euler- Cesaro bounded, convergent and null difference sequences and studied their somere properties. Then, in [2], we introduced the sequence spaces [l(infinity)] and [c](e.r) and [c(0)](e.r), of Euler-Riesz bounded, convergent and null difference sequences by using the composition of the Euler mean E-1 and Riesz mean R-q with backward difference operator Delta. The main purpose of this study is to introduce the sequence space [l(p)](e.r), of Euler-Riesz p absolutely convergent series, where 1

This work proposes a finite element method emphasizing with quartic-trigonometric basis functions for finding the numerical solution of nonlinear Burgers’ equation. The computational scheme is constructed by a discretized space-time hybrid approach using B-spline functions. This methodology produces a system of time-dependent differential equations which is integrated by finite elements technique. The experimental cases including graphical patterns of each wave interaction are simulated by the current computational algorithm. In addition, the method establishes the capacity to provide highly efficient solutions with relative ease of computation. Investigation of the stability analysis shows that the current computational method serves an unconditional stable numerical scheme. © 2022, The Author(s), under exclusive licence to Islamic Azad University. 2-s2.0-85132856901

This work was partially supported by the Turkish Academy of Sciences (TUBA). In this work, the application of "the modified reductive perturbation method" is extended to the generalized symmetric regularized long-wave equation for strongly dispersive case and the contribution of higher order terms in the perturbation expansion is obtained. It is shown that the first order term in the perturbation expansion is governed by the generalized nonlinear Schrödinger quation whereas the second order term is governed by the generalized linear Schrödinger equation with a nonhomogeneous term. A travelling wave type of solution to these evolution equations is also given. Turkish Academy of Sciences Publisher's Version Q2 WOS:000288059700016

Given a Lie group $G$, we elaborate the dynamics on $T^*T^*G$ and $T^*TG$, which is given by a Hamiltonian, as well as the dynamics on the Tulczyjew symplectic space $TT^*G$, which may be defined by a Lagrangian or a Hamiltonian function. As the trivializations we adapted respect the group structures of the iterated bundles, we exploit all possible subgroup reductions (Poisson, symplectic or both) of higher order dynamics., arXiv admin note: substantial text overlap with arXiv:1503.06568

In this paper, Wong-Zakai approximation methods are presented for some stochastic differential equations in engineering sciences. Wong-Zakai approximate solutions of the equations are analyzed and the numerical results are compared with results from popular approximation schemes for stochastic differential equations such as Euler-Alartiyama and Milstein methods. Several differential equations from engineering problems containing stochastic noise are investigated as numerical examples. Results show that Wong-Zakai method is a reliable tool for studying stochastic differential equations and can he used as an alternative for the known approximation techniques for stochastic models. WOS:000637582200014 2-s2.0-85101626407

In this study, a fractional Bernstein series solution method has been submitted to solve the fractional-order biological population model with one carrying capacity. The numerical method has been implemented by an effective algorithm written on the computer algebraic system Maple 15. An error-bound analysis is performed by using a process similar to the RK45 method. An error estimation technique relating to residual function is presented. In the numerical application, the variations in the population of prey and predator with respect to time and situations of these two species relative to each other are plotted. The outputs obtained from our method are compared with the homotopy perturbation Sumudu transform method and reproducing kernel Hilbert space method. The approximate solutions gained from the Bernstein series method are consistent with those of other methods. The advantage of our method is that it requires less computational cost compared with methods involving more complex operations.

Higher order differential equations (ODE) has an important role in the modelling process. It is also much significant which the method is used for the solution. In this study, in order to get the approximate solution of a nonhomogeneous initial value problem, reproducing kernel Hilbert space method is used. Reproducing kernel functions have been obtained and the given problem transformed to the homogeneous form. The results have been presented with the graphics. Absolute errors and relative errors have been given in the tables.

Let $$\gamma$$ be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that $$f(x)x^{-r}$$ is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie–Hellman mapping $$d(\gamma ^a)=\gamma ^{a^2}$$ , $$a=0,1,\ldots ,n-1$$ , and show that any mapping of small index coincides with d only on a small subset of G. Moreover, we prove similar results for the bivariate Diffie–Hellman mapping $$D(\gamma ^a,\gamma ^b)=\gamma ^{ab}$$ , $$a,b=0,1,\ldots ,n-1$$ . In the special case that G is a subgroup of the multiplicative group of a finite field we present improvements.

