International Journal of Mathematics for Industry

This paper introduces an asymptotic expansion for the smooth solution of a semi-linear partial differential equation. Our scheme is based on Itô’s formula, Taylor’s expansion, nonlinear Feynman–Kac formula and some algebras.

In this paper, we analyzed the redistribution of dopants in a multilayer structure during the overgrowing of a doped area. The effect of changing of regimes of overgrowing of the doped area on the redistribution of dopant was analyzed. An analytical approach for analysis of the redistribution of dopant was introduced. The approach makes it possible to take into account the changing parameters of the considered process in space and time together, as well as the nonlinearity of the mass transport.

Let G be a connected graph. A subset $S=\{v_1,v_2,\dots ,v_k\}$ of $V(G)$ is called a resolving set of G if the code of any vertex $u\in V(G)$ with respect to S is different from the code of any other vertex where code of u with respect to S denoted by $C_s(u)$ is defined as $C_s(u)=(d(u,v_1),d(u,v_2),\dots ,d(u,v_k))$. Resolving set was earlier studied in the name of locating set by Slater and Harary and Melter too studied this concept. The minimum cardinality of a resolving set is called the metric dimension (locating number). A vertex $x\in V(G)$ in a connected graph G is said to resolve two vertices $u,v\in V$ if $d(u,x)\ne d(v,x).$ Clearly, x resolves $x,v.$ A subset S of $V(G)$ is a resolving set of G if for any two distinct vertices $u,v\in V,$ there exists a vertex $x\in S$ such that x resolves $u,v.$ Motivated by this equivalent definition, a study of resolving chain and maximal resolving set is initiated in this paper. Also, study of total resolving sets is initiated.

The notion of competition graphs has a wide range of applications and has been utilized in designing various ecological problems and real-world competitions. Here, the idea of m-step inverse fuzzy mixed competition graphs has been introduced and some of its basic properties have been discussed. Also, some ideas associated with the concept of m-step inverse fuzzy mixed competition graphs, such as m-step inverse fuzzy neighborhood graph, inverse fuzzy mixed economic competition graph, and m-step inverse fuzzy mixed economic competition graph have been defined. Our motive was to focus on the real-life problems that can be properly addressed with graphs involving directed and undirected edges. In this case, the concept of an m-step fuzzy competition graph is not appropriate for solving these problems. However, the idea of m-step inverse fuzzy mixed competition graphs is more relevant to address them. An application of m-step inverse fuzzy mixed competition graphs has been identified as a suitable solution to a real-life problem.

Corruption is an old and complex phenomenon that affects many countries in different forms. Despite many countries using various techniques to control corruption transmission, the problem is still present. Hence, we developed and analyzed a nonlinear deterministic mathematical model that explains the dynamics of corruption to investigate the effect of control measures on the transmission of corruption. Depending on the corruption status we classified the total population into five compartments, which is an extension of the SCJH model, and techniques of reducing corruption transmission were also examined. We have computed the basic reproduction number $R_0$ of the model by using the next-generation matrix method and analyzed the stability of the equilibrium point. The stability analysis reveals that corruption-free equilibrium (CFE) is both locally and globally asymptotically stable for $R_0<1$, but the corruption-endemic equilibrium (CEE) is both locally and globally stable for $R_0>1$. In the model, we established a sufficient condition for the presence of forward bifurcation. We used the normalized forward sensitivity approach to compute the most sensitive parameters that reduce the spread of corruption. The model was simulated by using ode45 code in MATLAB and results indicate that the effective transmission contact rate $\beta$ and taking of adequate penalties rate $\gamma$ reduce corruption. Therefore, this study recommends that reducing the effective transmission contact rate and taking adequate penalty rate were better strategies to reduce the spread of corruption.

