Higinio Ramos

• PhD
• Professor at University of Salamanca

This work aims to provide approximate solutions for singularly perturbed problems with periodic boundary conditions using quintic B-splines and collocation. The well-known Shishkin mesh strategy is applied for mesh construction. Convergence analysis demonstrates that the method achieves parameter-uniform convergence with fourth-order accuracy in th...

This article is concerned with the construction and analysis of efficient uniformly convergent methods for a class of parabolic systems of coupled singularly perturbed reaction–diffusion problems with discontinuous source term. Due to the discontinuity in the source term, the solution to this problem exhibits interior layers along with boundary lay...

The convergence order of an iterative method used to solve equations is usually determined by using Taylor series expansions, which in turn require high-order derivatives, which are not necessarily present in the method. Therefore, such convergence analysis cannot guarantee the theoretical convergence of the method to a solution if these derivative...

This paper focuses on studying the oscillatory properties of a distinctive class of second-order advanced differential equations with distributed deviating arguments in a noncanonical case. Utilizing the Riccati method and the comparison method with first-order equations, in addition to other analytical methods, we have established criteria to test...

In this paper, we address the study of the oscillatory properties of the solutions of a class of third-order delay differential equations. The primary objective of this study is to provide new relationships that can be employed to obtain criteria for excluding increasing positive solutions and decreasing positive solutions so that the resulting cri...

We investigated a nonlinear singularly perturbed elliptic reaction-diffusion coupled system having non-smooth data networked by a k-star graph. We considered all possible boundary conditions at the free boundary located at the tail of the edge and imposed the continuity condition with Kirchhoff's junction law at the junction point of the -star gr...

In this paper, an optimal decomposition algorithm is introduced to solve a kind of nonlinear fourth-order Emden-Fowler equations (EFEs) that appear in many applied fields. Transforming the Emden-Fowler equation into a Volterra integral equivalent equation allows us to deal with the singularity at the endpoint x = 0. This conversion also helps to re...

In this work, only two independent conditions for the oscillation of all solutions of even-order delay differential equations in the non-canonical case are established. Using comparison techniques with first- and second-order delay differential equations, we obtain easy-to-apply criteria that improve previous results in the literature. In addition,...

In this article, we first convert a second order singularly perturbed boundary value problem (SPBVP) into a pair of initial value problems, which are solved later using exponential time differencing (ETD) Runge–Kutta methods. The stability analysis of the proposed scheme is addressed. Some linear and non-linear problems have been solved to study th...

We are focused on the numerical treatment of a singularly perturbed degenerate parabolic convection–diffusion problem that exhibits a parabolic boundary layer. The discretization and analysis of the problem are done in two steps. In the first step, we discretize in time and prove its uniform convergence using an auxiliary problem. In the second ste...

In this paper, we develop a novel higher-order compact finite difference scheme for solving systems of Lane-Emden-Fowler type equations. Our method can handle these problems without needing to remove or modify the singularity. To construct the method, initially, we create a uniform mesh within the solution domain and develop a new efficient compact...

This paper introduces and analyzes some new highly efficient iterative procedures for approximating fixed points of contractive-type mappings. The stability, data dependence, strong convergence, and performance of the proposed schemes are addressed. Numerical examples demonstrate that the newly introduced schemes produce approximations of great acc...

Investigation of the solutions of the coupled viscous Burgers system is crucial for realizing and understanding some physical phenomena in applied sciences. Particularly, Burgers equations are used in the modeling of fluid mechanics and nonlinear acoustics. In the present study, a modified meshless quadrature method based on radial basis functions...

This article presents an efficient one‐step hybrid block method (OHBM) to solve third‐order initial value problems (IVPs). The proposed method is derived using interpolation and collocation techniques of the assumed exact solution and its derivatives. The theoretical properties of the OHBM are analyzed. An embedding‐like technique is utilized and e...

This paper solves a class of Fredholm integro-differential equations involving a small parameter with integral boundary conditions numerically. The solution to the problem possesses boundary layers at both end boundaries. A central difference scheme is used for approximating the derivatives. In contrast, the trapezoidal rule is used for the integra...

