Showing papers on "Integro-differential equation published in 2017"
TL;DR: In this article, the authors studied the phase shift of the B-type Kadomtsev-Petviashvili-Boussinesq (B-KPV) equation.
Abstract: We study two (3
$$+$$
1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type Kadomtsev–Petviashvili–Boussinesq equation. We use the simplified Hirota’s method to conduct this study and to find the general phase shift of these equations. We obtain one- and two-soliton solutions, for each equation, with the coefficients of the three spatial variables are left as free parameters. However, we also develop special conditions on the coefficients of the spatial variables guarantee the existence of three-soliton solutions for each of these two equation.
161 citations
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TL;DR: Using the $\psi-$Hilfer fractional derivative, this work presents a study of the Hyer-Ulam-Rassias stability and the Hyers- Ulam stability of the fractional Volterra integral-differential equation by means of fixed-point method.
Abstract: Using the $\psi-$Hilfer fractional derivative, we present a study of the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the fractional Volterra integral-differential equation by means of fixed-point method.
100 citations
TL;DR: In this paper, the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition is studied.
Abstract: In this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition. Our results are based on the Banach contraction mapping principle and the Krasnoselskii fixed point theorem. Some examples are also given to illustrate our results.
90 citations
TL;DR: In this paper, the variable coefficient Jacobian elliptic function was extended to solve non-linear differential-difference sine-Gordon equation by introducing a negative power and some variable coefficients in the ansatz.
Abstract: In modern textile engineering, non-linear differential-difference equations are often used to describe some phenomena arising in heat/electron conduction and flow in carbon nanotubes. In this paper, we extend the variable coefficient Jacobian elliptic function method to solve non-linear differential-difference sine-Gordon equation by introducing a negative power and some variable coefficients in the ansatz, and derive two series of Jacobian elliptic function solutions. When the modulus of Jacobian elliptic function approaches to 1, some solutions can degenerate into some known solutions in the literature.
70 citations
TL;DR: In this article, the authors apply operational matrices method based on orthonormal Bernstein polynomials (OBPs) to solve fractional stochastic integro-differential equations.
65 citations
TL;DR: The proposed high-order numerical scheme to solve the space–time tempered fractional diffusion-wave equation is developed and the unconditional stability and convergence of the developed method are proved.
60 citations
TL;DR: The shifted Legendre spectral collocation method is proposed to solve stochastic fractional integro-differential equations (SFIDEs) and it reduces SFIDEs into a system of algebraic equations.
55 citations
22 Apr 2017
TL;DR: In this paper, the solvability of Darboux problems for nonlinear fractional partial integro-differential equations with uncertainty under Caputo gH-fractional differentiability is studied in the infinity domain.
Abstract: In this paper, the solvability of Darboux problems for nonlinear fractional partial integro-differential equations with uncertainty under Caputo gH-fractional differentiability is studied in the infinity domain J
∞
= [0,∞) × [0,∞). New concepts of Hyers-Ulam stability and Hyers-Ulam-Rassias stability for these problems are also investigated through the equivalent integral forms. A computational example is presented to demonstrate our main results.
49 citations
TL;DR: A general framework to find the mild solutions for impulsive fractional integro-differential equations is established, which will provide an effective way to deal with such problems.
45 citations
37 citations
TL;DR: In this paper, the authors examined the unique solvability for an inverse boundary value problem to recover the coefficient and boundary regime of a nonlinear integro-differential equation with degenerate kernel.
Abstract: This article examines questions of unique solvability for an inverse boundary value problem to recover the coefficient and boundary regime of a nonlinear integro-differential equation with degenerate kernel We propose a novel method of degenerate kernel for the case of inverse boundary value problem for the considered ordinary integro-differential equation of second order By the aid of denotation, the integro-differential equation is reduced to a system of algebraic equations Solving this system and using additional conditions, we obtained a system of two nonlinear equations with respect to the first two unknown quantities and a formula for determining the third unknown quantity We proved the single-value solvability of this system using the method of successive approximations
TL;DR: A finite-difference lattice Boltzmann (LB) model for nonlinear isotropic and anisotropic convection-diffusion equations is proposed and has a second-order convergence rate in space, and generally it is also more accurate than the standard LB model.
TL;DR: In this article, the uniqueness and existence of positive solutions for the fractional integro-differential equation with the integral boundary value problem with the Krasnoselskii fixed point theorem were investigated.
