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The integration of internationally sustainable practices into supply chain management methodologies is known as 'green supply chain management'. Reducing the supply chain's overall environmental impact is the main objective in order to improve corporate connections and the social, ecological, and economic ties with other nations. To accomplish appropriate and accurate measures to address the issue of emergency decision-making, the paper is divided into three major sections. First, the $ \left(p, q\right) $-fractional linear Diophantine fuzzy set represents a new generalization of several fuzzy set theories, including the Pythagorean fuzzy set, $ q $-rung orthopair fuzzy set, linear Diophantine fuzzy set, and $ q $-rung linear Diophantine fuzzy set, with its key features thoroughly discussed. Additionally, aggregation operators are crucial for handling uncertainty in decision-making scenarios. Consequently, algebraic norms for $ \left(p, q\right) $-fractional linear Diophantine fuzzy sets were established based on operational principles. In the second part of the study, we introduced a range of geometric aggregation operators and a series of averaging operators under the $ \left(p, q\right) $-fractional linear Diophantine fuzzy set, all grounded in established operational rules. We also explained some flexible aspects for the invented operators. Furthermore, using the newly developed operators for $ \left(p, q\right) $-fractional linear Diophantine fuzzy information, we constructed the multi-attribute decision-making ($ MADM $) technique to assess the green supply chain management challenge. Last, we compared the ranking results of the produced approaches with the obtained ranking results of the techniques using several numerical instances to demonstrate the validity and superiority of the developed techniques. Finally, a few comparisons between the findings were made.

This study explores the application of generalized conformable derivatives in modeling hotel demand dynamics in Mexico, using the Gompertz-type model. The research focuses on customizing conformable functions to fit the unique characteristics of the Mexican hotel industry, considering the Tourist Area Life Cycle (TALC) model and aiming to enhance forecasting accuracy. The parameter adjustment in all cases was made by designing a convex function, which represents the difference between the theoretical model and real data. Results demonstrate the effectiveness of the generalized conformable derivative approach in predicting hotel demand trends, showcasing its potential for improving decision-making processes in the Mexican hospitality sector. The comparison between the logistic and Gompertz models, in both integer and fractional versions, provides insights into the suitability of these modeling techniques for capturing the dynamics of hotel demand in the studied regions.

We prove a version of the fundamental theorems of Morse theory in the setting of finite partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the Morse-Pitcher inequalities in that context.

In this work, we introduce generalized Raina fractional integral operators and derive Chebyshev-type inequalities involving these operators. In a first stage, we obtain Chebyshev-type inequalities for one product of functions. Then we extend those results to account for arbitrary products. Also, we establish some inequalities of the Chebyshev type for functions whose derivatives are bounded. In addition, we derive an estimate for the Chebyshev functional by applying the generalized Raina fractional integral operators. As corollaries of this study, some known results are recaptured from our general Chebyshev inequalities. The results of this work may prove useful in the theoretical analysis of numerical models to solve generalized Raina-type fractional-order integro-differential equations.

Integral inequalities play a crucial role in both theoretical and applied mathematics. Because of the relevance of these notions, we have discussed a new class of introduced generalized convex function called as coordinated left and right convex interval-valued function (coordinated LR-convex IVF) using the pseudo-order relation (≤p). On interval space, this order relation is defined. First, a pseudo-order relation is used to show Hermite-Hadamard type inequality (HH type inequality) for coordinated LR-convex IVF. Second for coordinated LR-convex IVF, Some HH type inequalities are also derived for the product of two coordinated LR-convex IVFs. Furthermore, we have demonstrated that our conclusions cover a broad range of new and well-known inequalities for coordinated LR-convex IVFs and their variant forms as special instances which are defined by Zhao et al. and Budak et al. Finally, we have shown that the inclusion relation "⊇" confidents to the pseudo-order relation "≤p" for coordinated LR-convex IVFs. The concepts and methodologies presented in this study might serve as a springboard for additional research in this field, as well as a tool for investigating probability and optimization research, among other things.

