**On solution of a nonlocal problem with dynamic boundary conditions for a loaded linear parabolic equation by straight-line methods**

Khankishiyev Z.

We consider a nonlocal problem with dynamic boundary conditions for a loaded linear parabolic equation. For this problem we prove the unique solvability in Sobolev's spaces and the maximum principle under some natural conditions. We suggest the numerical straight-lines method for the finding of the solution of the problem. The convergence of the straight-lines method to the exact solution is also proved.

Cite this paper:

Khankishiyev Z., On solution of a nonlocal problem with dynamic boundary conditions for a loaded linear parabolic equation by straight-line methods. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 5, No. 1, Jan-Jun, pp.75-96, 2017

Cores D., Figueroa J.

The well-known Conjugate Gradient (CG) method minimizes a strictly convex quadratic function for solving large-scale linear system of equations when the coefficient matrix is symmetric and positive definite. In this work we present and analyze a non-quadratic convex function for solving any large-scale linear system of equations regardless of the characteristics of the coefficient matrix. For finding the global minimizers, of this new convex function, any low-cost iterative optimization technique could be applied. In particular, we propose to use the low-cost globally convergent Spectral Projected Gradient (SPG) method, which allow us to extend this optimization approach for solving consistent square and rectangular linear system, as well as linear feasibility problem, with and without convex constraints and with and without preconditioning strategies. Our numerical results indicate that the new scheme outperforms state-of-the-art iterative techniques for solving linear systems when the symmetric part of the coefficient matrix is indefinite, and also for solving linear feasibility problems.

Cite this paper:

Cores D., Figueroa J., A convex optimization approach for solving large scale linear systems. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 5, No. 1, Jan-Jun, pp.51-74, 2017

**Legendre collocation method and its convergence analysis for the numerical solutions of the conductor-like screening model for real solvents integral equation**

Hesameddini E., Shahbazi M.

In this paper, a reliable algorithm for solving the nonlinear Hammerstein integral equation arising from chemical phenomenon is presented. The conductor-like screening model for real solvents (COSMO-RS) integral equation will be solved by the shifted Legendre collocation method. This method approximates the unknown function with Legendre polynomials. The merits of this algorithm lie in the fact that, on the one hand, the problem will be reduced to a nonlinear system of algebraic equations. On the other hand, we show that the efficiency and accuracy of the shifted Legendre collocation method for solving these equations are remarkable. Also, this method is using a simple computational manner and its error analysis will be discussed by illustrating some theorems. Finally, two numerical experiments are given to confirm the superiority and efficiency of presented method with respect to some other well-known methods such as the Bernstein collocation method, Haar wavelet method and Sinc collocation method.

Cite this paper:

Hesameddini E., Shahbazi M., Legendre collocation method and its convergence analysis for the numerical solutions of the conductor-like screening model for real solvents integral equation. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 5, No. 1, Jan-Jun, pp.33-49, 2017

**Linear programming model for solution of matrix game with payoffs trapezoidal intuitionistic fuzzy number**

D. Hunwisai, P. Kumam

In this work, we considered two-person zero-sum games with fuzzy payoffs and matrix games with payoffs of trapezoidal intuitionistic fuzzy numbers (TrIFNs). The concepts of TrIFNs and their arithmetic operations were used. The cut-set based method for matrix games with payoffs of TrIFNs was also considered. Compute the interval-type value of any alfa-constrategies by simplex method for linear programming. The proposed method is illustrated with a numerical example.

Cite this paper:

Hunwisai D., Kumam P., Linear programming model for solution of matrix game with payoffs trapezoidal intuitionistic fuzzy number. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 5, No. 1, Jan-Jun, pp.7-30, 2017

M.A. Fortes

In the last few years, several techniques to fill holes of a given surface by means of minimal energy surfaces have been proposed. In all cases, the filling patches are obtained by minimizing an `energy functional' defined in a vector space of spline functions over the Powell-Sabin triangulation associated to a $\Delta^1$-type triangulation of a given domain D. The energy functional and the space of spline functions are defined in order to the filling patch fulfills certain geometric features. In this work we present, for the first time, a general framework to include most of techniques above referred. Under this general new frame, we review the main filling-holes techniques developed until now, we give their main characteristics, the computation aspects as well as some graphical examples.

Cite this paper:

Fortes M.A., Hole-filling techniques by using minimal energy surfaces. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 2, Jul-Dec, pp.133-164, 2016

J.-P. Chehab

We consider the numerical computation of matrix functions f(x) via matrix ODE integration. The solution is modeled as an asymptotic steady state of a proper differential system. The framework we propose, allows to define flows of sparse matrices leading to sparse approximations to f(x). We discuss of this approach giving stability and approximation results in a general case. We apply our method to the factorization of matrices (LU, Cholesky) as well as the computation of the square root. Numerical illustrations are presented.