Abstract: In this paper, we study the uniqueness and existence of positive solutions for the fractional integro-differential equation with the integral boundary value problem. By means of the Banach contraction principle and the Krasnoselskii fixed point theorem, the sufficient conditions on the uniqueness and existence of positive solutions are investigated. An example is given to illustrate the main results.
TL;DR: An efficient matrix method based on shifted Legendre polynomials for the solution of non-linear volterra singular partial integro-differential equations(PIDEs) is proposed and analyzed.
TL;DR: The designed schemes are unconditionally stable and have the global truncation error O ( τ 2 + h 2 ) , being theoretically proved and numerically verified.
TL;DR: An existence result of mild solutions for the control systems under the mixed Lipschitz and Caratheodory conditions is proved and a new set of sufficient conditions for the approximate controllability of the systems are discussed.
TL;DR: It turns out that the -fractional diffusionwave equation inherits some properties of both the conventional diffusion equation and of the wave equation, and in the one- and two-dimensional cases, the fundamental solution can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of theventional diffusion equation.
Abstract: In this paper, a multi-dimensional -fractional diffusionwave equation is introduced and the properties of its fundamental solution are studied. This equation can be deduced from the basic continuous time random walk equations and contains the Caputo time-fractional derivative of the order /2 and the Riesz space-fractional derivative of the order so that the ratio of the derivatives orders is equal to one half as in the case of the conventional diffusion equation. It turns out that the -fractional diffusionwave equation inherits some properties of both the conventional diffusion equation and of the wave equation. In particular, in the one- and two-dimensional cases, the fundamental solution to the -fractional diffusionwave equation can be interpreted as a probability density function and the entropy production rate of the stochastic process governed by this equation is exactly the same as the case of the conventional diffusion equation. On the other hand, in the three-dimensional case this equation describes a kind of anomalous wave propagation with a time-dependent propagation phase velocity.
TL;DR: In this article, a mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel was considered, and the Fourier method of variable separation was employed for this equation.
Abstract: A mixed problem for a certain nonlinear third-order intregro-differential equation of the pseudoparabolic type with a degenerate kernel is considered. The method of degenerate kernel is essentially used and developed and the Fourier method of variable separation is employed for this equation. A system of countable systems of algebraic equations is first obtained; after it is solved, a countable system of nonlinear integral equations is derived. The method of sequential approximations is used to prove the theorem on the unique solvability of the mixed problem.
TL;DR: Sinc collocation method is considered to obtain the numerical solution of pantograph Volterra delay-integro-differential equation (VDIDE) and this numerical method reduces the VDIDE to an explicit system of algebraic equation.
Abstract: In this article, Sinc collocation method is considered to obtain the numerical solution of pantograph Volterra delay-integro-differential equation VDIDE. This numerical method reduces the VDIDE to an explicit system of algebraic equation. Convergence analysis is given and shows that Sinc solution produces an error of order , where k>0 is a constant. Moreover, Sinc method is applied to the test examples to illustrate accuracy and implementation of the method.
TL;DR: In this paper, the static and dynamic Green's functions for one-, two-and three-dimensional infinite domains within the formalism of peridynamics were derived, making use of Fourier transforms and Laplace transforms.
Abstract: We derive the static and dynamic Green’s functions for one-, two- and three-dimensional infinite domains within the formalism of peridynamics, making use of Fourier transforms and Laplace transforms. Noting that the one-dimensional and three-dimensional cases have been previously studied by other researchers, in this paper, we develop a method to obtain convergent solutions from the divergent integrals, so that the Green’s functions can be uniformly expressed as conventional solutions plus Dirac functions, and convergent nonlocal integrals. Thus, the Green’s functions for the two-dimensional domain are newly obtained, and those for the one and three dimensions are expressed in forms different from the previous expressions in the literature. We also prove that the peridynamic Green’s functions always degenerate into the corresponding classical counterparts of linear elasticity as the nonlocal length tends to zero. The static solutions for a single point load and the dynamic solutions for a time-dependent point load are analyzed. It is analytically shown that for static loading, the nonlocal effect is limited to the neighborhood of the loading point, and the displacement field far away from the loading point approaches the classical solution. For dynamic loading, due to peridynamic nonlinear dispersion relations, the propagation of waves given by the peridynamic solutions is dispersive. The Green’s functions may be used to solve other more complicated problems, and applied to systems that have long-range interactions between material points.
