1,721 research outputs found
A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations
In this paper we introduce a numerical method for solving nonlinear Volterra
integro-differential equations. In the first step, we apply implicit trapezium
rule to discretize the integral in given equation. Further, the Daftardar-Gejji
and Jafari technique (DJM) is used to find the unknown term on the right side.
We derive existence-uniqueness theorem for such equations by using Lipschitz
condition. We further present the error, convergence, stability and bifurcation
analysis of the proposed method. We solve various types of equations using this
method and compare the error with other numerical methods. It is observed that
our method is more efficient than other numerical methods
Asymptotic solutions of forced nonlinear second order differential equations and their extensions
Using a modified version of Schauder's fixed point theorem, measures of
non-compactness and classical techniques, we provide new general results on the
asymptotic behavior and the non-oscillation of second order scalar nonlinear
differential equations on a half-axis. In addition, we extend the methods and
present new similar results for integral equations and Volterra-Stieltjes
integral equations, a framework whose benefits include the unification of
second order difference and differential equations. In so doing, we enlarge the
class of nonlinearities and in some cases remove the distinction between
superlinear, sublinear, and linear differential equations that is normally
found in the literature. An update of papers, past and present, in the theory
of Volterra-Stieltjes integral equations is also presented
A 3-dimensional singular kernel problem in viscoelasticity: an existence result
Materials with memory, namely those materials whose mechanical and/or
thermodynamical behaviour depends on time not only via the present time, but
also through its past history, are considered. Specifically, a three
dimensional viscoelastic body is studied. Its mechanical behaviour is described
via an integro-differential equation, whose kernel represents the relaxation
modulus, characteristic of the viscoelastic material under investigation.
According to the classical model, to guarantee the thermodynamical
compatibility of the model itself, such a kernel satisfies regularity
conditions which include the integrability of its time derivative. To adapt the
model to a wider class of materials, this condition is relaxed; that is,
conversely to what is generally assumed, no integrability condition is imposed
on the time derivative of the relaxation modulus. Hence, the case of a
relaxation modulus which is unbounded at the initial time t = 0, is considered,
so that a singular kernel integro-differential equation, is studied. In this
framework, the existence of a weak solution is proved in the case of a three
dimensional singular kernel initial boundary value problem.Comment: 15 page
Singular Kernel Problems in Materials with Memory
In recent years the interest on devising and study new materials is growing since they are widely used in different applications which go from rheology to bio-materials or aerospace applications. In this framework, there is also a growing interest in understanding the behaviour of materials with memory, here considered. The name of the model aims to emphasize that the behaviour of such materials crucially depends on time not only through the present time but also through the past history. Under the analytical point of view, this corresponds to model problems represented by integro-differential equations which exhibit a kernel non local in time. This is the case of rigid thermodynamics with memory as well as of isothermal viscoelasticity; in the two different models the kernel represents, in turn, the heat flux relaxation function and the relaxation modulus. In dealing with classical materials with memory these kernels are regular function of both the present time as well as the past history. Aiming to study new materials integro-differential problems admitting singular kernels are compared. Specifically, on one side the temperature evolution in a rigid heat conductor with memory characterized by a heat flux relaxation function singular at the origin, and, on the other, the displacement evolution within a viscoelastic model characterized by a relaxation modulus which is unbounded at the origin, are considered. One dimensional problems are examined; indeed, even if the results are valid also in three dimensional general cases, here the attention is focussed on pointing out analogies between the two different materials with memory under investigation. Notably, the method adopted has a wider interest since it can be applied in the cases of other evolution problems which are modeled by analogue integro-differential equations. An initial boundary value problem with homogeneneous Neumann boundary conditions is studied.In recent years the interest on devising and study
new materials is growing since they are widely used in different applications
which go from rheology to bio-materials or aerospace applications.
In this framework,
there is also a growing interest in understanding the behaviour of materials with memory, here
considered. The name
of the model aims to emphasize that the behaviour of
such materials crucially depends on time not only
through the present time but also through the past history. Under the
analytical point of view, this corresponds to model problems represented by
integro-differential
equations which exhibit a kernel non local in time. This is the case of rigid
thermodynamics with memory as well as of isothermal viscoelasticity; in the two different
models the kernel represents, in turn, the heat flux relaxation function and
the relaxation modulus. In dealing with
classical materials with memory these kernels are regular function of both the present
time as wel
Parallel algorithm with spectral convergence for nonlinear integro-differential equations
We discuss a numerical algorithm for solving nonlinear integro-differential
equations, and illustrate our findings for the particular case of Volterra type
equations. The algorithm combines a perturbation approach meant to render a
linearized version of the problem and a spectral method where unknown functions
are expanded in terms of Chebyshev polynomials (El-gendi's method). This
approach is shown to be suitable for the calculation of two-point Green
functions required in next to leading order studies of time-dependent quantum
field theory.Comment: 15 pages, 9 figure
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