Generalized Cauchy Problem Research Articles

Method of Implicit Functions in the Solution of Multiparameter Nonlinear Spectral Problems

We propose a new numerical method for the solution of multiparameter nonlinear spectral problems of dimension m for the holomorphic operator functions defined in Banach spaces. We introduce the notion of generalized Cauchy problem, which is reduced to the solution of a system of m − 1 partial differential equations of the first order with common initial condition. We also present examples of solving of two- and three-parameter spectral problems.

On a generalized fractional differential Cauchy problem

Qualitative results for abstract problems are very important in understanding mathematical analysis on which any application is possible. The focus of this paper is twofold: first, we investigate the existence and uniqueness of mild solutions to a generalized Cauchy problem for the nonlinear differential equation with non-local conditions in a Banach space $X$. This is achievable using some fixed point theorems in infinite dimensional spaces. Secondly, we study the stability results of the system in the sense of Ulam-Hyers-Rassias. Our results improve and generalize most recent related results in the literature.

Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \mathrm{i}\partial_t q(x, t)+\partial_{x}^2q(x, t)+2\sigma q^{2}(x, t)\overline{q(-x, t)} = 0 $ with initial data $ q(x, 0)\in H^{1, 1}(\mathbb{R}) $. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $ H^{1, 1} $ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

A Weierstrass type representation for translating solitons and singular minimal surfaces

In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in R3. As application we study when the euclidean Gauss map has a harmonic argument and solve a general Cauchy problem in this class of surfaces.

On the Generalized Cauchy Problem for the Evolutionary Equation with Operator of Fractional Differentiation

On the Generalized Cauchy Problem for the Evolutionary Equation with Operator of Fractional Differentiation

On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem

In this paper we deal with construction and analysis of a multiwavelet spectral element scheme for a generalized Cauchy type problem with Caputo fractional derivative. Numerical schemes for this type of problems, often suffer from the draw-back of spurious oscillations. A common remedy is to render the problem to an equivalent integral equation. For the generalized Cauchy type problem, a corresponding integral equation is of nonlinear Volterra type. In this paper we investigate wellposedness and convergence of a stabilizing multiwavelet scheme for a, one-dimensional case (in [a, b] or [0, 1]), of this problem. Based on multiwavelets, we construct an approximation procedure for the fractional integral operator that yields a linear system of equations with sparse coefficient matrix. In this setting, choosing an appropriate threshold, the number of non-zero coefficients in the system is substantially reduced. A severe obstacle in the convergence analysis is the lack of continuous derivatives in the vicinity of the inflow/ starting boundary point. We overcome this issue through separating a J (mesh)-dependent, small, neighborhood of a (or origin) from the interval, where we only take L_2-norm. The estimate in this part relies on Chebyshev polynomials, viz. As reported by Richardson( Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms, Oxford University, UK, 2012) and decreases, almost, exponentially by raising J. At the remaining part of the domain the solution is sufficiently regular to derive the desired optimal error bound. We construct such a modified scheme and analyze its wellposedness, efficiency and accuracy. The robustness of the proposed scheme is confirmed implementing numerical examples.

Adopting Feynman–Kac Formula in Stochastic Differential Equations with (Sub-)Fractional Brownian Motion

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.

Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

A class of time-fractional Dirac type operators

By using a Witt basis, a new class of time-fractional Dirac type operators with time-variable coefficients is introduced. These operators lead to considering a wide range of fractional Cauchy problems. Solutions of the considered general fractional Cauchy problems are given explicitly. The representations of the solutions can be used efficiently for analytic and computational purposes. We apply the obtained representation of a solution to recover a variable coefficient solution of an inverse fractional Cauchy problem. Some concrete examples are given to show the diversity of the obtained results.

On the Generalized Cauchy Problem for One Class of Differential Equations of Infinite Order

We establish the solvability of a nonlocal multipoint (in time) problem regarded as a generalization of the Cauchy problem for the evolution equation with pseudodifferential operator (differentiation operator of infinite order) with initial conditions from the space of generalized functions of the ultradistribution type.

Multidimensional inverse Cauchy problems for evolution equations

We discuss inverse problems to finding the time-dependent coefficient for the multidimensional Cauchy problems for both strictly hyperbolic equations and polyharmonic heat equations. We also extend our techniques to the general inverse Cauchy problems for evolution equations.

On a Problem of Multidimensional Tauberian Theory

In many Tauberian theorems, the asymptotic properties of functions were investigated with respect to a predefined function (usually in the scale of regularly varying functions). In this paper, we address an alternative problem: Given a generalized function, does it have asymptotics with respect to some regularly varying function? We find necessary and sufficient conditions for the existence of quasiasymptotics of those generalized functions whose Laplace transforms have a bounded argument in a tube domain over the positive orthant. Moreover, we point out a regularly varying function with respect to which quasiasymptotics exists. It turns out that the modulus of a holomorphic function in a tube domain over the positive orthant in the purely imaginary subspace on rays emanating from the origin behaves as a regularly varying function. We use the obtained results to find the quasiasymptotics of the generalized Cauchy problem for convolution equations whose kernels are passive operators.

