Reproducing kernel method in Hilbert spaces for solving the linear and nonlinear four-point boundary value problems

Numerical solution of the multiterm time‐fractional diffusion equation based on reproducing kernel theory

“…From 1980, Cui and coworkers [21,22] are pioneers in linear and nonlinear numerical analysis using the reproducing kernel Hilbert space method. Recently, a lot of research works have been devoted to the application of reproducing kernel Hilbert space method to solve several linear and nonlinear problems such as singularly perturbed delay initial value problems [23], differential equations of Bratu-type with fractional order [24], mixed boundary value problems [25], Rayleigh-type equations [26], Feredholm-Volterra integro-differential equations, second-order periodic boundary value problems [27], functional integral equations [28], variational problems depending on indefinite integrals [29], nonlinear delay differential equations of fractional order [30], nonlocal initial-boundary value problems for parabolic and hyperbolic integro-differential equations [31], Black-Scholes equation [32], multi-order fractional differential equations [33], nonlinear systems of PDEs [34], parabolic problems with nonclassical conditions [35], linear and nonlinear four-point boundary value problems [36], to name a few. Also, the book [36] provides an excellent overview of the existing reproducing kernel methods for solving various model problems such as integral and integro-differential equations.…”

A reproducing kernel Hilbert space method for solving the nonlinear three‐point boundary value problems

“…() In recent years, the reproducing kernel methods have been used to solve all kinds of differential equations, providing new description forms for exact and approximate solutions of many classical equations. For example, the use of reproducing kernel methods for solving a variety of boundary value problems was discussed by Arqub and Al‐Smadi,() Qing and Niu, Cui and Geng,() Li and Wu,() Geng, Lin and Niu,() Foroutan et al, and Momani et al…”

“…The reproducing kernel Hilbert space method have successfully been applied to several boundary value problems, such as two‐point boundary value problems of fourth‐order mixed integro‐differential equations, nonlinear initial value problems, singular nonlinear second‐order periodic boundary value problems, three‐point boundary value problems of nonlinear fractional differential equations, nonlinear multipoint boundary value problems, nonlinear problems with nonlinear boundary conditions, and third‐order partial differential equations with three‐point boundary value problems. ()…”

Reproducing kernel Hilbert space method for nonlinear boundary‐value problems

“…Sahihi et al have used this technique for solving singularly perturbed differential‐difference equations with boundary layer behavior. For more details other studies …”