In this study, we first show that the existing arithmetic operations of trapezoidal fuzzy numbers do not satisfy the basic properties. Then, for trapezoidal fuzzy numbers, we define new arithmetic operations. Furthermore, we demonstrate that the existing Simplex method for addressing fully fuzzy linear programming problems has some drawbacks. Finally, we provide a new strategy for solving fully fuzzy linear programming problems and compare our results with the existing methods. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional. 2-s2.0-85123434717

In this paper, the fuzzy approximate solutions for the fuzzy Hybrid differential equation emphasizing the type of generalized Hukuhara differentiability of the solutions are obtained by using the two-dimensional Muntz-Legendre wavelet method. To do this, the fuzzy Hybrid differential equation is transformed into a system of linear algebraic equations in a matrix form. Thus, by solving this system, the unknown coefficients are obtained. The convergence of the proposed method is established in detail. Numerical results reveal that the two-dimensional Muntz-Legendre wavelet is very effective and convenient for solving the fuzzy Hybrid differential equation. © 2023 World Scientific Publishing Company. 2-s2.0-85128519600

WOS:000619048200001 In this paper, we aim to understand the characteristics of dynamical behaviour for a higher order Kirchhoff type systems with logarithmic nonlinearities. Based on the potential well method, the main ingredient of this study is to construct several conditions for initial data leading to the solution global existence in case of E (0) < d. On the other hand, we proved that if the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of energy.

Some efficient radial basis functions (RBFs) have a free parameter called the shape parameter that controls the shape of approximating function. This parameter is mainly selected by trial and error related to the problem. Because of the significant role of this value in the accuracy and stability of the RBF method, we propose an algorithm to choose a good value as a shape parameter. In this work, we focus on the interpolating of scattered data approximation and attempt to find a better value of the shape parameter for RBF interpolation. The proposed algorithm selects a good value for the shape parameter by minimizing a cost function which imitates the error between the radial interpolant and the unknown function f that the data points sampled from the function f. The algorithm can be applied to a multidimensional problem of any dimension and for any radial basis function. Also, many experiments including interpolation of one and two dimensional data sets as well as zooming pictures consider investigating the efficiency of this procedure. Also, we show that this algorithm consistently finds a good value for a shape parameter α.

Several problems in the field of science and technology are modeled with information about the situation that is ambiguous, imprecise, or incomplete. That is, the information about values of parameters, functional relationships, or initial conditions is not given in precise. In these circumstance, existing analytic or numerical methods can be applied only to the selected behavior of the system. For example, by fixing the values of unknown parameters to some credible values. On the basis of partial knowledge, it is impossible to describe the behavior of complete system. Thus, fuzzy differential equations arise in many dynamical models. In modeling of several real-world problems, differential equations frequently involve multi-agent, multi-index, multi-objective, multi-attribute, multi-polar information or uncertainty rather than a single bit. These type of differential equations cannot be well represented by means of fuzzy differential equations or bipolar fuzzy differential equations. Therefore, the theory of m-polar fuzzy sets can be applied to differential equations to handle the problems which have multi-polar information. The aim of this paper is to study differential equation in m-polar fuzzy environment. A fourth-order Runge–Kutta method to solve m-polar FIVPs is presented. The consistency, stability and convergence of suggested method are discussed to ensure its efficiency and validity. Since it requires no higher order function derivatives, the suggested method is straightforward to implement. Euler and Euler modified methods have global truncations errors of O(h) and O(h2) respectively whereas the suggested Runge–Kutta’s global truncation errors of O(h4). Numerical examples are provided to compare the proposed method with Euler and modified Euler methods in terms of global truncation errors (GTE). The numerical findings suggest that the purposed method has an adequate level of accuracy. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional. 2-s2.0-85128893019

We introduce a new class of fuzzy control functions and define the concept of matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability. Then, we apply Radu–Mihet method derived from an alternative fixed point theorem to investigate existence, uniqueness and matrix-valued fuzzy Mittag–Leffler–Hypergeometric–Wright stability of a a class of P-Hilfer fractional differential equations in matrix-valued fuzzy Banach space. Next, we show the main results for unbounded domains. An example is given to illustrate the Mittag–Leffler–Hypergeometric–Wright stability for a fractional system. © 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional. 2-s2.0-85133556237

Using the trial equation method (TEM) and modified trial equation method (MTEM), firstly, we find the analytical solutions of the conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE), and finally, we present numerical results in tables and charts. © 2022 Safoura Rezaei Aderyani et al. 2-s2.0-85124734800