Cite this paper:

Chehab J.-P., Sparse approximations of matrix functions via numerical integration of ODEs. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 2, Jul-Dec, pp.95-131, 2016

**Derivative-free method for bound constrained nonlinear monotone equations and its application in solving steady state reaction-diffusion problems**

O. Batta, W. La Cruz, G. Noguera

We present a derivative-free algorithm for solving bound constrained systems of nonlinear monotone equations. The algorithm generates feasible iterates using in a systematic way the residual as search direction and a suitable step-length closely related to the Barzilai-Borwein choice. A convergence analysis is described. We also present one application in solving problems related with the study of reaction-diffusion processes that can be described by nonlinear partial differential equations of elliptic type. Numerical experiences are included to highlight the efficacy of proposed algorithm.

Cite this paper:

Batta O., La Cruz W., Noguera G., Derivative-free method for bound constrained nonlinear monotone equations and its application in solving steady state reaction-diffusion problems. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 2, Jul-Dec, pp.71-93, 2016

**Constrained optimization with integer and continuous variables using inexact restoration and projected gradients**

E.G. Birgin, R.D. Lobato, J.M. Martínez

Inexact restoration (IR) is a well established technique for continuous minimization problems with constraints that can be applied to constrained optimization problems with specific structures. When some variables are restricted to be integer, an IR strategy seems to be appropriate. The IR strategy employs a restoration procedure in which one solves a standard nonlinear programming problem and an optimization procedure in which the constraints are linearized and techniques for mixed-integer (linear or quadratic) programming can be employed.

Cite this paper:

Birgin E.G., Lobato R.D., Martínez J.M., Constrained optimization with integer and continuous variables using inexact restoration and projected gradients. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 2, Jul-Dec, pp.55-70, 2016

**Modified Spectral Projected Subgradient Method: Convergence Analysis and Momentum Parameter Heuristics**

M. Loreto, S. Clapp, C. Cratty, B. Page

The Modified Spectral Projected Subgradient (MSPS) was proposed to solve Langrangen Dual Problems, and its convergence was shown when the momentum term was zero. The MSPS uses a momentum term in order to speed up its convergence. The momentum term is built on the multiplication of a momentum parameter and the direction of the previous iterate. In this work, we show convergence when the momentum parameter is a non-zero constant. We also propose heuristics to choose the momentum parameter intended to avoid the Zigzagging Phenomenon of Kind I. This phenomenon is present in the MSPS when at an iterate the subgradient forms an obtuse angle with the previous direction. We identify and diminish the Zigzagging Phenomenon of Kind I on Setcovering problems, and compare our numerical results to those of the original MSPS algorithm.

Cite this paper:

Loreto M., Clapp S., Cratty C., Page B., Modified Spectral Projected Subgradient Method: Convergence Analysis and Momentum Parameter Heuristics. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 2, Jul-Dec, pp.27-54, 2016

**A globally convergent method for nonlinear least-squares problems based on the Gauss-Newton model with spectral correction**

D.S. Gonçalves, S.A. Santos

This work addresses a spectral correction for the Gauss-Newton model in the solution of nonlinear least-squares problems within a globally convergent algorithmic framework. The nonmonotone line search of Zhang and Hager is the chosen globalization tool. We show that the search directions obtained from the corrected Gauss-Newton model satisfy the conditions that ensure the global convergence under such a line search scheme. A numerical study assesses the impact of using the spectral correction for solving two sets of test problems from the literature.

Cite this paper:

Gonçalves D.S., Santos S.A., A globally convergent method for nonlinear least-squares problems based on the Gauss-Newton model with spectral correction. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 2, Jul-Dec, pp.7-26, 2016

B. Davvaz, V. Leoreanu-Fotea, F. Feng

A ternary hyperoperation on a set H is a 3-ary hyperoperation, which associates a subset of H with any three elements of H. In this paper, we give examples of ternary hyperoperations associated with redox reactions. We observe that for Ag, Cu, Am and Au the ternary hyperoperations are weak associative and so their algebraic structures are H

_{v}-semigroups.
Cite this paper:

Davvaz B., Leoreanu-Fotea V., Feng F., Redox reactions as experimental examples of ternary weak algebraic hyperstructures. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 1, Jan-Jun, pp.39-55, 2016

M. Remili, M. Rahmane

The aim of the present paper is to establish a new result, which guarantees the asymptotic stability of zero solution and square integrability of solutions and their derivatives to nonlinear

differential equations of fourth order.

differential equations of fourth order.

