Open Journal of Mathematical Analysis (OMA)

In this work, we propose a deep learning approach for identifying parameters (initial condition, a coefficient in the diffusion term and source function) in parabolic partial differential equations (PDEs) from scattered final observations in space and noisy a priori knowledge. In Particular, we approximate the unknown solution and parameters by four deep neural networks trained to satisfy the differential operator, boundary conditions, a priori knowledge and observations. The proposed algorithm is mesh-free, which is key since meshes become infeasible in higher dimensions due to the number of grid points explosion. Instead of forming a mesh, the neural networks are trained on batches of randomly sampled time and space points. This work is devoted to the identification of several parameters of PDEs at the same time. The classical methods require a total a priori knowledge which is not feasible. While they cannot solve this inverse problem given such partial data, the deep learning method allows them to resolve it using minimal a priori knowledge.

The work that we have done in this paper is the coupling method between the local fractional derivative and the Natural transform (we can call it the local fractional Natural transform), where we have provided some essential results and properties. We have applied this method to some linear local fractional differential equations on Cantor sets to get nondifferentiable solutions. The results show this transform’s effectiveness when we combine it with this operator.

In quantum-plank calculus, \(q\)-derivatives and \(h\)-derivatives are fundamental factors. Recently, a composite form of both derivatives is introduced and called \(q-h\)-derivative. This paper aims to present a further generalized notion of derivatives will be called \((q,p-h)\)-derivatives. This will produce \(q\)-derivative, \(h\)-derivative, \(q-h\)-derivative and \((p,q)\)-derivative. Theory based on all aforementioned derivatives can be generalized via this new notion. It is expected, this paper will be useful and beneficial for researchers working in diverse fields of sciences and engineering.

Leibniz’s rule for the \(n\)-th derivative of a product is a very well-known and extremely useful formula. This article introduces an analogous explicit formula for the \(n\)-th derivative of a quotient of two functions. Later, we use this formula to derive new partition identities and to develop expressions for some particular \(n\)-th derivatives.

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for an attraction-repulsion chemotaxis model with logistic source term of Eq. (1) in bounded convex domains \(\Omega\subset\mathbb{R}^{n},~ n\geq1\), with smooth boundary. It is shown that if the ratio \(\frac{\mu}{\chi \alpha-\xi \gamma}\) is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by \((u_{1},u_{2},u_{3})=(1,~\frac{\alpha}{\beta},~\frac{\gamma}{\eta})\) is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data \((u_{10},u_{20},u_{30})\) such that \(u_{10}\not\equiv0\), the above problem possesses uniquely determined global classical solution \((u_{1},u_{2},u_{3})\) with \((u_{1},u_{2},u_{3})|_{t=0}=(u_{10},u_{20},u_{30})\) which satisfies \(\left\|u_{1}(\cdot,t)-1\right\|_{L^{\infty}(\Omega)}\rightarrow{0},~~
\left\|u_{2}(\cdot,t)-\frac{\alpha}{\beta}\right\|_{L^{\infty}(\Omega)}\rightarrow{0},\left\|u_{3}(\cdot,t)-\frac{\gamma}{\eta}\right\|_{L^{\infty}(\Omega)}\rightarrow{0}\,,\) \(\mathrm{as}~t\rightarrow{\infty}\).

The purpose of this paper is to emphasize the role of the Bayesian Vector Autoregressive models (VAR) in macroeconomic analysis and forecasting. To help the policy-makers to do better, the Bayesian VAR models are considered more robust and valuable because they put in the model the mathematician’s beliefs or priors and the data. By using BVAR(1), we get the main results: (i)the best out sample point forecasts; (ii) the exchange rate shock contributes more to inflation; (iii) the inflation shock has high effects on exchange rate innovation. These results are due to the dollarization of this small open economy.

This article considers the limit cycles of a class of Kukles polynomial differential systems of the form Eq. (5). We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center \(\dot{x}=y, \dot{y}=-x,\) by using the averaging theory of first and second order.

This research presents the solution of the generalized version of Abel’s integral equation, which was computed considering the first and second kinds. First, Abel’s integral equation and its generalization were described using fractional calculus, and the properties of Orthogonal polynomials were also described. We then developed a technique of solution for the generalized Abel’s integral equation using infinite series of orthogonal polynomials and utilized the numerical method to approximate the generalized Abel’s integral equation of the first and second kind, respectively. The Riemann-Liouville fractional operator was used in these examples. Our technique was implemented in MAPLE 17 through some illustrative examples. Absolute errors were estimated. In addition, the occurred errors between using orthogonal polynomials for solving Abel’s integral equations of order \(0\ <\ \alpha \ <\ 1\) and the exact solutions show that the orthogonal polynomials used were highly effective, reliable and can be used independently in situations where the exact solution is unknown which the numerical experiments confirmed.

Stochastic differential equations (SDEs) are a powerful tool for modeling certain random trajectories of diffusion phenomena in the physical, ecological, economic, and management sciences. However, except in some cases, it is generally impossible to find an explicit solution to these equations. In this case, the numerical approach is the only favorable possibility to find an approximative solution. In this paper, we present the mean and mean-square stability of the Non-standard Euler-Maruyama numerical scheme using the Vasicek and geometric Brownian motion models.

Using opoola differential operator, we defined a subclass \(S^{n}_{p}(\lambda,\alpha,\gamma,\delta)\) of the class of multivalent or p-valent functions. Several properties of the class were studied, such as coefficient inequalities, hadamard product, radii of close-to-convex, star-likeness, convexity, extreme points, the integral mean inequalities for the fractional derivatives, and further growth and distortion theorem are given using fractional calculus techniques.