Solution Of Cauchy Problem Research Articles

Global Existence of Solution for Cauchy Problem of Two-Dimensional Boussinesq-Type Equation

In this paper, we consider the Cauchy problem of two-dimensional Boussinesq-type equations utt-Δu-Δutt+Δ2u=Δfu. Under the assumptions that fu is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence of global weak solution.

Numerical solution of parabolic Cauchy problems in planar corner domains

An iterative method for the parabolic Cauchy problem in planar domains having a finite number of corners is implemented based on boundary integral equations. At each iteration, mixed well-posed problems are solved for the same parabolic operator. The presence of corner points renders singularities of the solutions to these mixed problems, and this is handled with the use of weight functions together with, in the numerical implementation, mesh grading near the corners. The mixed problems are reformulated in terms of boundary integrals obtained via discretization of the time-derivative to obtain an elliptic system of partial differential equations. To numerically solve these integral equations a Nyström method with super-algebraic convergence order is employed. Numerical results are presented showing the feasibility of the proposed approach.

Strong analytic solutions of fractional Cauchy problems

Fractional derivatives can be used to model time delays in a diffusion process. When the order of the fractional derivative is distributed over the unit interval, it is useful for modeling a mixture of delay sources. In some special cases distributed order derivative can be used to model ultra-slow diffusion. We extend the results of Baeumer and Meerschaert (3) in the single order fractional derivative case to distributed order fractional derivative case. In particular, we develop the strong analytic solutions of distributed order fractional Cauchy problems.

Continuous and integrable solutions of a nonlinear Cauchy problem of fractional order with nonlocal conditions

Abstract In this article, we discuss the existence of at least one solution as well as uniqueness for a nonlinear fractional differential equation with weighted initial data and nonlocal conditions. The existence of at least one L 1 and continuous solution will be proved under the Caratheodory conditions via a classical fixed point theorem of Schauder. An example is also given to illustrate the efficiency of the main theorems.

Time asymptotic description of an abstract Cauchy problem solution and application to transport equation

In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in L 1-space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.

On a New Formulas for a Direct and Inverse Cauchy Problems of Heat Equation

In this paper a solution of the direct Cauchy problems for heat equation is founded in the form of Hermite polynomial series. A well-known classical solution of direct Cauchy problem is represented as Poisson's integral. The author reveals, the formulas obtained by him for solution of the inverse Cauchy problems have a symmetry with respect to the formulas for corresponding direct Cauchy problems. Obtained formulas for solution of the inverse problems can serve as a basis for reg-ularizing computational algorithms while well-known classical formula for the solution of the inverse Cauchy problem can't be a basis for regu-larizing computational algorithms.

Weak solutions of the isothermal bipolar hydrodynamic model for semiconductors with large data

This paper is devoted to weak solutions of Cauchy problem to the isothermal bipolar hydrodynamic model with large data. The model takes the bipolar Euler-Poisson form, with electric field and relaxation terms added to the momentum equations. Using Glimm scheme to the hyperbolic part and the standard theory to the ordinary differential equations, we first construct the approximation solutions, then from the facts that the total charge is quasi-conservation, we can obtain a uniform estimate of the total variation of the electric field, which allows to prove the L∞ estimate of densities and velocities, and the convergence of the scheme. Then we can prove the global existence of weal solution to Cauchy problem with large data.

An integral equations method combined minimum norm solution for 3D elastostatics Cauchy problem

In this paper, we establish new density results for the equilibrium equations. Based on the denseness result of the elastic potential functions, the Cauchy problem for the equilibrium equations is investigated. For this ill-posed problem, we construct a regularizing solution using the single-layer potential function. The well-posedness of the regularizing solution as well as the convergence property is rigorously analyzed. The advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented. The method combines minimum norm solution with Morozov discrepancy principle to solve an inverse problem. Convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method. The numerical convergence, accuracy, and stability of the method with respect to the discretisation about the integral equations on pseudo-boundary and the distance between the pseudo-boundary and the real boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples.

Existence of Solutions of a Partial Integrodifferential Equation with Thermostat and Time Delay

This paper deals with the mathematical analysis of a retarded partial integrodifferential equation that belongs to the class of thermostatted kinetic equations with time delay. Specifically, the paper is devoted to the proof of the existence and uniqueness of mild solutions of the related Cauchy problem. The main result is obtained by employing integration along the characteristic curves and successive approximations sequence arguments. Applications and perspective are also discussed within the paper.

$L^1$ Contraction for Bounded (Nonintegrable) Solutions of Degenerate Parabolic Equations

We obtain new $L^1$ contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or nonlocal diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local $L^1$ contraction results for scalar conservation laws. A key ingredient in the proofs is a new and nontrivial construction of a subsolution of a fully nonlinear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and, in the nonlocal case, new existence and uniqueness results.

