Vol. 12, No. 2 (2023): Regular Issue

In this paper, we establish sufficient conditions on existence and uniqueness of solutions for a class of nonlinear impulsive nabla fractional difference equations of order $\alpha$, $0 < \alpha \leq 1$, associated with non–periodic boundary conditions. The right hand side of the proposed equation may grow linearly, or sublinearly in its second argument. We employ the classical fixed point theorem of Schaefer, and the Nonlinear Alternative to prove the existence and uniqueness of solutions. Further, we study stability of solutions in sense of Ulam--Hyers by the help of generalized Gronwall Inequality. To demonstrate the validity and applicability of the established results, we provide a couple of particular examples.

After motivation we give a complete background on needed $\psi $-Hilfer fractional Calculus. Then we produce $\psi $-Hilfer fractional left and right Taylor formulae. We give also important $\psi $-Hilfer fractional left and right representation integral formulae regarding $\psi $-Hilfer left and right fractional derivatives. Then we give extensive applications of our $% \psi $-Hilfer fractional results to left and right $\psi $-Hilfer fractional Ostrowski, Opial and Poincaré type integral inequalities. We create the space for more future forthcoming results.

In this paper we are investigating to figure out the exponential growth of solutions with $L_{p}$-norm of a viscoelastic Klein-Gordon wave equation with strong damping, source and delay terms.

In our recent work, we study three fractional order nonlinear evolution equations by sine-cosine method, a class of traveling wave solutions with significant physical structures are obtained. The solutions include periodic solutions, soliton solutions, bell shaped solutions with the estimated values of involved parameters. The significance of contemplating and applying such equations originates from a dynamical point of view of the complexities of non linear physical models. Some real time problems of nonlinear physical worlds can be realized such as fractional Zoomeran equation helps to understand time evolution of single scalar field and novelty of boomerons and trappons, numerous wave phenomenas in solid state physics, plasma physics and quantum filed theory can be understood by fractional Hirota-Ramani equation and fractional Zarkhov-Kuznetsov-Benjamin-Bona-Mohany equation helps to understand the propagation of long range gravity waves in fluid mechanics and waves in plasma. The importance of trigonometric and hyperbolic solution in fractional calculus discussed. By the implementation of MATLAB graphical plots are shown.

This manuscript is devoted to proving some results concerning the existence of solutions for a class of initial value problems for nonlinear fractional Hybrid differential equations and Generalized Hilfer fractional derivative. The result is based on a fixed point theorem due to Dhage. Further, some examples are provided for the justification of our main results.

The aim of the paper is to present an analysis of special random impulsive fractional differential equations involving Fredholm and Volterra integrals. This paper is mainly focused to the existence, uniqueness and stability of special random impulsive fractional differential equations with local initial conditions and nonlocal initial conditions separately. Such an approach enabled the generalisation of equations with local initial conditions and also helps in obtaining more practical results. To test the effectiveness of our results, we provide examples.

The dynamics of a delayed multiparameter nonlinear Mathieu equation: $$ \ddot{x}+(\delta+\epsilon\alpha \cos{t})x+\epsilon\gamma x^3=\epsilon\beta\int_{-\infty}^{t}{x(\tau)\xi e^{-\xi(t-\tau)}}d\tau,$$ is investigated in the neighborhood of $\delta=1/4$. Three different features interact here: a distributed delay, cubic nonlinearity and 2:1 parametric resonance. The averaging method is used to obtain a slow flow that is analyzed for stability and bifurcations, and the resulting predictions are compared against actual system responses. In particular, we find regimes where: i. the slow flow has a zero stable fixed point (implying Amplitude Death), or ii. the slow flow goes to a stable non-zero fixed point (implying periodic solutions), or iii. the slow flow goes to a stable periodic solution at large times (corresponding to a quasiperiodic system response). All of these types of behavior would be very difficult to isolate otherwise, except by intensive numerical searching of the multiparameter space. However, there are also parameter regimes where the slow flow predictions may occasionally disagree with the actual system response $x(t)$ in cases where that has large amplitude or exhibits bounded aperiodicity. The reasons for these discrepancies are also carefully considered.

In this article, q-homotopy analysis transformation method (q-HATM) has been applied to solve {fractional partial differential equations}. {The} q-HATM is a well known method, which is the outcome of {the} conjunction of q- Homotopy analysis method and Laplace transform. Which provides the solution of such problems in a very easy manner. In our analysis, we derive the approximate analytical results of the non-linear fractional differential equation. And it shows that this method is more likely to converge for a series solution.

