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This article aims to study a model of flame propagation in a nonhomogeneous medium with a p-Laplacian operator. The intention with such operator is to model the effects of slow and fast diffusion, that can appear in a nonhomogeneous media, depending on the pressure driven conditions. In addition, the authors introduce a general form in the reaction term, that introduces the flame chemical kinetics.

To introduce the governing equations, the authors depart from previously reported models in flame propagation, but the authors consider a new modeling approach based on a p-Laplacian operator.

The authors provide evidences of regularity and uniqueness of solutions. Afterward, the authors introduce profiles of stationary solutions based on the definition of a Hamiltonian for the newly discussed model. Eventually, the authors obtain exponential profiles solutions with the help of a scaling, that transforms the model into a nonlinear Hamilton–Jacobi equation.

The new model has not been previously reported in the literature. The authors consider that the mathematical properties of a p-laplacian (in particular the property known as finite propagation) is of inherent interest to model pressure drive flames with slow or fast diffusion. Indeed, the authors’ approach has the value of providing an operator that can fit better to model flame propagation. In addition, the authors introduce a general form of chemical kinetics, to make the authors’ model further general.

The paper deals with the existence of positive solutions for a coupled system of nonlinear fractional differential equations with p-Laplacian operator and involving both right Riemann–Liouville and left Caputo-type fractional derivatives. The existence results are obtained by the help of Guo–Krasnosel'skii fixed-point theorem on a cone in the sublinear case. In addition, an example is included to illustrate the main results.

Fixed-point theorems.

No finding.

The obtained results are original.

The purpose of this paper is to find a doubly nonlinear parabolic equation of fast diffusion in a bounded domain.

For positive and bounded initial data, the authors study the initial zero-boundary value problem.

The findings of this study showed the complete extinction of a continuous weak solution at a finite time.

The extinction time is studied earlier but for the Laplacian case. The authors presented the finite extinction time for the case of p-Laplacian.

In this paper, the authors give a new version of the sub-super solution method and prove the existence of positive solution for a (p, q)-Laplacian system under weak assumptions than usually made in such systems. In particular, nonlinearities need not be monotone or positive.

The authors prove that the sub-super solution method can be proved by the Shcauder fixed-point theorem and use the method to prove the existence of a positive solution in elliptic systems, which appear in some problems of population dynamics.

The results complement and generalize some results already published for similar problems.

The result is completely new and does not appear elsewhere and will be a reference for this line of research.

In this article, the authors discuss the existence and multiplicity of solutions for an anisotropic discrete boundary value problem in T-dimensional Hilbert space. The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.

The approach is based on variational methods especially on the three critical points theorem established by B. Ricceri.

The authors study the existence of results for a discrete problem, with two boundary conditions type. Accurately, the authors have proved the existence of at least three solutions.

An other feature is that problem is with non-local term, which makes some difficulties in the proof of our results.

This paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions. With regard to this nonlinear boundary value problem, three popular fixed point theorems, namely, Krasnoselskii’s theorem, Leray–Schauder’s theorem and Banach contraction principle, are employed to theoretically prove and guarantee three novel theorems. The main outcomes of this work are verified and confirmed via several numerical examples.

In order to accomplish our purpose, three fixed point theorems are applied to the problem under consideration according to some conditions that have been established to this end. These theorems are Krasnoselskii's theorem, Leray Schauder's theorem and Banach contraction principle.

In accordance to the applied fixed point theorems on our main problem, three corresponding theoretical results are stated, proved, and then verified via several numerical examples.

The existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions are studied. To the best of the authors’ knowledge, this work is original and has not been published elsewhere.

This paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.

This paper considered theoretical and numerical methodologies.

This paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.

The novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

The purpose of this paper is the study of existence and multiplicity of solutions for a nonlinear discrete boundary value problems involving the p-laplacian.

The approach is based on variational methods and critical point theory.

Theorem 1.1. Theorem 1.2. Theorem 1.3. Theorem 1.4.

The paper is original and the authors think the results are new.

The objective of this study is to model the propagating front in the interaction of gases in an aircraft fuel tank. To this end, we introduce a nonlinear parabolic operator, for which solutions are shown to be regular.

The authors provide an analytical expression for the propagating front, that shifts any combination of oxygen and nitrogen, in the tank airspace, into a safe condition to avoid potential explosions. The analytical exercise is validated with a real flight.

According to the flight test data, the safe condition, of maximum 7% of oxygen, is given for a time t = 45.2 min since the beginning of the flight, while according to our analysis, such a safe level is obtained for t = 41.42 min. For other safe levels of oxygen, the error between the analytical assessment and the flight data was observed to be below 10%.

The interaction of gases in a fuel tank has been little explored in the literature. Our value consists of introducing a set of nonlinear partial differential equations to increase the accuracy in modeling the interaction of gasses, which has been typically done via algebraic equations.

Multi‐point boundary value problems have important roles in the modelling of various problems in physics and engineering. This paper aims to present the solution of ordinary differential equations with multi‐point boundary value conditions by means of a semi‐numerical approach which is based on the homotopy analysis method.

The convergence of the obtained solution is expressed and some typical examples are employed to illustrate validity, effectiveness and flexibility of this procedure. This approach, in contrast to perturbation techniques, is valid even for systems without any small/large parameters and therefore it can be applied more widely than perturbation techniques, especially when there do not exist any small/large quantities.

Unlike other analytic techniques, this approach provides a convenient way to adjust and control the convergence of approximation series. Some applications will be briefly introduced.

The paper shows how an important boundary value problem is solved with a semi‐analytical method.