Nonlinear Analysis: Modelling and Control
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Optimal harvesting in a unidirectional consumer–resource mutualisms system with size structure in the consumer
Full text linkThis paper considers the optimal harvesting problem for a size-structured model of unidirectional consumer–resource mutualisms in which the consumer species has both positive and negative effects on the resource species, while the resource has only a positive effect on the consumer. First, we show the existence of a unique nonnegative solution of the system and give the continuous dependence of solutions on the control variable. Next, the adjoint system is derived, which is necessary for optimality and the existence of a unique optimal policy. Then necessary conditions for optimality are established via the normal cone and adjoint system. Moreover, the existence of a unique optimal strategy is proved via Ekeland’s variational principle and fixed-point reasoning in convex analysis. Finally, we use numerical simulations to verify the main results and find other dynamic properties of the system
Solitons and other solutions of perturbed nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index
Full text linkWe analytically study the exact solitary wave solutions of the perturbed nonlinear Biswas–Milovic equation with Kudryashov’s law of refractive index, which describes the propagation of pulses of various types in optical fiber. We apply three efficient and reliable schemes, specifically, the simple equation method, the (G\u27/G)-expansion method, and the new Kudryashov method. These approaches lead to a range of solitons and other solutions comprising of the bright solitons, dark solitons, singular solitons, periodic, rational, and exponential solutions. These solutions are also presented graphically. Furthermore, all obtained solutions are verified by symbolic computations
Turing instability and pattern formation of a fractional Hopfield reaction–diffusion neural network with transmission delay
Full text linkIt is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation. Recently, some studies indicate that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we mainly investigate the Turing pattern in a delayed reaction–diffusion neural network with Caputo-type fractional derivative. In particular, we find that this fractional neural network can form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern, which implies that temporal fractional derivative contributes to pattern formation. Numerical simulations show that both fractional derivative and time delay have influence on the shape of Turing patterns
Solvability of a system of integral equations in two variables in the weighted Sobolev space W(1,1)-omega(a,b) using a generalized measure of noncompactness
Full text linkIn this paper, we deal with the existence of solutions for a coupled system of integral equations in the Cartesian product of weighted Sobolev spaces E = Wω1,1 (a,b) x Wω1,1 (a,b). The results were achieved by equipping the space E with the vector-valued norms and using the measure of noncompactness constructed in [F.P. Najafabad, J.J. Nieto, H.A. Kayvanloo, Measure of noncompactness on weighted Sobolev space with an application to some nonlinear convolution type integral equations, J. Fixed Point Theory Appl., 22(3), 75, 2020] to applicate the generalized Darbo’s fixed point theorem [J.R. Graef, J. Henderson, and A. Ouahab, Topological Methods for Differential Equations and Inclusions, CRC Press, Boca Raton, FL, 2018]
Comparative exploration on bifurcation behavior for integer-order and fractional-order delayed BAM neural networks
Full text linkIn the present study, we deal with the stability and the onset of Hopf bifurcation of two type delayed BAM neural networks (integer-order case and fractional-order case). By virtue of the characteristic equation of the integer-order delayed BAM neural networks and regarding time delay as critical parameter, a novel delay-independent condition ensuring the stability and the onset of Hopf bifurcation for the involved integer-order delayed BAM neural networks is built. Taking advantage of Laplace transform, stability theory and Hopf bifurcation knowledge of fractional-order differential equations, a novel delay-independent criterion to maintain the stability and the appearance of Hopf bifurcation for the addressed fractional-order BAM neural networks is established. The investigation indicates the important role of time delay in controlling the stability and Hopf bifurcation of the both type delayed BAM neural networks. By adjusting the value of time delay, we can effectively amplify the stability region and postpone the time of onset of Hopf bifurcation for the fractional-order BAM neural networks. Matlab simulation results are clearly presented to sustain the correctness of analytical results. The derived fruits of this study provide an important theoretical basis in regulating networks
Global dynamics of solutions for a sixth-order parabolic equation describing continuum evolution of film-free surface
Full text linkThis paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces H3(Ω) and fractional-order spaces Hk(Ω), where 0 ≤ k < ∞
Index spaces and standard indices in metric modelling
Full text linkWe analyze the basic structure of certain metric models, which are constituted by an index I acting on a metric space (D; d) representing a relevant property of the elements of D. We call such a structure (D; d; I) an index space and define on it normalization and consistency constants that measure to what extent I is compatible with the metric d. The “best” indices are those with such constants equal to 1 (standard indices), and we show an approximation method for other indices using them. With the help of Lipschitz extensions, we show how to apply these tools: a new model for the triage process in the emergency department of a hospital is presented
Fractional uncertain differential equations with general memory effects: Existences and alpha-path solutions
Full text linkGeneral fractional calculus is popular recently. Fractional uncertain differential equations (FUDEs) with general memory effects are proposed in this paper. Firstly, existence and uniqueness theorems of solution for general fractional uncertain differential equations (GFUDEs) is presented, and the analytic solution of a linear one is given. Then the concept of α-path is introduced, and relationship between solution of GFUDE and corresponding α-path is also discussed. In addition, a theorem is proved to obtain the expected value of a monotonic function related to solutions of GFUDEs. Finally, a numerical example is given to better understand the significance of general memory effects. This paper provides more types of FUDEs to better describe some phenomena in uncertain environments
Exponentials of general multivector in 3D Clifford algebras
Full text linkClosed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics
