Block Boundary Value Methods Research Articles

Solving Nonlinear Delay-Differential-Algebraic Equations with Singular Perturbation via Block Boundary Value Methods

Solving Nonlinear Delay-Differential-Algebraic Equations with Singular Perturbation via Block Boundary Value Methods

Numerical approximation to nonlinear delay-differential–algebraic equations with proportional delay using block boundary value methods

This paper focuses on the numerical solutions of nonlinear delay-differential–algebraic equations with proportional delay, which are transformed into nonlinear delay- differential–algebraic equation with constant delay through exponential transformation. Block boundary value methods are extended to solve this type of equation, and their unique solvability, convergence, and stability are rigorously proved, respectively. The computational effectiveness of the methods and correctness of the theoretical result are illustrated through numerical examples.

Block boundary value methods for solving linear neutral Volterra integro-differential equations with weakly singular kernels

A class of block boundary value methods is constructed for the solution of linear neutral Volterra integro-differential equations with weakly singular kernels. Under suitable conditions on the data, it is shown that these methods yield optimal convergence rates when implemented on special graded meshes. Furthermore, these methods are easily extended to solve linear Volterra integral equations of the 2nd kind with weakly singular kernels. Numerical experiments confirm the theoretical results and the accuracy of the methods, and a comparison with piecewise polynomial collocation methods is provided.

Block boundary value methods for linear weakly singular Volterra integro-differential equations

A class of block boundary value methods (BBVMs) is constructed for linear weakly singular Volterra integro-differential equations (VIDEs). The convergence and stability of these methods is analysed. It is shown that optimal convergence rates can be obtained by using special graded meshes. Numerical examples are given to illustrate the sharpness of our theoretical results and the computational effectiveness of the methods. Moreover, a numerical comparison with piecewise polynomial collocation methods for VIDEs is given, which shows that the BBVMs are comparable in numerical precision.

Convergence and global stability analysis of fractional delay block boundary value methods for fractional differential equations with delay

In this paper, a new numerical scheme termed as fractional delay block boundary value methods (FDBBVMs) is proposed. It is an extended version of the block boundary value methods (BBVMs). The proposed scheme is used to find numerical solutions of the fractional delay differential equations including Caputo fractional derivative of βth order with 0<β<1 and a constant delay term. The estimation of the fractional-order derivative term is obtained by combining the mth-order Lagrange interpolating polynomial along with the pth-order BBVMs, and the constant delay term is dealt with certain modifications in the BBVM. Further, the convergence analysis of the proposed scheme is discussed and it is observed that the FDBBVM is convergent with order min{p,m−β+1}. Moreover, the scheme is shown to be globally stable and its computational efficiency and accuracy has been illustrated with the help of numerical examples.

Convergence of block boundary value methods for solving delay differential algebraic equations with index-1 and index-2

In this paper, we propose block boundary value methods to solve the initial value problem of delay differential algebraic equations. For the problems with index-1 and index-2, we give the error estimate of the proposed methods respectively. It is shown that, under certain conditions, the convergence order of the proposed methods is consistent with the underlying one in the case of ordinary differential equations. Finally, some numerical experiments are carried out to demonstrate the effectiveness of the theoretical results.

Convergence and stability of extended BBVMs for nonlinear delay-differential-algebraic equations with piecewise continuous arguments

Delay-differential-algebraic equations have been widely used to model some important phenomena in science and engineering. Since, in general, such equations do not admit a closed-form solution, it is necessary to solve them numerically by introducing suitable integrators. The present paper extends the class of block boundary value methods (BBVMs) to approximate the solutions of nonlinear delay-differential equations with algebraic constraint and piecewise continuous arguments. Under the classical Lipschitz conditions, convergence and stability criteria of the extended BBVMs are derived. Moreover, a couple of numerical examples are provided to illustrate computational effectiveness and accuracy of the methods.

Compact block boundary value methods for semi‐linear delay‐reaction–diffusion equations with algebraic constraints

AbstractIn the present paper, we study a class of linear approximation methods for solving semi‐linear delay‐reaction–diffusion equations with algebraic constraint (SDEACs). By combining a fourth‐order compact difference scheme with block boundary value methods (BBVMs), a class of compact block boundary value methods (CBBVMs) for SDEACs are suggested. It is proved under some suitable conditions that the CBBVMs are convergent of order 4 in space and order p in time, where p is the local order of the used BBVMs, and are globally stable. With several numerical experiments for Fisher equation with delay and algebraic constraint, the computational effectiveness and theoretical results of CBBVMs are further illustrated.

Numerical approximation to a class of nonlinear hybrid system with distributed delay via block boundary value methods

This paper deals with the numerical approximation to a class of nonlinear hybrid system with distributed delay (HSDD). By combining block boundary value methods (BBVMs) and reducible quadrature rules based on the underlying boundary value methods (BVMs), a kind of extended BBVMs are proposed for solving nonlinear HSDD. Under some suitable conditions, an extended BBVM is proved to be uniquely solvable, globally stable and convergent of order p, where p is consistency order of the used underlying BVM. The presented numerical experiments further illustrate the obtained theoretical results and computational effectiveness of the extended BBVMs.