The importance of convex and non-convex functions in the study of optimization is widely established. The concept of convexity also plays a key part in the subject of inequalities due to the behavior of its definition. The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this study, first, Hermite-Hadamard type inequalities for LR-p-convex interval-valued functions (LR-p-convex-I-V-F) are constructed in this study. Second, for the product of p-convex various Hermite-Hadamard (HH) type integral inequalities are established. Similarly, we also obtain Hermite-Hadamard-Fejér (HH-Fejér) type integral inequality for LR-p-convex-I-V-F. Finally, for LR-p-convex-I-V-F, various discrete Schur's and Jensen's type inequalities are presented. Moreover, the results presented in this study are verified by useful nontrivial examples. Some of the results reported here for be LR-p-convex-I-V-F are generalizations of prior results for convex and harmonically convex functions, as well as p-convex functions.

We prove that a diffeological space is diffeomorphic to the categorical join of any generating family of plots.

Abstract We investigate a model of spatio-temporal spreading of human immunodeficiency virus HIV-1. The mathematical model considers the presence of various components in a human tissue, including the uninfected CD4+T cells density, the density of infected CD4+T cells, and the density of free HIV infection particles in the blood. These three components are nonnegative and bounded variables. By expressing the original model in an equivalent exponential form, we propose a positive and bounded discrete model to estimate the solutions of the continuous system. We establish conditions under which the nonnegative and bounded features of the initial-boundary data are preserved under the scheme. Moreover, we show rigorously that the method is a consistent scheme for the differential model under study, with first and second orders of consistency in time and space, respectively. The scheme is an unconditionally stable and convergent technique which has first and second orders of convergence in time and space, respectively. An application to the spatio-temporal dynamics of HIV-1 is presented in this manuscript. For the sake of reproducibility, we provide a computer implementation of our method at the end of this work.

The objective of the current paper is to incorporate the new class and concepts of convexity and Hermite–Hadamard inequality with the fuzzy Riemann integral operators because almost all classical single-valued and interval-valued convex functions are special cases of fuzzy-number valued convex mappings. Therefore, a new class of nonconvex mapping in the fuzzy environment has been defined; up and down h-pre-invex fuzzy-number valued mappings (U.D h-pre-invex F-N∙V∙Ms). With the help of this newly defined class, some new versions of Hermite–Hadamard (HH) type inequalities have been also presented. Moreover, some related inequalities such as HH Fejér- and Pachpatte-type inequalities for U∙D h-pre-invex F-N∙V∙Ms are also introduced. Some exceptional cases have been discussed, which can be seen as applications of the main results. We have provided some nontrivial examples. Finally, we also discuss some future scopes.

Abstract A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method.

In this work, we propose a mathematical model that describes liver evolution and concentrations of alanine aminotransferase and aspartate aminotransferase in a group of rats damaged with carbon tetrachloride. Carbon tetrachloride was employed to induce cirrhosis. A second groups damaged with carbon tetrachloride was exposed simultaneously a plant extract as hepatoprotective agent. The model reproduces the data obtained in the experiment reported in [Rev. Cub. Plant. Med. 22(1), 2017], and predicts that using the plants extract helps to get a better natural recovery after the treatment. Computer simulations show that the extract reduces the damage velocity but does not avoid it entirely. The present paper is the first report in the literature in which a mathematical model reliably predicts the protective effect of a plant extract mixture in rats with cirrhosis disease. The results reported in this manuscript could be used in the future to help in fighting cirrhotic conditions in humans, though more experimental and mathematical work is required in that case.

In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications.

In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate formula of the fractional derivative of a power function ζp in terms of the RFE kernel. Next, by using the spectral collocation method (SCM) based on the shifted Vieta–Lucas polynomials (VLPs), the fractional differential system is reduced to a set of algebraic equations. We provide a theoretical convergence analysis for the numerical approach, and the accuracy is verified by evaluating the residual error function through some concrete examples. The results are then contrasted with those derived using the fourth-order Runge-Kutta (RK4) method.