The CSFS is an extension of CPFS and CPicFS. The design addresses vagueness more explicitly. It extends the real subset’s range to complex with the use of unit disc. Due to its coverage in three directions: complex membership functions, complex abstinence membership function, and complex non-membership functions, CSFS has more flexibility than CFSs, CIFSs, CPFSs, and CPicFS. It can be applied in various ways through fuzzy control. The primary objective of this study is to present CSFGS, which exhibits exceptional performance in tackling challenging problem, particularly ones that involve multiple relationships. GSs and CSFSs are combined in this concept. As a result, many problems are combined with different relations. The AM of CSFGS and degree of vertex presence have been found. The lower and upper constraints on the energy of CSFGS are derived by calculating the energy of CSFGS. Furthermore, we have provided some details regarding the LE of the ${\sigma }_J$-dominant CSFGS. We also propose the notion of isomorphic and identical ${\sigma }_J$-dominating CSFGS. Finally, we have discussed a real-world example based on temperature variation and climatic data analysis in an environment with a LE of the ${\sigma }_J$-dominant CSFGS, to demonstrate the applicability of the generated results. Also, an algorithm has been developed to clarify the fundamental workings of our application.

This research develops a model describing a relation between a predator and a prey where the prey species are divided into two classes i.e., susceptible prey species and the infected prey species. Furthermore, the susceptible prey species are also provided with a refuge. After this, the research expands the formulated model into a model with Caputo fractional derivative. A detailed analysis of the solution of Caputo fractional model for the predator–prey relation with a sickness in prey and a refuge to susceptible prey is done. Numerical solutions along with the simulations are briefed for the Caputo fractional model. Lastly, the phase portrait diagrams are also given for better understanding.

A simple model of the insurance coverage for the damage of COVID-19 is introduced. Concerning the estimation of the numbers of patients and/or deaths, we employ the Cramér–Lundberg model for the risk process, which is combined with the discrete SIR model. Under various premium principles, we are able to design suitable insurance. Numerical research with the data of Tokyo region is also performed.

This paper expressed the fractional kinetic equation (KE) in terms of the incomplete $\aleph$-function, I-function, Fox H-function, and Meijer’s G-function. It assesses the importance of fractional KE in diverse scientific and engineering contexts, highlighting its relevance across various scenarios. In this paper, we investigate the results of Katugampola’s kinetic fractional equations with the product of the generalized M-series and incomplete $\aleph$-function. The $\tau$-Laplace transform is applied for obtaining the desired results. Utilizing an M-series, the universality of this series is harnessed to deduce solutions for a fractional KE. Furthermore, a graphical representation of the obtained solutions’ behaviors is presented, enhancing the impact of the findings.

In this paper, we estimate the maximal annealing time of dopant and/or radiation defects during the manufacturing of elements of integrated circuits by dopant diffusion and ion implantation. We introduce an analytical approach to estimate the maximal continuance of the diffusion process. We analyzed the influence of parameters of considered technological processes on the value of their maximal continuance of diffusion process.

The quality of products in manufacturing engineering depends first of all on the machining processes. One of the crucial economic components of the production processes is the metal-cutting machining technologies that form the quality of manufactured objects. The physics of metal-cutting technologies is complex and cannot be described by one mathematical model because of different natures. Researchers derived an empirical formula of their tool life that describes the properties of the metal-cutting tools depending on many mechanical factors mentioned above. Engineers investigated and worked the metal-cutting regimes for manufacturing processes that comprise recommendations for the tool life. The components of the tool life are obtained by the practical tests at the research laboratories that give the approximated results. Practice shows the recommendations of the handbooks for metal-cutting regimes are not always optimal for manufacturing processes. The machine tools’ work in the manufacturing environment differs from the research laboratories. This work considers the analytical optimization of the cutting speed for the threading machine tool by the criterion of its maximal productivity rate.

In this study, a precise and analytical method, namely Shehu transform decomposition method (STDM), is applied to examine multi-dimensional fractional diffusion equations, which will describe density dynamics in a material undergoing diffusion. The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative. The proposed approach combines the Shehu transformation (ST) with the Adomian decomposition method (ADM), employing Adomain polynomials to handle nonlinear terms. Fixed point theorem has been used to prove the uniqueness and existence of the solution of governing differential equation. To validate the accuracy of proposed method, we present the analytical solutions of three applications of multi-dimensional fractional diffusion equations. Furthermore, we compute the graphical and numerical results using MATLAB to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.