In this paper, we consider a Bresse-Timoshenko type system of second sound with distributed delay term. Under suitable assumptions, we establish the global well-posedness of the initial and boundary value problem by using the Faedo-Galerkin approximations and some energy estimates. By using the energy method, we show exponential stability results for the system with distributed term delay acting in angular rotation, respectively. This extends earlier results in the literature.

We present the dynamics of the predator-prey model of Leslie type by introducing fear and gestation delay in the prey population to get a more realistic model. It is assumed that the predator consumes prey in the form of Beddington DeAngelis functional response. For all positive equilibrium points, the existence and local stability analysis is discussed. The condition for the local stability of coexisting equilibrium point for both delayed and non-delayed model is provided by using the Routh-Hurwitz criterion. The global stability property of the coexisting equilibrium point is analyzed with the help of constructing a suitable Lyapunov function. Also, the model shows bifurcation behavior, particularly Hopf-bifurcation at coexisting equilibrium point for both delayed and non-delayed model are proven analytically. Also, the analytical results are verified numerically in each section.

In this paper, we established some new results concerning the existence and uniqueness of the solutions of nonlinear Volterra-Fredholm integro-differential equations of Caputo fractional order. These new results are obtained by using the Leray-Schauder nonlinear alternative, Krasnoselskii's and Banach fixed point theorems. In addition, we investigate generalized Ulam stability for this fractional system.

In this paper, we consider nonlinear neutral q-fractional difference equations which have important applications in many domains of life sciences. By using the Krasnoselskii fixed point theorem, sufficient conditions for the existence of solutions are established, also the uniqueness of solutions is given. As an application of the main theorems, we provide the existence and uniqueness of the discrete $q$-fractional Lotka-Volterra model of neutral type. Our main theorems are important results because are extend and generalize the results in the literature.

Stationary patterns in a predator-prey model with Holling-III functional response and harvesting policy are investigated in this work. For the proposed model, nonnegativity, uniformly boundedness, permanence, stability, the existence and direction of Hopf bifurcation are analyzed. For the spatial system, the existence conditions for the Turing instability are also established. Then using weakly nonlinear analysis, amplitude equations near critical values of the Turing instability are derived. Different kinds of solutions can be shown analytically by analyzing amplitude equations. Based on numerical analysis, the stationary patterns can be found, such as hexagonal patterns, stripe patterns and mixed states of hexagonal pattern and stripe pattern. Numerical simulations are well consistent with theoretical results. It is further noted that the behavior of harvesting is a factor for existence and stability of equilibria, the occurrence of the transcritical bifurcation, pattern formation and the permanence of the system.

Our current research is motivated by the new interesting generalization (Srivastava et al \cite{1}) of a couple of contour-type Mellin-Barnes integral representations of their incomplete $H$-functions $\gamma^{m,\;n}_{p,\;q}(z)$ and $\Gamma^{m,\;n}_{p,\;q}(z)$, and incomplete $\overline{H}$-functions $\overline{\gamma}^{m,\;n}_{p,\;q}(z)$ and $\overline{\Gamma}^{m,\;n}_{p,\;q}(z)$. By virtue of the gamma functions of incomplete type, that is $\gamma(s,x)$ and $ \Gamma(s,x)$, we introduced here a class of the incomplete $I$-functions $^{\gamma}I^{m,\;n}_{p,\;q}(z)$ and $^{\Gamma}I^{m,\;n}_{p,\;q}(z)$ which leads to a natural extension of a class of $I$-functions. The aim of the present insvestigation is to analyze and examine some impressive properties of these incomplete $I$-functions, inclusive of formulas for decomposition, reduction, derivative and several integral transformations, etc. Further, as the application of newly defined functions, we also formulate and solve a generalized fractional kinetic equation in terms of these incomplete $I$-functions. For the corresponding incomplete $\overline{I}$-functions, we demonstrate the simply determinable extensions of the outcome shown here which also hold. We also raising these effects in certain useful specific forms and established results as well.

In this paper, we have proposed an $SEIHRV$ mathematical model of the pandemic COVID-19 using a system of ordinary differential equations. The mathematical modelling is a vital tool to make the use of imposing a strategy in order to fight against this pandemic. We are obtained a boundedness of the system and steady state of the solutions. The basic reproduction number is computed and used as a threshold to negotiate the asymptotic behavior of the mathematical model. Our analytical and numerical results show a close faith of the basic reproduction number on epidemic parameters. Also, our model delineates the various transmission route in the infection dynamics and an exertion the foreword of the environmental reservoir in the devolution and the dispersion of this disease.