The Boiti–Leon–Mana–Pempinelli Equation (BLMPE) is an essential mathematical model describing wave propagation in incompressible fluid dynamics. In the present manuscript, a novel generalization of the BLMPE is introduced, called herein the functional BLMPE (F-BLMPE), which involves different functions, including exponential, logarithmic and monomaniacal functions. In these cases, the F-BLMPE reduces to an explicit form in the dependent variable. In addition to this, it is worth deriving approximate similarity solutions of the F-BLMPE with constant coefficients using the extended unified method (EUM). In this method, nonlinear partial differential equation (NLPDE) solutions are expressed in polynomial and rational forms through an auxiliary function (AF) with adequate auxiliary equations. Exact solutions are estimated using formal solutions substituted into the NLPDEs, and the coefficients of the AF of all powers are set equal to zero. This approach is valid when the NLPDE is integrable. However, this technique is not valid for non-integrable equations, and only approximate solutions can be found. The maximum error can be controlled by an adequate choice of the parameters in the residue terms (RTs). Multiple similarity solutions are derived, and the ME is depicted in various examples within this work. The results found here confirm that the EUM is an efficient method for solving NLPDEs of the F-BLMPE type.

In this study, we provide an efficient simulation to investigate the behavior of the solution to the Brusselator system (a biodynamic system) with the Rabotnov fractional-exponential (RFE) kernel fractional derivative. A system of fractional differential equations can be used to represent this model. The fractional-order derivative of a polynomial function tp is approximated in terms of the RFE kernel. In this work, we employ shifted Vieta–Lucas polynomials in the spectral collocation technique. This process transforms the mathematical model into a set of algebraic equations. By assessing the residual error function, we can confirm that the provided approach is accurate and efficient. The outcomes demonstrate the effectiveness and simplicity of the technique for accurately simulating such models.

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings (UDET-convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for UDET-convex FNVMs. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems.

This work aims at providing a concise review of various agri-food models that employ fractional differential operators. In this context, various mathematical models based on fractional differential equations have been used to describe a wide range of problems in agri-food. As a result of this review, we found out that this new area of research is finding increased acceptance in recent years and that some reports have employed fractional operators successfully in order to model real-world data. Our results also show that the most commonly used differential operators in these problems are the Caputo, the Caputo–Fabrizio, the Atangana–Baleanu, and the Riemann–Liouville derivatives. Most of the authors in this field are predominantly from China and India.

The symmetric function class interacts heavily with other types of functions. One of these is the pre-invex function class, which is strongly related to symmetry theory. This paper proposes a novel fuzzy fractional extension of the Hermite-Hadamard, Hermite-Hadamard-Fejér, and Pachpatte type inequalities for up and down pre-invex fuzzy-number-valued mappings. Using fuzzy fractional operators, several generalizations have been developed, where well-known results fit as particular cases. Additionally, some non-trivial examples are included to support the discussion and the applicability of the key findings. The approach appears trustworthy and effective for dealing with various nonlinear problems in science and engineering. The findings are general and may constitute contributions to complex waveform theory.

This article investigates the propagation of a deadly human disease, namely leprosy. At the outset, the mathematical model is transformed into a fractional-order model by introducing the Caputo differential operator of arbitrary order. A result is established, which ensures the positivity of the fractional-order epidemic model. The stability of the continuous model at different points of equilibria is investigated. The basic reproduction number, R0, is obtained for the leprosy model. It is observed that the leprosy system is locally asymptotically stable at both steady states when R0<1. On the other hand, the fractional-order system is globally asymptotically stable when R0>1. To find the approximate solutions for the continuous epidemic model, a non-standard numerical scheme is constructed. The main features of the non-standard scheme (such as positivity and boundedness of the numerical method) are also confirmed by applying some benchmark results. Simulations and a feasible test example are presented to discern the properties of the numerical method. Our computational results confirm both the analytical and the numerical properties of the finite-difference scheme.

In this study, an integer-order rabies model is converted into the fractional-order epidemic model. To this end, the Caputo fractional-order derivatives are plugged in place of the classical derivatives. The positivity and boundedness of the fractional-order mathematical model is investigated by applying Laplace transformation and its inversion. To study the qualitative behavior of the non-integer rabies model, two steady states and the basic reproductive number of the underlying model are worked out. The local and global stability is investigated at both the steady states of the fractional-order epidemic model. After analytic treatment, a structure-preserving numerical template is constructed to numerically solve the fractional-order epidemic model. Moreover, the positivity, boundedness and symmetry of the numerical scheme are examined. Lastly, numerical experiment and simulations are accomplished to substantiate the significant traits of the projected numerical design. Consequences of the study are highlighted in the closing section.

