Solution of Certain Admissible Control Systems Characterised by Nonlinear Integral and Integro-Differential Equations within a Bounded Banach Space via the Laplace Adomian Decomposition Algorithm

Abstract

A system of linear/nonlinear integral and integro-differential equations that exists in a bounded Banach space is described in the paper along with the acceptable control functions involved. We obtain a dual space of the p-integrable space and the solution to its dual from the theory. The system\'s kernel produces an admissible control function. Considering these factors and the existence of Laplace transform and its inverse conditions stipulated, we solved some various systems of nonlinear Volterra integro-differential equations using the Laplace-Adomian Decomposition Algorithm. The Laplace-Adomian Decomposition Algorithm is used without the need of restrictive assumptions, discretization, transformation or the guess of initial term. When compared to the current/exact solution, numerical results show that the Laplace-Adomian Decomposition Algorithm is very promising and effective particularly for polynomial, exponential and differential nonlinearity. The method developed by the author with an extension, modification and new approach to initial and boundary value problem named Shooting Type Laplace Adomian Decomposition Algorithm (STLADA). The paper describes a system of linear/nonlinear integral and integro-differential equations that occur in a bounded Banach space, along with the appropriate control functions involved. From the theory, we derive a dual space of the p-integrable space and the solution to its dual. An acceptable control function is generated by the system\'s kernel. We used the Laplace-Adomian Decomposition Algorithm, an analytical approach to solve a number of different systems of nonlinear Volterra integro-differential equations in light of these parameters as well as the presence of the Laplace transform and its inverse conditions. Restrictive assumptions, discretisation, transformation, and initial term guesswork are not required when using the Laplace-Adomian Decomposition Algorithm. Numerical results indicate that the Laplace-Adomian Decomposition Algorithm is highly promising and efficient when compared to the current/exact solution, especially for polynomial, exponential, and differential nonlinearity. The Shooting Type Laplace Adomian Decomposition Algorithm (STLADA) is an extension, modification, and novel technique to the boundary value problems that the author created.

Introduction

Integro-differential equations are critical in modeling real-world phenomena and reformulating mathematical problems. Analytical solutions are valuable for understanding underlying physical behaviors, while numerical methods provide approximate solutions when exact results are difficult. Nonlinear integro-differential equations can be solved using methods like Taylor Series, Adomian Decomposition, Variational Iteration, Homotopy Perturbation, and Laplace-Adomian Decomposition Algorithm (LADA). Recent developments focus on efficient, high-accuracy techniques such as the Wavelet-Galerkin method, Lagrange interpolation, and Tau methods.

This paper proposes using LADA to solve systems of nonlinear Volterra integro-differential equations. LADA is versatile, efficient, and produces results very close to exact solutions without assuming a starting term. The method can handle linear or nonlinear equations of any order and is applicable to both integral and integro-differential systems.

The study also considers control systems described by nonlinear integral and integro-differential equations. Admissible control functions are chosen from a closed ball in the Lp([0,T])L^p([0,T])Lp([0,T]) space, satisfying boundedness, continuity, and Lipschitz conditions. Systems with finite energy constraints are modeled with integral restrictions on controls. Conditions on system functions αiα_iαi? and KiK_iKi?, as well as parameters λ,p,μλ, p, μλ,p,μ, ensure existence and uniqueness of solutions. The paper defines a conjugate operator L∗(λ)L^*(λ)L∗(λ) to analyze system behavior and proves that under specified conditions, continuous admissible control functions exist, guaranteeing well-posed solutions for the system.

Conclusion

This work used the Laplace-Adomian Decomposition Algorithm to explore three systems of nonlinear and one system of linear Volterra integro-differential equations. This approach does not choose the initial term at random; rather, it takes it directly from the procedure. The numerical findings demonstrate how quickly LADA converges when compared to alternative techniques. It just takes a small number of iterations, and the outcome is really near to the precise answer. This approach is simple to apply and may be applied to a wide range of general form problems. Without undergoing any transformations or making any restrictive assumptions, this method is applied directly to the system of equations. A large class of linear and nonlinear initial and boundary value problems have analytic solutions that can be found using LADA, which is a highly powerful tool. Maple is used to perform the computations.

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