Single-term Walsh Series Research Articles

Single Term Walsh Series Method for the System of Nonlinear Delay Volterra Integro-Differential Equations Describing Biological Species Living Together

In this article, the single term Walsh series (STWS) method is presented to solve the system of nonlinear delay Volterra integro-differential equations, which arises in biology. The STWS method reduces the model problem into recursive system of nonlinear algebraic equations. The solutions of the algebraic equations lead to the solution of the problem. Illustrative examples related to the biological species living together are experimented with the STWS method. The numerical results are compared with an existing method. That shows the effectiveness and applicability of the method. The STWS method is working well for the modeled problem. It will give more accuracy when the choice of m increases.

Numerical Solutions of Non-Linear System of Higher Order Volterra Integro-Differential Equations using Generalized STWS Technique

In this article, we deal with non-linear system of higher order Volterra integro-differential equations and their numerical solutions using the Single Term Walsh Series (STWS) method. The connections between STWS coefficients of the unknown functions and its derivatives are derived. The non-linear system of Volterra integro-differential equations are converted into a system of non-linear algebraic equations using the Single Term Walsh Series coefficients. Solving these system of algebraic equations, we obtain the discrete numerical solutions of the non-linear Volterra integro-differential equations. Numerical examples are presented to show the efficiency and applicability of this STWS method for solving the non-linear system of higher order Volterra integro-differential equations.

English

English

STWS approach for Hammerstein system of nonlinear Volterra integral equations of the second kind

ABSTRACTIn this paper, single-term Walsh series (STWS) method is applied to obtain the numerical solutions of Hammerstein systems of nonlinear Volterra integral equations of second kind (HSNVIES). Using the properties of the STWS method, HSNVIES can be easily converted into solvable recursive system of algebraic equations. Solutions obtained from the recursive system of algebraic equations are the solutions of the HSNVIES. Illustrative examples are provided with numerical solutions and the efficiency of this STWS method is also compared with the existing methods.

Single-Term Walsh Series Approach for the System of Linear and Non-linear Volterra Integral Equations of First Kind

In this article, we apply the Single Term Walsh Series (STWS) method to solve the systems of Volterra integral equations of first kind (SVIEF). Because of the ill-posedness of the first kind problem, the SVIEF is converted into system of Volterra integral equations of second kind (SVIES). The STWS method for the SVIES with variable coefficient is presented. Numerical examples for linear and non-linear problems are experimented to show the efficiency and simplicity of the STWS method. The obtained numerical solutions (results) have been compared with the existing method.

Numerical Solutions of Delay Volterra Integral Equations Using Single-term Walsh Series Approach

In this article, we study the numerical solutions of the delay Volterra integral equations and the system of delay Volterra integral equations using Single Term Walsh Series (STWS) method. The delay Volterra integral equation and its system is converted into a recursive linear algebraic equations and its systems. Solving this recursive algebraic equations and its systems, we find the discrete solutions to the delay Volterra integral equations and its systems respectively. Numerical examples are presented to show the computational efficiency of this STWS method for delay Volterra integral equations and its systems.

English

English

System of linear second order Volterra integro-differential equations using Single Term Walsh Series technique

In this article, we deal with a system of linear second order Volterra integro-differential equations of the second kind and their numerical solutions using Single Term Walsh Series (STWS) technique. The given system of linear second order Volterra integro-differential equations of the second kind is converted into a linear system of algebraic equations. Solving this linear system of algebraic equations, we find the discrete solutions to the system of linear second order Volterra integro-differential equations of the second kind. Numerical examples are presented and the numerical results are compared with the one of the spectral methods, Chebyshev polynomial method to show the efficiency of the STWS method.

Method of Solving Linear System of Volterra Integro-Differential Equations Using the Single Term Walsh Series

In this article, we study the system of linear Volterra integro-differential equations and their numerical solutions using the Single Term Walsh Series (STWS) approach. STWS approach converts the linear system of Volterra integro-differential equations into a system of linear algebraic equations. From this system of algebraic equations, we find the discrete solutions to the linear system of integro-differential equations. The obtained numerical solutions are compared with the exact solutions and also with the results of the Adomian Decomposition Method. We also exhibit the maximum absolute errors, for each example, with different choice of m in the STWS method.

Single-term Walsh series method for solving Volterra's population model

Single-term Walsh series are developed to approximate the solution of the Volterra’s population model. Volterra’s model is a nonlinear integro-difierential equation where the integral term represents the efiect of toxin. Properties of Single-term Walsh series are presented and are utilized to reduce the computation of the Volterra’s population model to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples. A comparison is made with existing results.

Single-Term Walsh Series method for systems of linear Volterra integral equations of the second kind

This paper generalizes Single-Term Walsh Series method for solving systems of linear Volterra integral equations of the second kind. Using the properties of Walsh functions, this method reduces these types of equations to a linear system of algebraic equations. In addition, this method (i) does not require any operational matrices, (ii) needs less computational effort, (iii) gives both piecewise constant and discrete solution, (iv) yields good agreement with exact solutions. Convergence results and error analysis are discussed. Numerical examples are presented to illustrate the computational efficiency of the method.