In this study, the authors present a uniform algebraic trigonometric tension B‐spline‐based differential quadrature method combined with an optimized hybrid block method to numerically solve the Rosenau–KdV–RLW equation. The discrete mass and energy have been calculated, showing that they are conserved, thus indicating the efficiency and accuracy o...

In this article, non-standard compact finite difference schemes are constructed for the numerical approximation to first- and second-order derivatives. The proposed compact schemes have eighth order of accuracy and are tri-diagonal in nature, making use of a stencil smaller than those of conventional tri-diagonal compact finite difference schemes o...

In this paper, we address a two-parameter singularly perturbed convection-reaction-diffusion 2-D problem. We also consider that the convection and source terms are discontinuous in space. Due to these discontinuities and the presence of perturbation parameters, solutions to such problems show boundary and interior layers. In this study, we have car...

This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the...

A rapidly converging domain decomposition algorithm is introduced for a time delayed parabolic problem with mixed type boundary conditions exhibiting boundary layers. Firstly, a space-time decomposition of the original problem is considered. Subsequently, an iterative process is proposed, wherein the exchange of information to neighboring subdomain...

A parameter-uniform implicit approach for two-parameter singularly perturbed boundary value problems is constructed. On the solution derivatives, sharp limits are presented. The solution is additionally divided into regular and singular components, limiting the derivatives of these components utilized in the convergence analysis. In the temporal di...

In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the d...

In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary dif...

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid sc...

For the approximate solution of the Kepler equations and some related problems, a fourth-order convergent functionally-fitted block hybrid Falkner method which is based on the concepts of interpolation and collocation of the fitting function given as a linear combination of {1, sin(ωx), cos(ωx), sinh(ωx), cosh(ωx)} is presented. The proposed method...

This paper presents a new hybrid block method formulated in variable stepsize mode to solve some first-order initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method is implemented considering an adaptive stepsize strategy to maintain the estimated error in each step w...

This paper introduces an efficient approach for solving Lane–Emden–Fowler problems. Our method utilizes two Nyström schemes to perform the integration. To overcome the singularity at the left end of the interval, we combine an optimized scheme of Nyström type with a set of Nyström formulas that are used at the fist subinterval. The optimized techni...

The search for efficient higher order methods is a constant goal in numerical analysis. In this paper, a higher order two-step hybrid block method is presented to directly solve second-order initial value problems in ordinary differential equations. In addition to the higher order, the proposed method has been formulated in variable step-size mode...

In the past decades, many applications related to applied physics, physiology and astrophysics have been modelled using a class of two-point singular boundary value problems (SBVPs). In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation for approximating a class of two-poin...

This article introduces a computational hybrid one-step technique designed for solving initial value differential systems of a first order, which utilizes second derivative function evaluations. The method incorporates three intra-step symmetric points that are calculated to provide an optimum version of the suggested scheme. By combining the hybri...

An intra-step block Falkner method whose coefficients depend on a parameter ω and the step length h is presented in this study for solving numerically second-order delay differential equations with oscillatory solutions. In the development of the method, the collocation and interpolation techniques were employed. The investigation of the properties...

This article aims at the development and analysis of a numerical scheme for solving a singularly perturbed parabolic system of n reaction-diffusion equations where m of the equations (with m < n) contain a perturbation parameter while the rest do not contain it. The scheme is based on a uniform mesh in the temporal variable and a piecewise uniform...

This work proposes a hybrid block numerical method of tenth order for the direct solution of fifth-order initial value problems. The formulas that constitute the block method are derived from a continuous approximation obtained through interpolation and collocation techniques. In order to obtain better accuracy, sixth-order derivatives are incorpor...

In this paper, we have developed a novel three step second derivative block method and coupled it with fourth order standard compact finite difference schemes for solving time dependent nonlinear partial differential equations (PDEs) of physical relevance. Two well-known problems viz. the FitzHugh–Nagumo equation and the Burgers’ equation have been...

In this work, we study the numerical solution for time fractional Black-Scholes model under jump-diffusion involving a Caputo differential operator. For simplicity of the analysis, the model problem is converted into a time fractional partial integro-differential equation with a Fredholm integral operator. The L1 discretization is introduced on a g...

