Vol. 13, No. 2 (2024): Regular Issue

A linear stability analysis determining the onset of convection of a chemically reacting fluid in a horizontal layer with rigid boundaries is performed. The governing dimensionless equations are solved using the normal modes, which leads to an eigenvalue problem for the onset of convection. The effects of the solute Rayleigh number, Prandtl number, Lewis number, and Damkohler number on the stability of the system are investigated. We find that the Damkohler number has a contrasting effect on stationary and oscillatory instability. The frequency decreases with $\chi $, but increases with $Le $ and $Pr$. The effect of increasing the Prandtl number is to advance the onset of convection. The Lewis number has a destabilizing effect on the onset of oscillatory convection.

A linear stability analysis determining the onset of convection of a chemically reacting fluid in a horizontal layer with rigid boundaries is performed. The governing dimensionless equations are solved using the normal modes, which leads to an eigenvalue problem for the onset of convection. The effects of the solute Rayleigh number, Prandtl number, Lewis number, and Damkohler number on the stability of the system are investigated. We find that the Damkohler number has a contrasting effect on stationary and oscillatory instability. The frequency decreases with $\chi $, but increases with $Le $ and $Pr$. The effect of increasing the Prandtl number is to advance the onset of convection. The Lewis number has a destabilizing effect on the onset of oscillatory convection.

We study the dynamic behavior of a discretized prey-predator system in this research. The model is formulated in terms of difference equations and derived by using the piecewise method, which takes into consideration the non-overlapping generations. The existence of fixed points as well as their local asymptotic stability are proved. Further, it is shown that the model experiences Neimark-Sacker bifurcation (NSB for short) and period-doubling bifurcation (PDB) in a small neighborhood of the coexistence fixed point under certain parametric conditions. This analysis utilizes bifurcation theory and the center manifold theorem. The chaos is stabilized using two different chaos control methods. Lastly, we verify our theoretical findings and provide more complex dynamics through computer analysis and numerical simulations.

We study the dynamic behavior of a discretized prey-predator system in this research. The model is formulated in terms of difference equations and derived by using the piecewise method, which takes into consideration the non-overlapping generations. The existence of fixed points as well as their local asymptotic stability are proved. Further, it is shown that the model experiences Neimark-Sacker bifurcation (NSB for short) and period-doubling bifurcation (PDB) in a small neighborhood of the coexistence fixed point under certain parametric conditions. This analysis utilizes bifurcation theory and the center manifold theorem. The chaos is stabilized using two different chaos control methods. Lastly, we verify our theoretical findings and provide more complex dynamics through computer analysis and numerical simulations.

This manuscript analyses the controllability of a certain classes of sobolev-type Volterra-Fredholm functional Integro-Differential Equations (SVFIDE) of fractional order via Caputo fractional derivative involving finite delay with initial condition. We have empolyed the precompactness of Arzela-Ascoli theorem along with the standard fixed point method to attain the desired result. Finally, we present an example to demonstrate the validity of our result.

This manuscript analyses the controllability of a certain classes of sobolev-type Volterra-Fredholm functional Integro-Differential Equations (SVFIDE) of fractional order via Caputo fractional derivative involving finite delay with initial condition. We have empolyed the precompactness of Arzela-Ascoli theorem along with the standard fixed point method to attain the desired result. Finally, we present an example to demonstrate the validity of our result.

In this paper, a preventive model for the treatment of an infectious disease in two animal population, is discussed. First, we study the well-posedness of the proposed mathematical model. Then we determine the disease-free equilibrium point and basic reproductive number of the model. Further, we perform the stability analysis for the disease-free equilibrium point of the model. Next, an optimal control problem is developed to minimize the cost of treating an infected population. Optimality conditions are derived using Pontryagin's principle. Numerical results are provided to validate the theoretical results.

In this paper, a preventive model for the treatment of an infectious disease in two animal population, is discussed. First, we study the well-posedness of the proposed mathematical model. Then we determine the disease-free equilibrium point and basic reproductive number of the model. Further, we perform the stability analysis for the disease-free equilibrium point of the model. Next, an optimal control problem is developed to minimize the cost of treating an infected population. Optimality conditions are derived using Pontryagin's principle. Numerical results are provided to validate the theoretical results.