Global 1 Estimation of the Cauchy Problem Solutions to the Navier-Stokes Equation

The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided. The paper also describes the time blowup of classical solutions for the Navier-Stokes equations by the smoothness assumption.

Global Estimation of the Cauchy Problem Solutions’ the Navier-Stokes Equation

The analytic properties of the scattering amplitude are discussed, and a representation of the potential is obtained using the scattering amplitude. A uniform time estimation of the Cauchy problem solution for the Navier-Stokes equations is provided.

To solve the inverse Cauchy problem in linear elasticity by a novel Lie-group integrator

In this paper, we propose a simple, iteration free and easy-to-implement numerical algorithm for the solution of inverse Cauchy problem in linear or nonlinear elasticity. The bottom of a finite rectangular plate is imposed by overspecified boundary data, and we seek unknown data on the top side. A spring-damping transform method (SDTM) is introduced to the Navier equations, such that after a discretization by the differential quadrature method, we can apply a novel Lie-group integrator, namely the mixed group-preserving scheme (MGPS), to solve them as an initial value problem. Several numerical examples including nonlinear ones are examined to show that the MGPS can overcome the ill-posed behaviour of the inverse Cauchy problem in elasticity, which has good efficiency and stability against the noisy disturbance, even with an intensity large up to and .

On solutions of a discretized heat equation in discrete Clifford analysis

The main purpose of this paper was to study solutions of the heat equation in the setting of discrete Clifford analysis. More precisely we consider this equation with discrete space and continuous time. Thereby we focus on representations of solutions by means of dual Taylor series expansions. Furthermore we develop a discrete convolution theory, apply it to the inhomogeneous heat equation and construct solutions for the related Cauchy problem by means of heat polynomials.

On the Reynolds number expansion for the Navier–Stokes equations

In a previous paper of ours (Morosi and Pizzocchero (2012) [1]) we have considered the incompressible Navier–Stokes (NS) equations on a d-dimensional torus Td, in the functional setting of the Sobolev spaces HΣ0n(Td) of divergence free, zero mean vector fields (n>d/2+1). In the cited work we have presented a general setting for the a posteriori analysis of approximate solutions of the NS Cauchy problem; given any approximate solution ua, this allows to infer a lower bound Tc on the time of existence of the exact solution u and to construct a function Rn such that ‖u(t)−ua(t)‖n⩽Rn(t) for all t∈[0,Tc). In certain cases it is Tc=+∞, so global existence is granted for u. In the present paper the framework of Morosi and Pizzocchero (2012) [1] is applied using as an approximate solution an expansion uN(t)=∑j=0NRjuj(t), where R is the Reynolds number. This allows, amongst else, to derive the global existence of u when R is below some critical value R∗ (increasing with N in the examples that we analyze). After a general discussion about the Reynolds expansion and its a posteriori analysis, we consider the expansions of orders N=1,2,5 in dimension d=3, with the initial datum of Behr, Nečas and Wu (2001) [11]. Computations of order N=5 yield a quantitative improvement of the results previously obtained for this initial datum in Morosi and Pizzocchero (2012) [1], where a Galerkin approximate solution was employed in place of the Reynolds expansion.

Cauchy problem for fractional evolution equations with Caputo derivative

This paper concerns the abstract nonlocal Cauchy problem of a class of fractional evolution equations with Caputo derivative. A suitable mild solution of evolution equations with Caputo derivative is introduced. In the cases C 0 semigroup is compact or noncompact, the existence theorems of mild solutions for the nonlocal Cauchy problem are established by means of fractional calculus, theory of Hausdorff measure of noncompactness and fixed point theorems.

Global existence and decay of solutions for the generalized bad Boussinesq equation

In this paper, we consider the global existence of solutions for the Cauchy problem of the generalized sixth order bad Boussinesq equation. Moreover, we show that the supremum norm of the solution decays algebraically to zero as (1 + t)−(1/7) when t approaches to infinity, provided the initial data are sufficiently small and regular.

Sharp Extensions for Convoluted Solutions of Abstract Cauchy Problems

In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of the algebraic structure (for the usual convolution product *) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of k-distribution semigroups to extend previous concepts of distribution semigroups.

Extremal solutions of Cauchy problems for abstract fractional differential equations

Abstract In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L 1-Lipschitz-like, L 1-Carathéodory-like and L 1-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.

Computing the Solution of Cauchy Problems of a Class of Nonlinear Heat Conduction Equation on Turing Machine

The computability of the solution for the Cauchy Problems of the nonlinear Heat Conduction equation is studied in this paper. A nonlinear map is defined from the initial value to the solution . We used the relevant knowledge of type-2 theory of effectivity, functional analysis and Sobolev space to prove that when , the solution operator of the initial problem for the Heat Conduction equation is computable under certain conditions. The conclusion enriches the theories of computability.