In this work, a phase- and amplification-fitted 5(4) diagonally implicit pair of Runge–Kutta–Nyström methods with four stages is developed to solve special second-order ordinary differential equations (ODEs), whose solutions are oscillatory. The local truncation errors of the formulas of the new pair are given, showing that the orders are maintaine...

An optimization of the sixth-order explicit Runge-Kutta-Nyström method with six stages derived by El-Mikkawy and Rahmo using the phase-fitted and amplification-fitted techniques with constant step-size is constructed in this paper. The new adapted method integrates exactly the common test: y = −w 2 y. The local truncation error of the new method is...

This research article introduces an efficient method for integrating Lane–Emden–Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nyström technique with a set of formulas applied at the initial step to circumvent the singularity...

In this work, by obtaining a new condition that excludes a class of positive solutions of a type of higher order delay differential equations, we were able to construct an oscillation criterion that simplifies, improves and complements the previous results in the literature. The adopted approach extends those commonly used in the study of second-or...

This study proposes an accurate approximation to the solution of second-order nonlinear two-point boundary value problems, including the well-known Bratu problem, using an iterative technique based on Green's function. The approach relies on constructing an equivalent integral representation incorporating Green's function. The proposed methodology...

A parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction-diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic B-spline basis functions with an exponentially graded mesh are used to solve a × system whose solution exhibits parabolic (or exponent...

We consider a singularly perturbed two-dimensional steady-state convection-diffusion problem with Robin boundary conditions. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ϵ, is a positive perturbation parameter, and so it may be arbitrarily small. Solutions to such problems presen...

The present article aims to study the oscillatory properties of a class of second-order dynamic equations on time scales. We consider during this study the noncanonical case, which did not receive much attention compared to the canonical dynamic equations. The approach adopted depends on converting the noncanonical equation to a corresponding canon...

This paper presents an efficient numerical approach for first-order delay differential equations containing a piece-wise constant delay. The strategy is based on a five-point hybrid block method that has been developed for ordinary differential equations. We will use the interpolation technique for the evaluation of delay terms that are not defined...

In this study, an effective iterative technique based on Green's function is proposed to solve a nonlinear fourth‐order boundary value problem (BVP) with nonlinear boundary conditions, which models an elastic beam. An iterative Green's function approach and a shooting method are integrated in the proposed method. The mathematical derivation is furt...

This research work deals with the development, analysis, and implementation of an adaptive optimized one‐step Nyström method for solving second‐order initial value problems of ODEs and time‐dependent partial differential equations. The new method is developed through a collocation technique with a new approach for selecting the collocation points....

In this paper, a generalization of a classical result by Gander concerning the characterization of third-order methods is addressed. New and classical methods are included in the family. In particular, a new construction of the well-known Chebyshev method is presented. Other methods, based on exponential and logarithmic fittings respectively, are r...

We construct and analyze a domain decomposition method to solve a class of singularly perturbed parabolic problems of reaction-diffusion type having Robin boundary conditions. The method considers three subdomains, of which two are finely meshed, and the other is coarsely meshed. The partial differential equation associated with the problem is disc...

In this paper, the numerical solution of a mixed derivative type Hunter–Saxton equation is addressed. A given equation is discretized transforming it into a system of ODEs with the use of a cubic trigonometric B-splines based differential quadrature method. The system is further solved using a fifth-order optimized one-step hybrid block method. Thr...

This work introduces a new one-step method with three intermediate points for solving stiff differential systems. These types of problems appear in different disciplines and, in particular, in problems derived from chemical reactions. In fact, the term “stiff”’ was coined by Curtiss and Hirschfelder in an article on problems of chemical kinetics (H...

This work aims at obtaining a numerical approximation to the solution of a two-parameter singularly perturbed convection-diffusion-reaction system of partial differential equations with discontinuous coefficients. This discontinuity, together with small values of the perturbation parameters, causes interior and boundary layers to appear in the solu...