TL;DR: In this paper, the integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function were investigated.
Abstract: In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example.
TL;DR: The hp-version error bounds of the collocation method under the $$H^1$$H1-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes are derived.
Abstract: In this paper, we present an hp-version Legendre–Jacobi spectral collocation method for the nonlinear Volterra integro-differential equations with weakly singular kernels. We derive hp-version error bounds of the collocation method under the $$H^1$$
-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. Numerical experiments demonstrate the effectiveness of the proposed method.
TL;DR: In this article, the existence and uniqueness of solutions for a nonlinear fractional integro-differential equation of pantograph type were investigated and an algorithm based on Sinc basis functions was used to approximate the solution numerically.
Abstract: In this paper, at first we investigate the existence and uniqueness of solutions for a nonlinear fractional integro-differential equation of pantograph type. Then using an algorithm based on Sinc basis functions, we approximate the solution numerically. The single and double exponential transformations are used, respectively. Some test problems have been solved to demonstrate the applicability and efficiency of the proposed methods.
TL;DR: In this article, the authors present a method for numerical approximation of fixed point operator, particularly for the mixed Volterra-Fredholm integro-differential equations, using rationalized Haar wavelets for approximate of integral.
Abstract: In this work, we present a method for numerical approximation of fixed point operator, particularly for the mixed Volterra–Fredholm integro-differential equations. The main tool for error analysis is the Banach fixed point theorem. The advantage of this method is that it does not use numerical integration, we use the properties of rationalized Haar wavelets for approximate of integral. The cost of our algorithm increases accuracy and reduces the calculation, considerably. Some examples are provided toillustrate its high accuracy and numerical results are compared with other methods in the other papers.
TL;DR: In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system.
Abstract: In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system. It is shown that many classical equations of motion, e.g., the Bagley–Torvik equation, can be derived from the developed equation. The Laplace transform is utilized to solve the generalized equation and the analytic solution under some special cases is derived. Example demonstrates the generalized transfer function of an arbitrary viscoelastic system.
TL;DR: In this paper, the Cauchy problem for the nonstationary radiative transfer equation with generalized matching conditions that describes the diffuse reflection and refraction on the interface was studied and the solvability of the initial-boundary value problem was proved.
Abstract: Under study is the Cauchy problemfor the nonstationary radiative transfer equation with generalized matching conditions that describes the diffuse reflection and refraction on the interface. The solvability of the initial-boundary value problem is proved. Some stabilization conditions for the nonstationary solution are obtained.
TL;DR: In this article, the generalized Kudryashov method and extended trial equation method were used to extract analytical solutions to the coupled sine-Gordon equation, which plays an important role in describing the propagation of nonlinear waves.
TL;DR: In this article, the existence of nonnegative solutions for a fractional integro-differential equation subject to multi-point boundary conditions, by using the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators, was investigated.
Abstract: We investigate the existence of nonnegative solutions for a fractional integro-differential equation subject to multi-point boundary conditions, by using the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem for the sum of two operators.
TL;DR: In this paper, the authors derived an exact integro-differential equation for the movement of the sharp interface between fresh and salt groundwater in horizontal, confined aquifers of infinite extend.
Abstract: We analyze the motion of a sharp interface between fresh and salt groundwater in horizontal, confined aquifers of infinite extend. The analysis is based on earlier results of De Josselin de Jong (Proc Euromech 143:75–82, 1981). Parameterizing the height of the interface along the horizontal base of the aquifer and assuming the validity of the Dupuit–Forchheimer approximation in both the fresh and saltwater, he derived an approximate interface motion equation. This equation is a nonlinear doubly degenerate diffusion equation in terms of the height of the interface. In that paper, he also developed a stream function-based formulation for the dynamics of a two-fluid interface. By replacing the two fluids by one hypothetical fluid, with a distribution of vortices along the interface, the exact discharge field throughout the flow domain can be determined. Starting point for our analysis is the stream function formulation. We derive an exact integro-differential equation for the movement of the interface. We show that the pointwise differential terms are identical to the approximate Dupuit–Forchheimer interface motion equation as derived by De Josselin de Jong. We analyze (mathematical) properties of the additional integral term in the exact interface motion formulation to validate the approximate Dupuit–Forchheimer interface motion equation. We also consider the case of flat interfaces, and we study the behavior of the toe of the interface. In particular, we give a criterion for finite or infinite speed of propagation.