In this article, we propose the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm–Volterra integrodifferential equations. The solution ...methodology is based on generating the orthogonal basis from the obtained kernel functions in which the constraint initial condition is satisfied, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their
r
-cut representation form in the Hilbert space
W
2
2
Ω
⊕
W
2
2
Ω
. Several computational experiments are given to show the good performance and potentiality of the proposed procedure. Finally, the utilized results show that the present method and simulated annealing provide a good scheduling methodology to solve such fuzzy equations.
The object of this article is to present the computational solution of the time-fractional Schrödinger equation subject to given constraint condition based on the generalized Taylor series formula in ...the Caputo sense. The algorithm methodology is based on construct a multiple fractional power series solution in the form of a rabidly convergent series with minimum size of calculations using symbolic computation software. The proposed technique is fully compatible with the complexity of this problem and obtained results are highly encouraging. Efficacious computational experiments are provided to guarantee the procedure and to illustrate the theoretical statements of the present algorithm in order to show its potentiality, generality, and superiority for solving such fractional equation. Graphical results and numerical comparisons are presented and discussed quantitatively to illustrate the solution.
The aim of this paper is to propose the Atangana–Baleanu fractional methodology for fathoming the Van der Pol damping model by using the reproducing kernel algorithm. To this end, we discuss the ...mathematical structure of this new approach and some other numerical properties of solutions. Furthermore, all needed requirements for characterizing solutions by applying the reproducing kernel algorithm are debated. In this orientation, modern trend and new computational algorithm in terms of analytic and approximate Atangana–Baleanu fractional solutions are proposed. Finally, numerical simulations in fractional emotion is constructed one next to the other with tabulated data and graphical portrayals.
The aim of this article is to introduce the reproducing kernel algorithm for obtaining the numerical solutions of fractional order systems of Dirichlet function types. The algorithm provide ...appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n-term of exact solutions, numerical solutions of linear and nonlinear time-fractional equations of homogeneous and nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds for the present algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems compared with other numerical methods.
Latterly, many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, to a series of time-fractional partial differential ...equations. Unlike the normal case derivative, the differential order in such equations is with a fractional order, which will lead to new challenges for numerical simulation. The purpose of this analysis is to introduce the reproducing kernel Hilbert space method for treating classes of time-fractional partial differential equations subject to Neumann boundary conditions with parameters derivative arising in fluid-mechanics, chemical reactions, elasticity, anomalous diffusion, and population growth models. The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. Numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n-term of the exact solutions. Finally, the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional Neumann problems.
Purpose
The purpose of this study is to introduce the reproducing kernel algorithm for treating classes of time-fractional partial differential equations subject to Robin boundary conditions with ...parameters derivative arising in fluid flows, fluid dynamics, groundwater hydrology, conservation of energy, heat conduction and electric circuit.
Design/methodology/approach
The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. This representation is given in the W(Ω) and H(Ω) inner product spaces, while the computation of the required grid points relies on the R(y,s) (x, t) and r(y,s) (x, t) reproducing kernel functions.
Findings
Numerical simulation with different order derivatives degree is done including linear and nonlinear terms that are acquired by interrupting the n-term of the exact solutions. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional models.
Research limitations/implications
Future work includes the application of the reproducing kernel algorithm to highly nonlinear time-fractional partial differential equations such as those arising in single and multiphase flows. The results will be published in forthcoming papers.
Practical implications
The study included a description of fundamental reproducing kernel algorithm and the concepts of convergence, and error behavior for the reproducing kernel algorithm solvers. Results obtained by the proposed algorithm are found to outperform in terms of accuracy, generality and applicability.
Social implications
Developing analytical and numerical methods for the solutions of time-fractional partial differential equations is a very important task owing to their practical interest.
Originality/value
This study, for the first time, presents reproducing kernel algorithm for obtaining the numerical solutions of some certain classes of Robin time-fractional partial differential equations. An efficient construction is provided to obtain the numerical solutions for the equations, along with an existence proof of the exact solutions based upon the reproducing kernel theory.
Numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science, which means increased attention to fractional calculus. ...This paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in Hilbert space within appropriate kernel functions. The algorithm relies on the reproducing kernel Hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the Fourier coefficients of orthogonalization process. The Caputo fractional derivatives are introduced to address these issues. Moreover, the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions. The superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples. The outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method.
The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of ...this research article is to present results on the numerical simulation for time‐fractional Tricomi and Keldysh equations of Dirichlet functions types in Hilbert space that were found in the transonic flows. Those resulting mathematical models are solved using the reproducing kernel algorithm which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations, and error bounds under some hypotheses which provide the theoretical basis of the proposed algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the algorithm are discussed through academic validations.
The aim of this article is to propose a new definition of fuzzy fractional derivative, so-called fuzzy conformable. To this end, we discussed fuzzy conformable fractional integral softly. Meanwhile, ...uniqueness, existence, and other properties of solutions of certain fuzzy conformable fractional differential equations under strongly generalized differentiability are also utilized. Furthermore, all needed requirements for characterizing solutions by equivalent systems of crisp conformable fractional differential equations are debated. In this orientation, modern trend and new computational algorithm in terms of analytic and approximate conformable solutions are proposed. Finally, the reproducing kernel Hilbert space method in the conformable emotion is constructed side by side with numerical results, tabulated data, and graphical representations.
In this paper, continuous genetic algorithm is introduced as an efficient solver for systems of second-order boundary value problems where smooth solution curves are used throughout the evolution of ...the algorithm to obtain the required nodal values of the unknown variables. The solution methodology is based on representing each derivative in the system of differential equations by its finite difference approximation. After that, the overall residue for all nodes in the given system of differential equations is formulated. The solution to the system of differential equations is then converted into the problem of minimizing the overall residue or maximizing the fitness function based on the nodal values generated from the genetic operators. Three numerical test problems including linear and nonlinear systems were analyzed to illustrate the procedure and confirm the performance of the proposed method. In addition to that, a convergence and sensitivity analysis to genetic operators and control parameters of the algorithm has been carried out. The numerical results show that the proposed algorithm is a robust and accurate procedure for solving systems of second-order boundary value problems. Furthermore, the obtained accuracy for the solutions using CGA is much better than the results obtained using some modern methods.
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