In recent years, there has been a significant amount of research on the extension of convex functions which are known as preinvex functions. In this paper, we have used this approach to generalize the preinvex interval-valued function in terms of (£1, £2)-preinvex interval-valued functions because of its extraordinary applications in both pure and applied mathematics. The idea of (£1, £2)-preinvex interval-valued functions is explained in this work. By using the Riemann integral operator, we obtain Hermite-Hadamard and Fejér-type inequalities for (£1, £2)-preinvex interval-valued functions. To discuss the validity of our main results, we provide non-trivial examples. Some exceptional cases have been discussed that can be seen as applications of main outcomes.

The goal of this study is to create new variations of the well-known Hermite–Hadamard inequality (HH-inequality) for preinvex interval-valued functions (preinvex I-V-Fs). We develop several additional inequalities for the class of functions whose product is preinvex I-V-Fs. The findings described here would be generalizations of those found in previous studies. Finally, we obtain the Hermite–Hadamard–Fejér inequality with the support of preinvex interval-valued functions. Some new and classical special cases are also obtained. Moreover, some nontrivial examples are given to check the validity of our main results.

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results.

Interval analysis distinguishes between different types of order relations. As a result of these order relations, convexity and nonconvexity contribute to different kinds of inequalities. Despite this, convex theory is commonly known to rely on Godunova–Levin functions because their properties make it more efficient for determining inequality terms than convex ones. The purpose of this study is to introduce the notion of cr-h-Godunova–Levin functions by using total order relation between two intervals. Considering their properties and widespread use, center-radius order relation appears to be ideally suited for the study of inequalities. In this paper, various types of inequalities are introduced using center-radius order (cr) relation. The cr-order relation enables us firstly to derive some Hermite–Hadamard (H.H) inequalities, and then to present Jensen-type inequality for h-Godunova–Levin interval-valued functions (GL-IVFS) using a Riemann integral operator. This kind of convexity unifies several new and well-known convex functions. Additionally, the study includes useful examples to support its findings. These results confirm that this new concept is useful for addressing a wide range of inequalities. We hope that our results will encourage future research into fractional versions of these inequalities and optimization problems associated with them.

Many fields of mathematics rely on convexity and nonconvexity, especially when studying optimization issues, where it stands out for a variety of practical aspects. Owing to the behavior of its definition, the idea of convexity also contributes significantly to the discussion of inequalities. The concepts of symmetry and convexity are related and we can apply this because of the close link that has grown between the two in recent years. In this study, harmonic convexity, also known as harmonic s-convexity for fuzzy number valued functions (F-NV-Fs), is defined in a more thorough manner. In this paper, we extend harmonically convex F-NV-Fs and demonstrate Hermite–Hadamard (H.H) and Hermite–Hadamard Fejér (H.H. Fejér) inequalities. The findings presented here are summaries of a variety of previously published studies.

Boundedness is an essential feature of the solutions for various mathematical and numerical models in the natural sciences, especially those systems in which linear or nonlinear preservation or stability features are fundamental. In those cases, the boundedness of the solutions outside a set of zero measures is not enough to guarantee that the solutions are physically relevant. In this note, we will establish a criterion for the boundedness of integrable solutions of general continuous and numerical systems. More precisely, we establish a characterization of those measures over arbitrary spaces for which real-valued integrable functions are necessarily bounded at every point of the domain. The main result states that the collection of measures for which all integrable functions are everywhere bounded are exactly all of those measures for which the infimum of the measures for nonempty sets is a positive extended real number.