In this article, we investigate a two-dimensional (2-D) singularly perturbed convection-reaction–diffusion elliptic type problem where two parameters ϵ and μ multiply the diffusion and convection terms, respectively. Furthermore, we assume that jump discontinuities exist in the source term along the x- and y-axis. Due to the presence of perturbatio...

In this article, a singularly perturbed second-order Fredholm integro-differential equation with a discontinuous source term is examined. An exponentially-fitted numerical method on a Shishkin mesh is applied to solve the problem. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Some numerical resu...

In this work, a new pair of diagonally implicit Runge–Kutta–Nyström methods with four stages is constructed. The proposed method has been derived to solve initial value problems of special second-order ordinary-differential equations. The principal local truncation error of the new method is obtained, and the main characteristics of the new method...

Two non-standard predictor-corrector type finite difference methods for a SIR epidemic model are proposed. The methods have useful and significant features, such as positivity, basic stabil- ity, boundedness and preservation of the conservation laws. The proposed schemes are compared with classical fourth order Runge–Kutta and non-standard differen...

The variational iteration method (VIM) has been in the last two decades, one of the most used semi-analytical techniques for approximating nonlinear differential equations. The notion of VIM is based on the identification of the Lagrange multiplier using the variational theory. The performance of the method is highly dependent on how the Lagrange m...

This manuscript presents a variable stepsize formulation of a pair of block methods to efficiently solve third-order IVP models of Lane-Emden-Fowler type equations. The main method is obtained considering two intermediate points. This method combines an appropriate set of formulas for dealing with the singularity at the left endpoint t=0. The propo...

This article addresses the development and analysis of an efficient optimized hybrid block method for integrating general second order initial value problems (IVPs) of ordinary differential equations (ODEs). The construction of the method is based on a combination of two methodologies, namely hybrid and block that result in an efficient implicit nu...

We construct an efficient hybrid numerical method for solving coupled systems of singularly perturbed linear parabolic problems of reaction‐diffusion type. The discretization of the coupled system is based on the use of an additive or splitting scheme on a uniform mesh in time and a hybrid scheme on a layer‐adapted mesh in space. It is proven that...

A parameter-uniform numerical scheme for a system of weakly coupled singularly perturbed reaction-diffusion equations of arbitrary size with appropriate boundary conditions is investigated. More precisely, quadratic $B$-spline basis functions with an exponentially graded mesh are used to solve a $\ell\times\ell$ system whose solution exhibits parab...

In this study, a trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström (RKN) pair with six stages is formulated, considering a previous method developed by El‐Mikkawy and Rahmo. The obtained adapted pair integrates exactly the usual test equation: y′′=−w2y$$ {y}^{\prime \prime }=-{w}^2y $$. The local truncation error of the new method is pres...

Implicit block methods for solving initial value problems in ordinary differential equations are well-known among the contemporary scientific community, since they are cost-effective, self-starting, consistent, stable, and usually converge fast when applied to solve particularly stiff models. These characteristics of block methods are the primary r...

In this paper, an efficient wavelet‐based numerical scheme is presented for the solution of fourth‐order singularly perturbed boundary value problems with discontinuous data. The fourth‐order derivative of the solution function is expanded in Haar series and then integrated to get the approximations for the lower order derivatives of the solution f...

In this paper, we evaluate and discuss different numerical methods to solve the Black–Scholes equation, including the θ-method, the mixed method, the Richardson method, the Du Fort and Frankel method, and the MADE (modified alternating directional explicit) method. These methods produce numerical drawbacks such as spurious oscillations and negative...

The generalized finite difference method is a meshless method for solving partial differential equations that allows arbitrary discretizations of points. Typically, the discretizations have the same density of points in the domain. We propose a technique to get adapted discretizations for the solution of partial differential equations. This strateg...

In this article, a family of one-step hybrid block methods having two intrastep points is developed for solving first-order initial value stiff differential systems that occur frequently in science and engineering. In each method of the family, an intrastep point controls the order of the main method and a second one has a control over the stabilit...