Extended block boundary value methods for neutral equations with piecewise constant argument

This paper is concerned with numerical analysis and computation for neutral equations with piecewise constant argument (NEPCA). The block boundary value methods (BBVMs) are extended to solve NEPCA. Under the conditions of preconsistency and p-order consistency, it is proved that the convergence order of the extended BBVMs can arrive at p. Moreover, a sufficient and necessary condition of the numerical asymptotical stability is derived for linear NEPCA. The presented numerical examples further testify the computational effectiveness of the methods and the obtained theoretical results.

Solving nonlinear functional–differential and functional equations with constant delay via block boundary value methods

This paper deals with the numerical solutions of nonlinear functional-differential and functional equations (FDFEs) with constant delay. The block boundary value methods (BBVMs) are extended to solve the FDFEs. Under the suitable conditions, it is shown that the extended BBVMs are uniquely solvable and globally stable. Moreover, the method can be convergent of order p whenever the Lipschitz condition holds and this method is preconsistent and p-order consistent. With several numerical examples, the theoretical results and computational validity of the extended BBVMs are further confirmed.

Convergence and stability of block boundary value methods applied to nonlinear fractional differential equations with Caputo derivatives

In this paper, by combining the p-order block boundary value methods with the m-th Lagrange interpolation, a class of new numerical methods for solving nonlinear fractional differential equations with the γ-order (0<γ<1) Caputo derivatives are obtained. It is proved under some appropriate conditions that the induced methods are convergent of order min⁡{p,m−γ+1} and globally stable. Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods.

The adapted block boundary value methods for singular initial value problems

This paper deals with the numerical methods for solving singular initial value problems. By adapting the block boundary value methods (BBVMs) for regular initial value problems, a class of adapted BBVMs are constructed for singular initial value problems. It is proved under some suitable conditions that the adapted BBVMs are uniquely solvable, stable and convergent of order p, where p is the consistence order of the methods. Several numerical examples are performed to verify the stability, efficiency and accuracy of the adapted methods. Moreover, a comparison between the adapted BBVMs and the IEM-based iterated defect correction methods is given. The numerical results show that the adapted BBVMs are comparable.

Block boundary value methods applied to functional differential equations with piecewise continuous arguments

This paper deals with a class of functional differential equations with piecewise continuous arguments. Block boundary value methods (BBVMs) are extended to solve this class of equations. It is shown under the Lipschitz condition that the order of convergence of an extended block boundary value method coincides with its order of consistency. Moreover, we study the linear stability of the extended methods and give the corresponding asymptotical stability criterion. In the end, with several numerical examples, the theoretical results and the computational effectiveness of the methods are further illustrated.

Galerkin-Chebyshev spectral method and block boundary value methods for two-dimensional semilinear parabolic equations

In this paper, we present a high-order accurate method for two-dimensional semilinear parabolic equations. The method is based on a Galerkin-Chebyshev spectral method for discretizing spatial derivatives and a block boundary value methods of fourth-order for temporal discretization. Our formulation has high-order accurate in both space and time. Optimal a priori error bound is derived in the weighted L?2$L^{2}_{\omega }$-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence properties of the method.

Stability analysis of block boundary value methods for neutral pantograph equation

This paper deals with the convergence and stability properties of block boundary value methods (BBVMs) for the neutral pantograph equation. Due to its unbounded time lags and limited computer memory, a change in the independent variable is used to transform a pantograph equation into a non-autonomous differential equation with a constant delay but variable coefficients. It is shown under the classical Lipschitz condition that a BBVM is convergent of order p if the underlying boundary value method is consistent with order p. Furthermore, it is proved under a certain condition that BBVMs can preserve the asymptotic stability of exact solutions for the neutral pantograph equation. Meanwhile, some numerical experiments are given to confirm the main conclusions.

Stability analysis of block boundary value methods for the neutral differential equation with many delays

This paper mainly deals with the asymptotic stability properties of block boundary value methods (B2VMs) for the neutral differential equation with many delays. For the time lagging arguments, a technique of Lagrange interpolation is considered. Under some certain conditions, it is proved that B2VMs can preserve the asymptotic stability of exact solutions for the neutral differential equation with many delays if and only if B2VMs are A-stable for ordinary differential equation. Moreover, some numerical experiments are given to confirm the main conclusion.

Stability analysis of extended block boundary value methods for linear neutral delay integro-differential equations

This paper is concerned with the stability of extended block boundary value methods (B2VMs) for the linear neutral delay integro-differential-algebraic equations (NDIDAEs) and the linear neutral delay integro-differential equations (NDIDEs). It is proved that every A-stable B2VM can preserve the asymptotic stability of the exact solution of NDIDAEs under some certain conditions. A necessary and sufficient condition of the B2VMs to be asymptotically stable for NDIDEs is also obtained. A few numerical experiments confirm the expected results.

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.

Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations

In this paper, we construct a class of extended block boundary value methods (B 2VMs) for Volterra delay integro-differential equations and analyze the convergence and stability of the methods. It is proven under the classical Lipschitz condition that an extended B 2VM is convergent of order p if the underlying boundary value methods (BVM) has consistent order p. The analysis shows that a B 2VM extended by an A-stable BVM can preserve the delay-independent stability of the underlying linear systems. Moreover, under some suitable conditions, the extended B 2VMs can also keep the delay-dependent stability of the underlying linear systems. In the end, we test the computational effectiveness by applying the introduced methods to the Volterra delay dynamical model of two interacting species, where the theoretical precision of the methods is further verified.