Slip MHD viscous flow over a stretching sheet – An exact solution

In this paper, the magnetohydrodynamic (MHD) flow under slip condition over a permeable stretching surface is solved analytically. The solution is given in a closed form equation and is an exact solution of the full governing Navier–Stokes equations. The effects of the slip, the magnetic, and the mass transfer parameters are discussed. Results show that there is only one physical solution for any combination of the slip, the magnetic, and the mass transfer parameters. The velocity and shear stress profiles are greatly influenced by these parameters.     

Existence and uniqueness results for a semilinear Black–Scholes type equation

This paper deals with the application of fixed point theorem to determine the source term of semilinear Black–Scholes type equation and thereby establish the existence and uniqueness of the solution. The proof mainly relies on the iteration method and the Schauder fixed point theorem.     

Corrigendum to “Existence and uniqueness results for a nonlinear differential equation arising in MHD Falkner-Skan flow” [Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 2272–2277]

We correct the hypothesis for which the existence and uniqueness theorems of Van Gorder and Vajravelu [Commun. Nonlinear Sci. Numer. Simulat. 15 (2010) 2272–2277] hold. This correction modifies the range of parameters valid under the given theorems.     

Existence and uniqueness of positive solutions to?m-point boundary value problem for?nonlinear fractional differential equation

In this paper, we consider the following nonlinear fractional m-point boundary value problem where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative. By the properties of the Green function, the lower and upper solution method and fixed-point theorem in partially ordered sets, some new existence and uniqueness of positive solutions to the above boundary value problem are established. As applications, examples are presented to illustrate the main results.     

The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet

In this work, the magnetohydrodynamic (MHD) boundary layer flow is investigated by employing the modified Adomian decomposition method and the Padé approximation. The series solution of the governing non-linear problem is developed. Comparison of the present solution is made with the existing solution and excellent agreement is noted.     

Existence and uniqueness result for a backward stochastic differential equation whose generator is Lipschitz continuous in y and uniformly continuous in z

In this note, we extend the result in Lepetier and San Martin (Stat. Probab. Lett. 32:425–430, 1997) by eliminating the condition that (g(t,0,0)) _t∈[0,T] is a bounded process. Furthermore, we prove that if g is Lipschitz continuous in y and uniformly continuous in z, and (g(t,0,0)) _t∈[0,T] is square integrable, then for each square integrable terminal condition ξ, there exists a unique square integrable adapted solution to the one-dimensional backward stochastic differential equation (BSDE) with the generator g, which generalizes the corresponding (one-dimensional) results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008) and Jia (Stat. Probab. Lett. 79:436–441, 2009).     

Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment

The existence and uniqueness of forced waves in a general reaction–diffusion equation with time delay under climate change is concerned in this paper. By using upper and lower solutions method, monotone iteration scheme combined with the strong maximum principle, we show that there exists a nondecreasing and unique wave front with the speed consistent with the habitat shifting speed. Our results indicate the propagation of both the leading and trailing edges of the comoving population wavefront lag behind the climate envelope, which drives the species to extinction. Three examples and their corresponding numerical simulations are also given to illustrate the universality of analytical conclusions.     

Existence and nonexistence of blow-up solutions for a Schrödinger equation involving a nonlinear operator

By introducing a growth condition and using an iterative technique, we establish the results for the nonexistence and existence of positive entire blow-up solutions for a Schrödinger equation involving a nonlinear operator. Our main results improve and extend some existing works. In addition, we also give an example to illustrate our results.     

Existence and Hyers–Ulam stability of solutions for a delayed hyperbolic partial differential equation

Periodica Mathematica Hungarica - In this paper, we first prove the existence and uniqueness of the solutions for a delayed hyperbolic partial differential equation by applying the progressive...     

Existence of a ground state and scattering for a nonlinear Schrödinger equation with critical growth

We study the energy-critical focusing nonlinear Schrödinger equation with an energy-subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig and Merle (Invent Math 166:645–675, 2006).     

Existence of positive solutions with peaks on a Clifford torus for a fractional nonlinear Schrödinger equation

Science China Mathematics - We consider the fractional nonlinear Schrödinger equation in this paper. Applying the finite reduction method, we prove that the equation has positive solutions...     

Existence and uniqueness for a nonlinear parabolic/Hamilton–Jacobi coupled system describing the dynamics of dislocation densities

We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a 1D system of a parabolic equation and a first order Hamilton–Jacobi equation that are coupled together. We examine an associated Dirichlet boundary value problem. We prove the existence and uniqueness of a viscosity solution among those assuming a lower-bound on their gradient for all time including the initial time. Moreover, we show the existence of a viscosity solution when we have no such restriction on the initial data. We also state a result of existence and uniqueness of entropy solution for the initial value problem of the system obtained by spatial derivation. The uniqueness of this entropy solution holds in the class of bounded-from-below solutions. In order to prove our results on the bounded domain, we use an “extension and restriction” method, and we exploit a relation between scalar conservation laws and Hamilton–Jacobi equations, mainly to get our gradient estimates.     

Existence and uniqueness for a quasilinear hyperbolic equation with σ-finite Borel measures as initial conditions

The aim of this paper is to discuss the Cauchy problem for a quasilinear hyperbolic equation of the form

     

Unsteady flow and heat transfer in a thin film of Ostwald–de Waele liquid over a stretching surface

In this paper, the effects of viscous dissipation and the temperature-dependent thermal conductivity on an unsteady flow and heat transfer in a thin liquid film of a non-Newtonian Ostwald–de Waele fluid over a horizontal porous stretching surface is studied. Using a similarity transformation, the time-dependent boundary-layer equations are reduced to a set of non-linear ordinary differential equations. The resulting five parameter problem is solved by the Keller–Box method. The effects of the unsteady parameter on the film thickness are explored numerically for different values of the power-law index parameter and the injection parameter. Numerical results for the velocity, the temperature, the skin friction and the wall-temperature gradient are presented through graphs and tables for different values of the pertinent parameter. One of the important findings of the study is that the film thickness increases with an increase in the power-law index parameter (as well as the injection parameter). Quite the opposite is true with the unsteady parameter. Furthermore, the wall-temperature gradient decreases with an increase in the Eckert number or the variable thermal conductivity parameter. Furthermore, the surface temperature of a shear thinning fluid is larger compared to the Newtonian and shear thickening fluids. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena.     

A uniqueness and Existence theorem for a singular third-order boundary value Problem on [0, ∞)

It is proved that the singular third-order boundary value problem y‴ = f(y), y(0) = 0, y(+∞) = 1, y′(+∞) = y″(+∞) = 0, has a unique solution. Here f(y) = (1 − y)^λg(y), λ > 0, g(y) is positive and continuous on (0, 1]. The problem arises in the study of draining and coating flows.