This article deals with the development of an optimized third-derivative hybrid block method for integrating general second order two-point boundary value problems (BVPs) subject to different types of boundary conditions (BCs) such as Dirichlet, Neumann or Robin. A purely interpolation and collocation approach has been used in order to develop the...

In this study, new asymptotic properties of positive solutions of the even-order neutral delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Using these properties, we obtain new criteria to exclude a class from the positive solutions...

This research paper is concerned with developing, analyzing, and implementing an adaptive optimized one-step block Nyström method for solving second-order initial value problems of ODEs and time-dependent partial differential equations. The new technique is developed through a collocation method with a new approach for selecting the collocation poi...

In this paper, we develop an optimized hybrid block method which is combined with a modified cubic B-spline method, for solving non-linear partial differential equations. In particular, it will be applied for solving three well-known problems, namely, the Burgers equation, Buckmaster equation and FitzHugh–Nagumo equation. Most of the developed meth...

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable i...

In this article, a numerical scheme is developed to solve singularly perturbed convection–diffusion type degenerates parabolic problems. The degenerative nature of the problem is due to the coefficient b(x,t)=b0(x,t)xp,p≥1 of the convection term. As the perturbation parameter approaches zero, the solution to this problem exhibits a parabolic bounda...

A functionally fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. The characteristics of the method are studied, particularly, it is shown that it has a third order of convergence for the gen...

The generalized finite difference method is a meshless method for solving partial differential equations that allows arbitrary discretizations of points. Typically, the discretizations have the same density of points in the domain. We propose a technique to get adapted discretizations for the solution of partial differential equations. This strateg...

In this paper, we construct a method with eight steps that belongs to the family of Obrechkoff methods. Due to the explicit nature of the new method, not only does it not require another method as predictor, but it can also be considered as a suitable predictive technique to be used with implicit methods. Periodicity and error terms are studied whe...

This manuscript presents a one-step method incorporating a second-derivative applied to obtain approximate solutions of first-order initial-value problems of ordinary and time-dependent partial differential equations. The new scheme is derived through interpolation and collocation approaches, and the characteristics of the obtained method are analy...

In this paper, an optimized pair of hybrid block techniques is presented and successfully applied to integrate second-order singular initial value problems of Lane-Emden-Fowler type emanating from applied sciences and engineering. An adaptive technique implementation is considered. One of the proposed one-step hybrid block techniques is obtained by...

In this article, a pair of hybrid block techniques is constructed and successfully applied to integrate Emden-Fowler third-order singular boundary problems. One of the proposed one-step hybrid block techniques is obtained by considering two intermediate points. The obtained method is then paired with a hybrid block of order three to bypass the sing...

In this paper a 5(3) pair of explicit trigonometrically adapted Runge-Kutta-Nyström methods with four stages is derived based on an explicit pair appeared in the literature. The new adapted method is able to integrate exactly the usual test equation: y′′=-w2y. The local truncation error of the new method is obtained, proving that the algebraic orde...

In this work, we aim at studying the asymptotic and oscillatory behavior of even-order neutral delay noncanonical differential equations. To the best of our knowledge, most of the related previous works are concerned only with neutral equations in the canonical case. Our new oscillation criteria essentially improve, simplify, and complement related...

This paper deals with the construction of a functionally fitted method for solving first-order differential systems whose solutions present an oscillatory behaviour. The method incorporates the second derivative to obtain better accuracies and is developed on the basis that it provides no errors when the true solution is a linear combination of som...

In the present scientific literature, block methods to solve stiff and nonlinear initial value problems are in great use due to their better stability features and smaller computational cost. Adaptive step-size versions of such methods, however, are not presented in many research articles, although they are more efficient than their fixed step-size...

In this article, we have considered an adaptive step-size formulation of an optimized block method for directly solving general second-order initial value problems of ODEs numerically. This formulation has been done using an embedded-type procedure resulting in an efficient method that performs much better compared to its counterpart fixed step-siz...

This manuscript presents an efficient pair of hybrid Nyström techniques to solve second-order Lane–Emden singular boundary value problems directly. One of the proposed strategies uses three off-step points. The obtained formulas are paired with an appropriate set of formulas implemented for the first step to avoid singularity at the left end of the...