A dynamical predator-prey model with constant prey harvesting, proportional harvesting in predator has been studied. The square root functional response has also been included in the system to characterise the behaviour of the prey herd when the average handling time is zero. The existence and local stability of the system's equilibria have been discussed. It is examined that the system has two sorts of bifurcations. The two forms of bifurcations were studied, and it was explored that the saddle-node bifurcation offers the highest sustainable yield. It has been observed that if the harvesting rate exceeds the maximum sustainable yield, the prey population is eliminated from the system, and the predator population is wiped out. However, if such harvesting rate is below than the sustainable yield, the prey population may be able to sustain. An unstable limit cycle around the interior equilibrium point has been found by investigating the Hopf bifurcation. To verify the results, further numerical simulations are run.

A dynamical predator-prey model with constant prey harvesting, proportional harvesting in predator has been studied. The square root functional response has also been included in the system to characterise the behaviour of the prey herd when the average handling time is zero. The existence and local stability of the system's equilibria have been discussed. It is examined that the system has two sorts of bifurcations. The two forms of bifurcations were studied, and it was explored that the saddle-node bifurcation offers the highest sustainable yield. It has been observed that if the harvesting rate exceeds the maximum sustainable yield, the prey population is eliminated from the system, and the predator population is wiped out. However, if such harvesting rate is below than the sustainable yield, the prey population may be able to sustain. An unstable limit cycle around the interior equilibrium point has been found by investigating the Hopf bifurcation. To verify the results, further numerical simulations are run.

In this paper, we study a system of shallow water equations with a rectilinear sloping bottom. Two fundamentally different cases are considered. In the first case, the Jacobi matrix of dependent variables with respect to independent variables is degenerate. This case has not been studied by anyone before. We have obtained all such solutions of the system. Each solution for specific values of its parameters is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the propagation velocity of a surface wave. The physical meaning of these solutions is indicated. In the second case, the Jacobi matrix of dependent variables with respect to independent variables is nondegenerate. This allows you to linearize the original system using a special hodograph transformation. Two exact solutions of this linear system are found: an invariant solution and a partially invariant solution. With the help of these solutions, we have obtained the exact solutions of the original system. Each solution for specific values of its parameters also is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the velocity of propagation of a surface wave. The physical meaning of these solutions is indicated. Using the method of $A$-operators, we have found all zero-order conservation laws for the original system with an inclined bottom.

In this paper, we study a system of shallow water equations with a rectilinear sloping bottom. Two fundamentally different cases are considered. In the first case, the Jacobi matrix of dependent variables with respect to independent variables is degenerate. This case has not been studied by anyone before. We have obtained all such solutions of the system. Each solution for specific values of its parameters is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the propagation velocity of a surface wave. The physical meaning of these solutions is indicated. In the second case, the Jacobi matrix of dependent variables with respect to independent variables is nondegenerate. This allows you to linearize the original system using a special hodograph transformation. Two exact solutions of this linear system are found: an invariant solution and a partially invariant solution. With the help of these solutions, we have obtained the exact solutions of the original system. Each solution for specific values of its parameters also is illustrated by a graph of the distribution of the excess of the free surface above the bottom and a graph of the distribution of the velocity of propagation of a surface wave. The physical meaning of these solutions is indicated. Using the method of $A$-operators, we have found all zero-order conservation laws for the original system with an inclined bottom.

In this article, a prey-predator model has been proposed considering predator induced fear to prey species. Furthermore, it is considered that predator species suffers from cannibalism. It is observed that both the parameter associated with predator induced fear and cannibalism play a crucial role in controlling the stability of system dynamics. The predator induced fear factor acts as a stabilizing factor while predator cannibalism phenomenon affects the system dynamics in absence of fear through the occurrence of multiple local bifurcations; but in presence of fear factor this cannibalism act cannot influence system dynamics. On the other hand, it is noticed that predator's birth rate due to cannibalism plays as a stable factor in system dynamics for a suitable range of this parameter. Additionally, it is observed that there is an important relationship between the phenomenon fear and cannibalism. Both the phenomenon individually affects the population biomass in a significant way.