A new recursive method for the analysis of linear time invariant dynamic systems via double-term triangular functions (DTTF) in state space environment

This paper presents a new recursive method for system analysis via double-term triangular functions (DTTF) in state space environment. The proposed method uses orthogonal triangular function sets and proves to be more accurate as compared to single term Walsh series (STWS) method with respect to mean integral square error (MISE). This has been established theoretically and comparison of error with respect to MISE is presented for clarity. A numerical example is treated to establish the proposed method. Relevant curves for the solutions of states of the dynamic system are also presented with plots of percentage error for DTTF-based analysis.

RK–Butcher algorithms for singular system-based electronic circuit

In this paper, a new method of analysis for a singular system-based electronic circuits using the Runge–Kutta (RK)–Butcher algorithm is presented for linear time-invariant and time-varying cases. The discrete solutions obtained using the RK–Butcher algorithms are compared with the exact solutions of the electronic circuit problem and are found to be very accurate. Stability regions for the Single term walsh series method and the RK–Butcher algorithms are presented. Error graphs for inductor currents and capacitor voltages are presented in a graphical form to show the efficiency of this method. This RK–Butcher algorithm can be easily implemented in a digital computer and the solution can be obtained for any length of time.

State Analysis of Time-Varying Singular Bilinear Systems by RK-Butcher Algorithms

The Runge-Kutta (RK)-Butcher algorithm is used to study the timevarying singular bilinear systems with the exact solutions. The results (discrete solutions) obtained using the Haar wavelets, Single-Term Walsh series (STWS) and RK-Butcher algorithms are compared with the exact solutions of the time-varying singular bilinear systems. It is found that the solution obtained using the RK-Butcher algorithm is closer to the exact solutions of the time-varying singular bilinear systems. The RK-Butcher algorithm can easily be implemented using a digital computer and the solution can be obtained for any length of time, which is an added advantage of this algorithm.

Generalized STWS technique for higher order time-varying singular systems

A study of different higher-order time-varying singular systems of initial value problems (IVPs) using the generalized single-term Walsh series (STWS technique) is presented. The existing STWS technique is used to solve IVPs of first- and second-order systems and is extended to determine discrete solutions for the systems of IVPs of any higher order n with p variables. The effectiveness of the technique is demonstrated by using it to find discrete solutions for time-varying singular systems of different orders.

Non-linear singular systems using RK–Butcher algorithms

The Runge–Kutta (RK)–Butcher algorithm is used to study time-invariant and time-varying non-linear singular systems. The results (discrete solutions) obtained using the RK method based on the arithmetic mean (RKAM), single-term Walsh series (STWS) and RK–Butcher algorithms are compared with the exact solutions of the non-linear singular systems for the time-invariant and time-varying cases. It is found that the solution obtained using the RK–Butcher algorithm is closer to the exact solutions of the non-linear singular systems. Stability regions for the RKAM, STWS and RK–Butcher algorithms are presented. Error graphs for discrete and exact solutions are presented in a graphical form to highlight the efficiency of this method. The RK–Butcher algorithm can easily be implemented using a digital computer and the solution can be obtained for any length of time for both time-invariant and time-varying cases for these non-linear singular systems, which is an added advantage of this algorithm.

Optimal control of time-varying singular systems using the RK–Butcher algorithm

The problem of optimal control of time-varying linear singular systems with quadratic performance index has been studied using the Runge–Kutta–Butcher algorithm. The results obtained using the Runge–Kutta (RK) method based on the arithmetic mean (RKAM) and the RK–Butcher algorithms are compared with the exact solutions of the time-varying optimal control of linear singular systems. It is observed that the result obtained using the RK–Butcher algorithm is closer to the true solution of the problem. Stability regions for the RKAM algorithm, the single-term Walsh series method and the RK–Butcher algorithms are presented. Error graphs for the simulated results and exact solutions are presented in graphical form to highlight the efficiency of the RK–Butcher algorithm. This algorithm can easily be implemented using a digital computer. An additional advantage of this method is that the solution can be obtained for any length of time for this type of optimal control of time-varying linear singular systems.

Error Free Butcher Algorithms for Linear Electrical Circuits

In this paper, an error-free Butcher algorithm is introduced to study the singular system of a linear electrical circuit for time invariant and time varying cases. The discrete solutions obtained using Runge-Kutta (RK)-Butcher algorithms are compared with the exact solutions of the electrical circuit problem and are found to be very accurate. Stability regions for the single term Walsh series (STWS) method and the RK-Butcher algorithm are presented. Error graphs for inductor currents and capacitor voltages are presented in a graphical form to show the efficiency of the RK-Butcher algorithm. This RK-Butcher algorithm can be easily implemented in a digital computer for any singular system of electrical circuits.

Single-term Walsh series method for the Volterra integro-differential equations

A method for the solution of Volterra integro-differential equations by using single-term Walsh series is presented. Properties of single-term Walsh series are utilized to reduce the computation of Volterra integro-differential equations to some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.

Single-Term Walsh Series Direct Method for the Solution of Nonlinear Problems in the Calculus of Variations

A direct method for solving variational problems using single-term Walsh series is presented. Two nonlinear examples are considered. In the first example the classical brachistochrone problem is examined. and in the second example a higher-order nonlinear problem is considered. The properties of single-term Walsh series are given and are utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement and yields accurate results.