Singular Partial Integro-differential Equations Research Articles

Bivariate Chebyshev polynomials of the fifth kind for variable-order time-fractional partial integro-differential equations with weakly singular kernel

The shifted Chebyshev polynomials of the fifth kind (SCPFK) and the collocation method are employed to achieve approximate solutions of a category of the functional equations, namely variable-order time-fractional weakly singular partial integro-differential equations (VTFWSPIDEs). A pseudo-operational matrix (POM) approach is developed for the numerical solution of the problem under study. The suggested method changes solving the VTFWSPIDE into the solution of a system of linear algebraic equations. Error bounds of the approximate solutions are obtained, and the application of the proposed scheme is examined on five problems. The results confirm the applicability and high accuracy of the method for the numerical solution of fractional singular partial integro-differential equations.

On Exact Solutions of a Class of Singular Partial Integro-Differential Equations

On Exact Solutions of a Class of Singular Partial Integro-Differential Equations

An operational matrix based scheme for numerical solutions of nonlinear weakly singular partial integro-differential equations

In this article, we introduce an operational matrix scheme based on two-dimensional wavelets for the Volterra weakly singular nonlinear partial integro-differential equations. By implementing two-dimensional wavelets approximations and its operational matrices of integration and differentiation along with collocation points, the weakly singular partial integro-differential equations are reduced into the system of nonlinear algebraic equations. Moreover, Bernoulli wavelet approximation and Legendre wavelet approximation have been used for inspecting the errors and convergence analysis of the given problems. Some numerical examples are included to establish the accuracy of the proposed scheme via Bernoulli wavelet approximation and Legendre wavelet approximation respectively. Additionally, comparisons of error values between the two wavelets have been presented.

Two-dimensional wavelets operational method for solving Volterra weakly singular partial integro-differential equations

In this article, we discuss a method for finding an approximate solution of a class of two-dimensional linear Volterra weakly partial integro-differential equations. The operational matrices of integration, differentiation, and product are utilized to reduce the solution of the weakly partial integro-differential equation to the linear algebraic system of equations. Furthermore, some useful theorems are discussed to establish the convergence analysis of the proposed technique. Some numerical examples are solved by applying the presented scheme to show the effectiveness and applicability of the proposed scheme and also a comparison of error values between the two wavelets has been presented. Furthermore, some figures are plotted to demonstrate the error analysis of the proposed scheme.

Two Dimensional Wavelets Collocation Scheme for Linear and Nonlinear Volterra Weakly Singular Partial Integro-Differential Equations

In this article, we have study a 2D Legendre and Chebyshev wavelets collocation scheme for solving a class of linear and nonlinear weakly singular Volterra partial integro-differential equations (PIDEs). The scheme is based on wavelets collocation for PIDEs with uniquely designed matrices over the Hilbert space defined on the domain $$\left( [0, 1]\times [0, 1]\right) $$ . Using piecewise approximation associated with 2D Legendre wavelet, 2D Chebyshev wavelet and its operational matrices, the considered PIDEs will be reduced into the corresponding system of linear and nonlinear algebraic equations. The corresponding linear and nonlinear system of equations solved by collocation scheme and well-known Newton–Raphson scheme at collocation points respectively. In addition, the convergence and error analysis of the numerical scheme is provided under several mild conditions. The numerical results are correlated with the exact solutions and the execution of the proposed scheme is determined by estimating the maximum absolute errors, $$l_{2}\hbox {-}norm$$ errors and $$l_{\infty }\hbox {-}norm$$ errors. The numerical result shows that the scheme is simply applicable, efficient, powerful and very precisely at small number of basis function. The main important applications of the proposed wavelets collocation scheme is that it can be applied on linear as well as nonlinear problems and can be applied on higher order partial differential equations too.

Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices

In this paper, we propose and analyze an efficient matrix method based on shifted Legendre polynomials for the solution of non-linear volterra singular partial integro-differential equations(PIDEs). The operational matrices of integration, differentiation and product are used to reduce the solution of volterra singular PIDEs to the system of non-linear algebraic equations. Some useful results concerning the convergence and error estimates associated to the suggested scheme are presented. illustrative examples are provided to show the effectiveness and accuracy of proposed numerical method.

Operational matrix approach for the solution of partial integro-differential equation

In this paper, an effective numerical method is introduced for the treatment of Volterra singular partial integro-differential equations. They are based on the operational and almost operational matrix of integration and differentiation of 2D shifted Legendre polynomials. The methods convert the singular partial integro-differential equation in to a system of algebraic equations. Convergence analysis and error estimates are derived for the proposed method. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Singular evolution integro-differential equations with kernels defined on bounded intervals

We study linear singular first-order integro-differential Cauchy problems in Banach spaces. The adjective “singular” means here that the integro-differential equation is not in normal form neither can it be reduced to such a form. We generalize some existence and uniqueness theorems proved in [5] for kernels defined on the entire half-line R + to the case of kernels defined on bounded intervals removing the strict assumption that the kernel should be Laplace-transformable. Particular attention is paid to single out the optimal regularity properties of solutions as well as to point out several explicit applications relative to singular partial integro-differential equations of parabolic and hyperbolic type.

Identification problems for singular integro-differential equations of parabolic type II

We recover unknown kernels, depending on time only, in linear singular first-order integro-differential Cauchy problems in Banach spaces. Singular means here that the integro-differential equation is not in normal form nor it can be reduced to such a form. For this class of problems we prove an existence and uniqueness theorem, in the framework of general Banach spaces, under the condition that the “resolvent operator” (cf. (1.5) admits a polar singularity at λ=0 (see Section 3)). Moreover, when the Banach space under consideration is reflexive, we can prove a local in time existence and uniqueness result when the “resolvent operator” decays as (1+| λ|) −1. Finally, we give a few applications to explicit singular partial integro-differential equations of parabolic type.

SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE AND INVERSE PROBLEMS

We prove a global existence and uniqueness result for the recovery of unknown scalar kernels in linear singular first-order integro-differential initial-boundary value problems in Banach spaces. To this end use is made of suitable weighted Lp-spaces. Finally, we give a few applications to explicit singular partial integro-differential equations of parabolic type.