In this article, a prey-predator model has been proposed considering predator induced fear to prey species. Furthermore, it is considered that predator species suffers from cannibalism. It is observed that both the parameter associated with predator induced fear and cannibalism play a crucial role in controlling the stability of system dynamics. The predator induced fear factor acts as a stabilizing factor while predator cannibalism phenomenon affects the system dynamics in absence of fear through the occurrence of multiple local bifurcations; but in presence of fear factor this cannibalism act cannot influence system dynamics. On the other hand, it is noticed that predator's birth rate due to cannibalism plays as a stable factor in system dynamics for a suitable range of this parameter. Additionally, it is observed that there is an important relationship between the phenomenon fear and cannibalism. Both the phenomenon individually affects the population biomass in a significant way.

In this paper, peristaltic transport of Jeffrey fluid in circular cylinder tube has been studied. A mathematical model has developed to study flow by dilating peristaltic waves under assumption of long wavelength and low Reynolds number approximations. The expressions for velocities, pressure gradient and pressure are obtained. The effects of dilation parameter and ratio of relaxation and retardation times are numerically discussed by graphs. It has concluded that pressure increases with the increases of dilation parameter but decreases when ratio of relaxation to retardation time increases.

In this paper, peristaltic transport of Jeffrey fluid in circular cylinder tube has been studied. A mathematical model has developed to study flow by dilating peristaltic waves under assumption of long wavelength and low Reynolds number approximations. The expressions for velocities, pressure gradient and pressure are obtained. The effects of dilation parameter and ratio of relaxation and retardation times are numerically discussed by graphs. It has concluded that pressure increases with the increases of dilation parameter but decreases when ratio of relaxation to retardation time increases.

This article investigates Jacobi stability analysis of unified chaotic system. The stability analysis of unified system has been discussed by using the Kosambi-Cartan-Chern (KCC) theory, which is based on differential geometry. The unified system is transformed into a pair of second-order differential equations. The five KCC invariants are obtained to study the dynamics of unified chaotic system. The deviation curvature tensor is obtained and it explains the stability of the system. Further, the dynamics of unified chaotic system near the equilibrium points is also discussed.

This article investigates Jacobi stability analysis of unified chaotic system. The stability analysis of unified system has been discussed by using the Kosambi-Cartan-Chern (KCC) theory, which is based on differential geometry. The unified system is transformed into a pair of second-order differential equations. The five KCC invariants are obtained to study the dynamics of unified chaotic system. The deviation curvature tensor is obtained and it explains the stability of the system. Further, the dynamics of unified chaotic system near the equilibrium points is also discussed.

One and two-parameter families of high-degree interpolating polynomials are constructed using the Lagrange polynomial, which is closest to a given ``nice" polynomial. The problem of finding such closest polynomial is formulated as an optimization problem using the area between polynomials. The explicitly computable optimizer is uniquely deduced. Numerical results are given for high order polynomial which is closest to target piecewise linear and piecewise cubic spline interpolations. These results demonstrate that despite of high degree, the optimized polynomial is less oscillatory and mimics the ``nice" non-oscillatory property of the target polynomial.

One and two-parameter families of high-degree interpolating polynomials are constructed using the Lagrange polynomial, which is closest to a given ``nice" polynomial. The problem of finding such closest polynomial is formulated as an optimization problem using the area between polynomials. The explicitly computable optimizer is uniquely deduced. Numerical results are given for high order polynomial which is closest to target piecewise linear and piecewise cubic spline interpolations. These results demonstrate that despite of high degree, the optimized polynomial is less oscillatory and mimics the ``nice" non-oscillatory property of the target polynomial.

Fractional Power Spaces and Karakostas' Fixed Point Theorem were used to prove the existence and uniqueness of solutions for a semilinear neutral evolution equation with impulses and nonlocal conditions in a Banach space. As an application of our main result, we consider a neutral type Burgers' equation.

Fractional Power Spaces and Karakostas' Fixed Point Theorem were used to prove the existence and uniqueness of solutions for a semilinear neutral evolution equation with impulses and nonlocal conditions in a Banach space. As an application of our main result, we consider a neutral type Burgers' equation.