In this paper, the distribution of heat in skin tissue during thermal treatment using the time-fractional Pennes model is studied. Temperature variations concerning several parameters such as derivative $(\alpha >0),$ time, blood perfusion, and transmitted power are described using the Legendre collocation technique. The best feature of the proposed technique is the conversion of fractional-order differential equations into a set of algebraic systems and the ability to approximate both spatial and temporal coordinates with a single basis. The obtained results and comparison with an exact solution are illustrated.

This paper explores operations on fuzzy incidence graphs (FIGs), focusing on join, Cartesian product, tensor product, and composition. Emphasizing strong fuzzy incidence graphs (SFIGs), the study examines strong incidence domination (SID) and the strong incidence domination number (SIDN) in these operations. Basic properties of FIGs resulting from these operations are studied, and bounds for the SIDN of the product of two SFIGs are established for Cartesian and tensor products. The research includes FIGs with strong join and composition, as well as complete fuzzy incidence graphs (CFIGs) and FIGs with effective pairs. Additionally, the paper discusses an application of SID in security allocation.

Machining processes focused primarily on the quality of surfaces and sizes of machined workpieces, as well as minimizing costs and tool wear. Existing publications related to maximizing the productivity of machining processes, present results that are in fact sub-optimal. The increase in machining regimes improves productivity but may reduce the quality of machined products. Mathematical optimization methods help find a balance between the benefits and drawbacks of machining processes. The tool life of cutters largely influences machining productivity and sets limits on machining speed. The normative tool life equations developed in research labs do not take into account the reliability of machine units and the specificities of technological processes. The normative machining speeds are suboptimal for manufacturing processes and may not maximize productivity. This paper presents a mathematical model for optimizing the cutting speed of a drilling machine tool to achieve its maximum productivity under realistic manufacturing conditions.

In this study, we derive the approximate analytical solution for the fractional coupled Whitham–Broer–Kaup (WBK) equations, a significant mathematical model for representing wave propagation in shallow water. The solution is obtained through the utilization of the q-homotopy analysis $\rho$-Laplace transform method (q-HAL${}_{\rho }$TM), a hybrid approach combining $\rho$-Laplace transformation and the homotopy analysis method. Homotopy polynomials are employed to address nonlinear terms, and the introduced algorithm incorporates the auxiliary parameter $\hslash$ to regulate and fine-tune the convergence region of the resulting series solution. Comparative numerical analyses are conducted with outcomes from the Adomian decomposition method (ADM), variational iteration method (VIM), and optimal homotopy asymptotic method (OHAM), demonstrating the superior accuracy of the proposed method. The method’s novelty and straightforward implementation establish it as a reliable and efficient analytical technique for solving both linear and nonlinear fractional partial differential equations.

Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. For available good data sets on mobile fish populations, the distribution is certainly not normal. The data can be fitted with a long-tailed stable Lévy distribution that results from diffusion by fractional Laplacian. Exact multidimensional solutions are developed here for realistic Fisher–Kolmogorov–Petrovski-Piscounov models with diffusion by fractional Laplacian. These can also account for a delay in the reaction term. It is then shown how to modify critical domain sizes of protected areas with Dirichlet and Robin boundary conditions for populations.

The $\beta$-encoder is an analog circuit that converts an input signal $x\in [0,1]$ into a finite bit stream $\{b_i\}$. The bits $\{b_i\}$ are correlated and therefore are not immediately suitable for random number generation, but they can be used to generate bits $\{a_i\}$ that are (nearly) uniformly distributed. In this paper, we study two such methods. In the first part the bits $\{a_i\}$ are defined as the digits of the base-2 representation of the original input x. Under the assumption that there is no noise in the amplifier we then study a question posed by Jitsumatsu and Matsumura on how many bits $b_1,\dots ,b_m$ are needed to correctly determine the first n bits $a_1,\dots ,a_n$. In the second part, we show this method fails for random amplification factors. Nevertheless, even in this case, nearly uniformly distributed bits can still be generated from $b_1,\dots ,b_m$ using modern cryptographic techniques.