Mostrando recursos 1 - 16 de 16

  1. Split-step collocation methods for stochastic Volterra integral equations

    Xiao, Y.; Shi, J.N.; Yang, Z.W.
    In this paper, a split-step collocation method is proposed for solving linear stochastic Volterra integral equations (SVIEs) with smooth kernels. The H\"older condition and the conditional expectations of the exact solutions are investigated. The solvability and mean-square boundedness of numerical solutions are proved and the strong convergence orders of collocation solutions and iterated collocation solutions are also shown. In addition, numerical experiments are provided to verify the conclusions.

  2. Blow up of fractional reaction-diffusion systems with and without convection terms

    Pérez, Aroldo
    Based on the study of blow up of a particular system of ordinary differential equations, we give a sufficient condition for blow up of positive mild solutions to the Cauchy problem of a fractional reaction-diffusion system, and, by a comparison between the transition densities of the semigroups generated by $\Delta _\alpha :=-(-\Delta )^{\alpha /2}$ and $\Delta _\alpha +b(x)\cdot \nabla $ for $1\lt \alpha \lt 2$, $d\geq 1$ and $b$ in the Kato class on $\mathbb {R}^d$, we prove that this condition is also sufficient for the blow up of a fractional diffusion-convection-reaction system.

  3. Solution estimates for a system of nonlinear integral equations arising in optometry

    Okrasiński, Wojciech; Płociniczak, Łukasz
    In this paper, we investigate a system of nonlinear integral equations that has previously been proposed in modelling of the human cornea. The main result of our work is a construction of lower and upper estimates that bound the components of the exact solution to the system being considered. These results generalize some of the recent work by other authors. We conclude the paper with a numerical verification of our analytical estimates.

  4. Coupled Volterra integral equations with blowing up solutions

    Mydlarczyk, Wojciech
    In this paper, a system of nonlinear integral equations related to combustion problems is considered. Necessary and sufficient conditions for the existence and explosion of positive solutions are given. In addition, the uniqueness of the positive solutions is shown. The main results are obtained by monotonicity methods.

  5. General and optimal decay in a memory-type Timoshenko system

    Messaoudi, Salim A.; Hassan, Jamilu Hashim
    This paper is concerned with the following memory-type Timoshenko system \[ \rho _1\varphi _{tt}-K(\varphi _x+\psi )_x=0 \] \[ \rho _2\psi _{tt}-b\psi _{xx}+K(\varphi _x+\psi )+ \displaystyle \int _0^tg(t-s)\psi _{xx}(s)\,ds=0, \] $(x,t)\in (0,L)\times (0,\infty )$, with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some constant $1\leq p\lt {3}/{2}$, \[ g'(t)\leq -\xi (t)g^p(t),\quad \mbox {for all }t\geq 0. \] We prove some decay results which generalize and improve many earlier results in the literature. In particular, our result gives the optimal decay for the case of polynomial stability.

  6. General decay for a laminated beam with structural damping and memory: The case of non-equal wave speeds

    Li, Gang; Kong, Xiangyu; Liu, Wenjun
    In previous work, Lo and Tatar studied the exponential decay for a laminated beam with viscoelastic damping acting on the effective rotation angle in the case of equal-speed wave propagations. In this paper, we continue consideration of the same problem in the case of non-equal wave speeds. In this case, the main difficulty is how to estimate the non-equal speed term. To overcome this difficulty, the second-order energy method introduced in Guesmia and Messaoudi seems to be the best choice for our problem. For a wide class of relaxation functions, we establish the general decay result for the energy without...

  7. A blow-up result to a delayed Cauchy viscoelastic problem

    Kafini, Mohammad; Mustafa, Muhammad I.
    In this paper, we consider a Cauchy problem for a nonlinear viscoelastic equation with delay. Under suitable conditions on the initial data and the relaxation function, in the whole space, we prove a finite-time blow-up result.