The analysis of a cantilever beam at any level is a mature subject, and there are a lot of elegant numerical techniques to perform large deflection analysis of non-prismatic cantilever beams. This work aims to develop a straightforward and computationally efficient method to solve large deflection problems of non-prismatic cantilever beams subjected to transverse loading. The modeled beams are assumed to be naturally straight, slender, inextensible, and follow a linear elastic material behavior. The governing differential equation obtained from Euler--Bernoulli beam theory, where the exact curvature expression is preserved, has been discretized using the first-order forward finite difference method. When the deflection is very high, the discretized equation results in a system of highly non-linear equations, and an iterative solution method must be used to solve the problem. This paper employs the bisection method to satisfy the fixed-end boundary condition, which consists of enforcing a slope angle of zero. The method is illustrated through a numerical example of a cantilever beam with a rectangular cross-section subjected to both concentrated and distributed loads. The results provided by this method were shown to be in excellent agreement with the results from alternative numerical methods found in the literature, which verifies the validity of the proposed method.

The analysis of a cantilever beam at any level is a mature subject, and there are a lot of elegant numerical techniques to perform large deflection analysis of non-prismatic cantilever beams. This work aims to develop a straightforward and computationally efficient method to solve large deflection problems of non-prismatic cantilever beams subjected to transverse loading. The modeled beams are assumed to be naturally straight, slender, inextensible, and follow a linear elastic material behavior. The governing differential equation obtained from Euler--Bernoulli beam theory, where the exact curvature expression is preserved, has been discretized using the first-order forward finite difference method. When the deflection is very high, the discretized equation results in a system of highly non-linear equations, and an iterative solution method must be used to solve the problem. This paper employs the bisection method to satisfy the fixed-end boundary condition, which consists of enforcing a slope angle of zero. The method is illustrated through a numerical example of a cantilever beam with a rectangular cross-section subjected to both concentrated and distributed loads. The results provided by this method were shown to be in excellent agreement with the results from alternative numerical methods found in the literature, which verifies the validity of the proposed method.

Glass is an amorphous non-crystalline solid and often transparent that has wide applications in architecture, tableware, optics, and optoelectronics. \newline In this article, first, the mathematical relationship between M-polynomial and topological indices such as Revan indices, Gourava indices, and Y-index are obtained. Then M-polynomial is computed for the molecular graph and the line graph of Glass. Finally, using M-polynomial, Revan indices, Gourava indices, and Y-index are calculated.

Glass is an amorphous non-crystalline solid and often transparent that has wide applications in architecture, tableware, optics, and optoelectronics. \newline In this article, first, the mathematical relationship between M-polynomial and topological indices such as Revan indices, Gourava indices, and Y-index are obtained. Then M-polynomial is computed for the molecular graph and the line graph of Glass. Finally, using M-polynomial, Revan indices, Gourava indices, and Y-index are calculated.

In this paper we study feedback stabilization of inhomogeneous semi-linear control system on a Hilbert state space. The paper gives a feedback control that ensure the strong in term of approximate observability like assumptions. Applications to heat equations are provided.

In this paper we study feedback stabilization of inhomogeneous semi-linear control system on a Hilbert state space. The paper gives a feedback control that ensure the strong in term of approximate observability like assumptions. Applications to heat equations are provided.

In this paper, the problems of observer-based control design and asymptotically stable for uncertain Atangana-Baleanu-Caputo fractional-order one-sided Lipschitz nonlinear systems have been considered. The purpose of this problem is that we design observer-based controller to ensure that the controlled system is asymtotically stable by using singular value decomposition method (SVD) and a well-behaved property of the Atangana-Baleanu-Caputo fractional calculus. The results are obtained in terms of LMI, which can be effectively solved in polynomial time by various computational tools. Finally, a numerical example has been presented to show the simplicity of our design method.

In this paper, the problems of observer-based control design and asymptotically stable for uncertain Atangana-Baleanu-Caputo fractional-order one-sided Lipschitz nonlinear systems have been considered. The purpose of this problem is that we design observer-based controller to ensure that the controlled system is asymtotically stable by using singular value decomposition method (SVD) and a well-behaved property of the Atangana-Baleanu-Caputo fractional calculus. The results are obtained in terms of LMI, which can be effectively solved in polynomial time by various computational tools. Finally, a numerical example has been presented to show the simplicity of our design method.