Reaction-diffusion-advection equations provide precise interpretations for many important phenomena in complex interactions between natural and artificial systems. This paper studies second-order semi-discretizations for the numerical solution of reaction-diffusion-advection equations modeling quenching types of singularities occurring in numerous applications. Our investigations particularly focus at cases where nonuniform spatial grids are utilized. Detailed derivations and analysis are accomplished. Easy-to-use and highly effective second-order schemes are acquired. Computational experiments are presented to illustrate our results as well as to demonstrate the viability and capability of the new methods for solving singular quenching problems on arbitrary grid platforms.

The purpose of this study is to define a new class of harmonically convex functions, which is known as left and right harmonically convex interval-valued function (LR-𝓗-convex IV-F), and to establish novel inclusions for a newly defined class of interval-valued functions (IV-Fs) linked to Hermite–Hadamard (H-H) and Hermite–Hadamard–Fejér (H-H-Fejér) type inequalities via interval-valued Riemann–Liouville fractional integrals (IV-RL-fractional integrals). We also attain some related inequalities for the product of two LR-𝓗-convex IV-Fs. These findings enable us to identify a new class of inclusions that may be seen as significant generalizations of results proved by Iscan and Chen. Some examples are included in our findings that may be used to determine the validity of the results. The findings in this work can be seen as a considerable advance over previously published findings.

The main objective of this study is to introduce new versions of fractional integral inequalities in fuzzy fractional calculus utilizing the introduced preinvexity. Due to the behavior of its definition, the idea of preinvexity plays a significant role in the subject of inequalities. The concepts of preinvexity and symmetry have a tight connection thanks to the significant correlation that has developed between both in recent years. In this study, we attain the Hermite-Hadamard (H·H) and Hermite-Hadamard-Fejér (H·H Fejér) type inequalities for preinvex fuzzy-interval-valued functions (preinvex F·I·V·Fs) via Condition C and fuzzy Riemann–Liouville fractional integrals. Furthermore, we establish some refinements of fuzzy fractional H·H type inequality. There are also some specific examples of the reported results for various preinvex functions deduced. To support the newly introduced ideal, we have provided some nontrivial and logical examples. The results presented in this research are a significant improvement over earlier results. This paper’s awe-inspiring notions and formidable tools may energize and revitalize future research on this worthwhile and fascinating topic.

In this paper, we discuss the Riemann–Liouville fractional integral operator for left and right convex interval-valued functions (left and right convex I∙V-F), as well as various related notions and concepts. First, the authors used the Riemann–Liouville fractional integral to prove Hermite–Hadamard type (𝓗–𝓗 type) inequality. Furthermore, 𝓗–𝓗 type inequalities for the product of two left and right convex I∙V-Fs have been established. Finally, for left and right convex I∙V-Fs, we found the Riemann–Liouville fractional integral Hermite–Hadamard type inequality (𝓗–𝓗 Fejér type inequality). The findings of this research show that this methodology may be applied directly and is computationally simple and precise.

In recent years, fractional calculus has witnessed tremendous progress in various areas of sciences and mathematics [...]

Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R0<1. A similar result is obtained for the endemic equilibrium when R0>1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

This article proposes two conformal Solow models (with and without migration), accompanied by simulations for six Organisation for Economic Co-operation and Development economies. The models are proposed by employing suitable Inada conditions on the Cobb–Douglas function and making use of the truncated M-derivative for the Mittag–Leffler function. In the exact solutions derived in this manuscript, two new parameters play an important role in the convergence towards, or the divergence from, the steady state of capital and per capita product. The economical dynamics of these nations are influenced by the intensity of the capital and labor factors, as well as the level of depreciation, the labor force rate and the level of saving.

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz sense. Initial and homogeneous Dirichlet boundary data are imposed on a multidimensional spatial domain. To approximate the solutions, we employ a finite difference methodology. We rigorously establish the existence of numerical solutions along with the main numerical properties. Concretely, we show that the scheme is consistent in both space and time as well as stable and convergent. Numerical simulations in the one-dimensional scenario are presented in order to show the performance of the scheme. For the sake of convenience, A MATLAB code of the numerical model is provided in the appendix at the end of this work.

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. 16: 892–910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order O(τ2−α+h4), in the case that 0<α<1 satisfies 3α≥32, which means that 0.369⪅α≤1. The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order 0<α<1 used for that scheme at tk+1/2.

We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks.