Vol. 13, No. 4 (2024): Regular Issue

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Caterpillar trees have been used in chemical graph theory to represent the molecular structures of hydrocarbons. One of the important graph invariants based on a minimum edge dominating matrix of a graph is the minimum edge dominating energy of the graph. The minimum edge dominating energy of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of the minimum edge dominating matrix of $G$ with respect to a minimum edge dominating set in $G$. In this paper, we obtain a minimum edge dominating set and the minimum edge domination number of caterpillar trees. Also, some results of the minimum edge dominating energy on caterpillars are given. We compute explicit formulas for the minimum edge dominating eigenvalues of these graphs.

A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Caterpillar trees have been used in chemical graph theory to represent the molecular structures of hydrocarbons. One of the important graph invariants based on a minimum edge dominating matrix of a graph is the minimum edge dominating energy of the graph. The minimum edge dominating energy of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of the minimum edge dominating matrix of $G$ with respect to a minimum edge dominating set in $G$. In this paper, we obtain a minimum edge dominating set and the minimum edge domination number of caterpillar trees. Also, some results of the minimum edge dominating energy on caterpillars are given. We compute explicit formulas for the minimum edge dominating eigenvalues of these graphs.

One of the most commonly used methods for elliptic models is the method of analysis of Sub/super solutions. A fractional powers of the Laplace operator is considered. By using sub-super solutions method, we show the existence of weak positive solution for class of elliptic systems in bounded domains. Our results are natural extensions from the previous recent papers.

One of the most commonly used methods for elliptic models is the method of analysis of Sub/super solutions. A fractional powers of the Laplace operator is considered. By using sub-super solutions method, we show the existence of weak positive solution for class of elliptic systems in bounded domains. Our results are natural extensions from the previous recent papers.

This research investigates the reachability of linear and non-linear fractional dynamical systems with distributed delays in control using the $\psi$-Hilfer pseudo-fractional derivative in g-calculus. Grammian matrices, which are characterized by Mittag-Leffler functions, are used to provide necessary and sufficient criteria for reachability in the linear case, while Schauder's fixed point theorem is used to create sufficient conditions for reachability in the nonlinear case. A couple of numerical results is offered to explain the theoretical results.

This research investigates the reachability of linear and non-linear fractional dynamical systems with distributed delays in control using the $\psi$-Hilfer pseudo-fractional derivative in g-calculus. Grammian matrices, which are characterized by Mittag-Leffler functions, are used to provide necessary and sufficient criteria for reachability in the linear case, while Schauder's fixed point theorem is used to create sufficient conditions for reachability in the nonlinear case. A couple of numerical results is offered to explain the theoretical results.

In the present paper, we investigate the existence of at least one weak solution to the Dirichlet boundary value problem involving the $p(x)$-Kirchhoff type equation with a reaction term depending also on the gradient (convection). Our result is obtained by means of the theory of topological degree and the theory of variable exponent Sobolev spaces.

In the present paper, we investigate the existence of at least one weak solution to the Dirichlet boundary value problem involving the $p(x)$-Kirchhoff type equation with a reaction term depending also on the gradient (convection). Our result is obtained by means of the theory of topological degree and the theory of variable exponent Sobolev spaces.

This paper is devoted to the study of Cauchy problem for viscoelastic damped wave models with nonlinear memory on the right-hand side. The main goal is to prove blow-up results for local (in time) energy solutions. The model that we consider is parabolic-like from the point of view of energy decay estimates of the corresponding linear Cauchy problem with a vanishing right-hand side. For this reason, we apply the test function method to prove our results.

This paper is devoted to the study of Cauchy problem for viscoelastic damped wave models with nonlinear memory on the right-hand side. The main goal is to prove blow-up results for local (in time) energy solutions. The model that we consider is parabolic-like from the point of view of energy decay estimates of the corresponding linear Cauchy problem with a vanishing right-hand side. For this reason, we apply the test function method to prove our results.

In this paper we study the Cauchy problem of the Debye-Hückel system with initial data in variable Fourier-Besov spaces. By using littlewood-Paley decomposition, we obtain the global well-posedness result for small initial data belong to critical variable exponent Fourier-Besov spaces. Moreover, we get the analyticity of global solutions.

In this paper we study the Cauchy problem of the Debye-Hückel system with initial data in variable Fourier-Besov spaces. By using littlewood-Paley decomposition, we obtain the global well-posedness result for small initial data belong to critical variable exponent Fourier-Besov spaces. Moreover, we get the analyticity of global solutions.

In this article, we present a novel variant of Hermite-Hadamard-Mercer's inequality for strongly convex functions via ABK-fractional integrals and the Jensen-Mercer's inequality. Further, we introduce some Hermite-Hadamard-Mercer's type inequalities for differentiable functions whose absolute value of the derivative is convex. Some fundamental inequalities, like Holder's inequality and Young's inequality, have been used to establish inequalities. Furthermore, we discuss special cases for our main results and obtain the new Hermite-Hadamard-Mercer's inequalities for convex functions using the ABK-fractional Integral Operator. Additionally, some applications for special means are also given.

In this article, we present a novel variant of Hermite-Hadamard-Mercer's inequality for strongly convex functions via ABK-fractional integrals and the Jensen-Mercer's inequality. Further, we introduce some Hermite-Hadamard-Mercer's type inequalities for differentiable functions whose absolute value of the derivative is convex. Some fundamental inequalities, like Holder's inequality and Young's inequality, have been used to establish inequalities. Furthermore, we discuss special cases for our main results and obtain the new Hermite-Hadamard-Mercer's inequalities for convex functions using the ABK-fractional Integral Operator. Additionally, some applications for special means are also given.

A nonlinear delayed fractional order SIR compartmental model with a Holling type II saturated incidence rate and treatment rate are explored in this manuscript in the Caputo order fractional derivative approach. A few findings for the new model's existence and uniqueness criterion, as well as non-negativity and boundedness of the solution, have been established. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium point $E_0$ when $R_0< 1$ and at epidemic equilibrium $E_1$ when $R_0 >1$. We have studied forward bifurcation at $E_0$ and Hopf bifurcation at $E_1$ theoretically as well as numerically of our proposed model. The stability behavior of the endemic equilibrium is also discussed, revealing that oscillatory and periodic solutions may appear via Hopf bifurcation when regarding delay as the bifurcation parameter. Analytical correlations between delay and other system characteristics are built to ensure the presence of stability situations. Additionally, the suggested model's solution is approximated using the fractional-order Taylor's technique. Both graphical presentations and numerical simulations might be carried out with the help of MATLAB.

A nonlinear delayed fractional order SIR compartmental model with a Holling type II saturated incidence rate and treatment rate are explored in this manuscript in the Caputo order fractional derivative approach. A few findings for the new model's existence and uniqueness criterion, as well as non-negativity and boundedness of the solution, have been established. The stability analysis of the model shows that the system is locally as well as globally asymptotically stable at disease-free equilibrium point $E_0$ when $R_0< 1$ and at epidemic equilibrium $E_1$ when $R_0 >1$. We have studied forward bifurcation at $E_0$ and Hopf bifurcation at $E_1$ theoretically as well as numerically of our proposed model. The stability behavior of the endemic equilibrium is also discussed, revealing that oscillatory and periodic solutions may appear via Hopf bifurcation when regarding delay as the bifurcation parameter. Analytical correlations between delay and other system characteristics are built to ensure the presence of stability situations. Additionally, the suggested model's solution is approximated using the fractional-order Taylor's technique. Both graphical presentations and numerical simulations might be carried out with the help of MATLAB.

Due to the lack of effective research methods, the study of fractional time-varying and boundary feedback for Lamé system requires special studies and the involvement of new methods, so there are few works in this direction. The paper is devoted to Lamé system with fractional time-varying and boundary feedback. A general description of the question of well posedness of problem, their stability, and a review of the results are given. The novelty and main contributions located in the interaction between different damping terms and show the impact of each one of them on the stability.

Due to the lack of effective research methods, the study of fractional time-varying and boundary feedback for Lamé system requires special studies and the involvement of new methods, so there are few works in this direction. The paper is devoted to Lamé system with fractional time-varying and boundary feedback. A general description of the question of well posedness of problem, their stability, and a review of the results are given. The novelty and main contributions located in the interaction between different damping terms and show the impact of each one of them on the stability.

In this paper, we consider a one-dimensional porous thermoelastic system with microtemperatures effects, past history term acting only on the porous equation and a delay term in the internal feedback. Under an appropriate assumptions on the kernel and between the weight of the delay and the weight of the damping, we prove the well-posedness of the system. Furthermore, we establish a general decay rate result for the energy, which allows a wider class of relaxation functions, and thus generalize some results in the literature. Finally, some numerical experiments are presented.

In this paper, we consider a one-dimensional porous thermoelastic system with microtemperatures effects, past history term acting only on the porous equation and a delay term in the internal feedback. Under an appropriate assumptions on the kernel and between the weight of the delay and the weight of the damping, we prove the well-posedness of the system. Furthermore, we establish a general decay rate result for the energy, which allows a wider class of relaxation functions, and thus generalize some results in the literature. Finally, some numerical experiments are presented.

In the present paper, we obtain generalized Weyl fractional $q$-integrals of the general class of $q$-polynomials and demonstrate their use by studying $q$-Konhouser biorthogonal polynomial, $q$-Jacobi polynomials and basic analogue of the Kampé de Fériet function. Polynomials have been obtained as a particular case of our major findings.

In the present paper, we obtain generalized Weyl fractional $q$-integrals of the general class of $q$-polynomials and demonstrate their use by studying $q$-Konhouser biorthogonal polynomial, $q$-Jacobi polynomials and basic analogue of the Kampé de Fériet function. Polynomials have been obtained as a particular case of our major findings.