New numerical approximation for Chua attractor with fractional and fractal-fractional operators

In this study, we present new numerical scheme for modified Chua attractor model with fractional operators. However we give numerical solution of the considered model with fractal-fractional operators. Also, we offer error analysis for a general Cauchy problem with fractional and fractal-fractional operators. For numerical solution of the considered equation, we use new numerical scheme which is established with an efficient polynomial known as Newton interpolation polynomial. The results are discussed with some examples and simulations.

Modified numerical methods for fractional differential equations

A new modified numerical scheme is proposed. This idea is inspired through the general idea of the Lagrange and the Adams–Bashforth. We present the derivation of the general steps for a modified numerical scheme for an equation with two different fractional derivatives. We present an application of such method in a general nonlinear Cauchy problem. The application is presented to some fractional differential equations. Additionally we realized that when the non-integer order is one, the well-known multi-step-Adams–Bashforth numerical scheme for classical differential equations is recovered.

Validity of fractal derivative to capturing chaotic attractors

Abstract Suggested independently with different definitions, fractal derivative and conformable derivative are α proportional. They have been applied in quit a few problems in many field of sciences in the last few years with great success. However, some researchers have pointed out some criticisms and even concluded that they were flawed. In this paper, while confirm the validity of the conformable and fractal derivatives and we present their applications to chaotic attractors. We considered a general non-linear Cauchy problem where the differential operator is that of fractal and conformable and present the derivation of conditions for which the existence and the uniqueness of the exact solution are reached. Several examples are considered, solved and numerical simulations depicting real world observations.

On the asymptotics of solutions of elliptic equations at the ends of non-compact Riemannian manifolds with metrics of a special form

We consider a linear elliptic differential equation defined on a Riemannian manifold that has an end on which the metric takes the form in appropriate coordinates. Here , , and is a smooth compact Riemannian manifold with metric . At the end , the coefficient takes the form . For ends of parabolic type with such metrics, we describe the property of asymptotic distinguishability of solutions of this equation. For ends of hyperbolic type, we prove a theorem on the admissible rate of convergence to zero for a difference of solutions of this equation. For both types of ends, we formulate versions of the generalized Cauchy problem with initial data at the infinitely remote point and study its solubility. The results obtained are new and, in the case of ends of parabolic type, somewhat unexpected.

Distributed-order fractional constitutive stress–strain relation in wave propagation modeling

Distributed order fractional model of viscoelastic body is used in order to describe wave propagation in infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem, corresponding to fractional wave equation, is studied. The explicit form of fundamental solution is calculated, and wave propagation speed, arising from solution's support, is found to be connected with the material properties at initial time instant. Existence and uniqueness of the fundamental solutions to the fractional wave equations corresponding to four thermodynamically acceptable classes of linear fractional constitutive models, as well as to power type distributed order model, are established and explicit forms of the corresponding fundamental solutions are obtained.

Solution of the Cauchy Problem for the Three-Dimensional Telegraph Equation and Exact Solutions of Maxwell’s Equations in a Homogeneous Isotropic Conductor with a Given Exterior Current Source

For the solution of the Cauchy problem for the linear telegraph equation in three-dimensional space, we derive a formula similar to the Kirchhoff one for the linear wave equation (and turning into the latter at zero conductivity). Additionally, the problem of determining the field of a given exterior current source in an infinite homogeneous isotropic conductor is reduced to a generalized Cauchy problem for the three-dimensional telegraph equation. The derived formula enables us to reduce this problem to quadratures and, in some cases, to obtain exact three-dimensional solutions with a propagating front, which are of great applied importance for testing numerical methods for solving Maxwell’s equations. As an example, we construct the exact solution of the field from a Hertzian dipole with an arbitrary time dependence of the current in an infinite homogeneous isotropic conductor.

Existence results for Isaacs equations with local conditions and related semilinear Cauchy problems

Our goal is to prove existence results for classical solutions to some general nondegenerate Cauchy problems which are natural generalizations of Isaacs equations. For the latter we are able to extend our results by admitting local conditions for coeffici

Estimation of accuracy of an asymptotic solution of the generalized Cauchy problem for the Boussinesq equation as applied to the potential model of tsunami with a “simple” source

Statement of the hydrodynamic problem in the framework of the potential tsunami model with “simple” source whose solution is chosen as the reference one. Generalized Cauchy problem for the Boussinesq equation and its reduction to the classical one. Analytical solution of the Cauchy problem for the Boussinesq equation. An explanation of the incorrectness of the formulation of the problem. Derivation of an approximate equation for the correct setting of the Cauchy problem. The known reference solution of the problem. An analytical solution of the correct problem and the derivation of its asymptotic representation in the “far zone.” Comparison of the graphs of the temporal history of wave height calculated by the formulas of the asymptotic and reference solutions. Estimation of the accuracy of the asymptotic solution by a three-level scale. Discussion. A remark concerning the referees.