  8. Blow-up of solutions for semilinear fractional Schrödinger equations

    Fino, A.Z.; Dannawi, I.; Kirane, M.
    We consider the Cauchy problem in $\mathbb {R}^N$, $N \geq 1$, for the semi-linear Schr\"odinger equation with fractional Laplacian. We present the local well-posedness of solutions in $H^{{\alpha }/{2}}(\mathbb {R}^N)$, $0\lt \alpha \lt 2$. We prove a finite-time blow-up result, under suitable conditions on the initial data.

  9. Existence of a solution for the problem with a concentrated source in a subdiffusive medium

    Chan, C.Y.; Liu, H.T.
    By using Green's function, the problem is converted into an integral equation. It is shown that there exists a $t_b$ such that, for $0\leq t\lt t_b$, the integral equation has a unique nonnegative continuous solution $u$; if $t_b$ is finite, then $u$ is unbounded in $[0, t_b)$. Then, $u$ is proved to be the solution of the original problem.

  10. On a semi-linear system of nonlocal time and space reaction diffusion equations with exponential nonlinearities

    Ahmad, B.; Alsaedi, A.; Hnaien, D.; Kirane, M.
    In this article, we investigate the local existence of a unique mild solution to a reaction diffusion system with time-nonlocal nonlinearities of exponential growth. Moreover, blowing-up solutions are shown to exist, and their time blow-up profile is presented.

  11. On the contributions of W. Edward Olmstead

    Kirk, C.M.; Ritter, L.R.
    With this article, we wish to honor the many contributions of our mentor, colleague and dear friend, Professor W. Edward Olmstead, on the occasion of his retirement from Northwestern University. Ed has spent over five decades at Northwestern University, first as a graduate student and then as a member of the faculty. During this time he completed his PhD, played a key role in the formation of the Department of Engineering Science and Applied Mathematics (ESAM), developed several courses in applied mathematics, participated in the education of numerous students, and made vast and important contributions in the field of applied...

  12. Introduction to the Special Issue honoring W.E. Olmstead

    Kirk, Colleen M.; Roberts, Catherine A.

  13. Generation of nonlocal fractional dynamical systems by fractional differential equations

    Cong, N.D.; Tuan, H.T.
    We show that any two trajectories of solutions of a one-dimensional fractional differential equation (FDE) either coincide or do not intersect each other. However, in the higher-dimensional case, two different trajectories can meet. Furthermore, one-dimensional FDEs and triangular systems of FDEs generate nonlocal fractional dynamical systems, whereas a higher-dimensional FDE does not, in general, generate a nonlocal dynamical system.

  14. Memory dependent growth in sublinear Volterra differential equations

    Appleby, John A.D.; Patterson, Denis D.
    We investigate memory dependent asymptotic growth in scalar Volterra equations with sublinear nonlinearity. In order to obtain precise results we extensively utilize the powerful theory of regular variation. By computing the growth rate in terms of a related ordinary differential equation we show that, when the memory effect is so strong that the kernel tends to infinity, the growth rate of solutions depends explicitly upon the memory of the system. Finally, we employ a fixed point argument for determining analogous results for a perturbed Volterra equation and show that, for a sufficiently large perturbation, the solution tracks the perturbation asymptotically,...

  15. Regularized integral formulation of mixed Dirichlet-Neumann problems

    Akhmetgaliyev, Eldar; Bruno, Oscar P.
    This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions, for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on the use of Green functions and integral equations, and it relies on the Fourier continuation method...

  16. Weak solutions for partial Pettis Hadamard fractional integral equations with random effects

    Abbas, Saïd; Albarakati, Wafaa; Benchohra, Mouffak; Zhou, Yong
    In this article, we apply M\"onch and Engl's fixed point theorems associated with the technique of measure of weak noncompactness to investigate the existence of random solutions for a class of partial random integral equations via Hadamard's fractional integral, under the Pettis integrability assumption.

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