Cite this paper:

Remili M., Rahmane M., Stability and square integrability of solutions of nonlinear fourth order differential equations. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 1, Jan-Jun, pp.21-37, 2016

**Two extensions of the Dai-Liao method with sufficient descent property based on a penalization scheme**

M. Fatemi, S. Babaie-Kafaki

To achieve the good features of the linear conjugate gradient algorithm in a recent extension of the Dai-Liao method, two adaptive choices for parameter of the extended method are proposed based on a penalization approach. It is shown that the suggested parameters guarantee the sufficient descent property independent to the line search and the objective function convexity. Furthermore, they ensure the global convergence of the related algorithm for uniformly convex objective functions. Using a set of unconstrained optimization test problems from the CUTEr library, effectiveness of the suggested choices are numerically compared with two other recently proposed adaptive choices. Results of comparisons show that one of the proposed choices is computationally promising in the sense of the Dolan-Moré performance profile.

Cite this paper:

Fatemi M., Babaie-Kafaki S., Two extensions of the Dai-Liao method with sufficient descent property based on a penalization scheme. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 4, No. 1, Jan-Jun, pp.7-19, 2016

J. Játem, J. Vanegas

The present article has a threefold purpose: First it is a survey of the algebraic structures of generalized Clifford-type algebras and shows the main results of the corresponding Clifford-type analysis and its application to boundary value problems known so far. Second it is aimed to implement algorithms to provide the fast and accurate computation of boundary value problems for inhomogeneous equations in the framework of the generalized Clifford analysis. Finally it is also aimed to encourage the development of a generalized discrete Clifford analysis.

Cite this paper:

Játem J., Vanegas J., Algebraic structures in generalized Clifford analysis and applications to boundary value problems. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 3, No. 2, Jul-Dec, pp.39-69, 2015

Y. Derfoufi, M.I. Mamouni

We introduce here the notion of loop motion planning algorithms and show that it yields to a homotopical invariant: the loop topological complexity, denoted throughout this paper by TC

^{LP}(-), which measures the algorithmic complexity of the motion of a drone as, for example, an unmanned airplane or a guided TV camera. Our main result states that TC(-) = TC^{LP}(-), where TC denotes the ordinary topological complexity introduced by M. Farber. Some interesting applications will emerge and will be discussed.
Cite this paper:

Derfoufi Y., Mamouni M.I., Loop topological complexity. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 3, No. 2, Jul-Dec, pp.31-36, 2015

D. Cores, L. Guenni, L. Torres

One of the main problems in hydrology is the time scale of the historical rainfall data, available from many meteorological data bases. Most of the rainfall data is given at a time scale coarser than the one needed for many applications in hydrology and environmental sciences, as the estimation of spatially continuous rainfall at finer time scales, for drainage systems design and extreme rainfall analysis. A method to disaggregate monthly rainfall to daily or finer temporal scale is very important in many applications. Many authors have addressed this problem by using some stochastic methods including several stochastic rainfall models. The lowering resolution methods must be low-cost and low-storage since the amount of rainfall data is large. The purpose of this work is to formulate this problem as a constrained optimization problem and solve it with a low-cost and low-storage deterministic optimization method. We modify the objective function proposed by Guenni and Bárdossy for solving the disaggregation rainfall problem and we use the low-cost spectral projected gradient (SPG) method. In contrast with the stochastic method, a deterministic approach will take into account important information, as for example the gradient of the objective function. The proposed method was applied to a data set from a rainfall network of the central plains of Venezuela, in which rainfall is highly seasonal and data availability at a daily time scale or even higher temporal resolution is very limited. The numerical results show that the SPG method for solving the disaggregation rainfall problem avoids daily precipitations outliers that might occur as an artifact of the simulation procedure and accurately reproduces the probability distribution. Also, the proposed model and methodology outperforms the one proposed by Guenni and Bárdossy (2002) in the sense that it reduces the absolute error value for the statistical properties from the observed data.

Cite this paper:

Cores D., Guenni L., Torres L., A deterministic optimization approach for solving the rainfall disaggregation problem. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 3, No. 2, Jul-Dec, pp.7-29, 2015

W. La Cruz

In this paper the use of recent residual algorithms for the simulation of the oxygen distribution in a tumor tissue in 2-D is proposed. The oxygen distribution in a tumor is considered a reaction-diffusion problem in steady state, whose mathematical model is a nonlinear partial differential equation, which is numerically solved using the traditional methods for systems of nonlinear equations (Newton's method, Broyden method, inexact Newton methods, etc.). Unlike of these traditional methods that require the use of derivatives and a large memory storage capacity, the proposed residual algorithms are derivative-free methods with low memory storage. The preliminary numerical results indicate that the proposed methods allows efficiently determine the distribution of oxygen in a tumor tissue to synthetic problems.

Cite this paper:

La Cruz W., Simulation of the oxygen distribution in a tumor tissue using residual algorithms. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 3, No. 1, Jan-Jun, pp.13-30, 2015

Y. Derfoufi, M.I. Mamouni

The topological study of the so-called "motion planning algorithms" emerged in the 2003-2004 with the works of M. Farber. We focus here on the topological study of the set of these algorithms, when the configuration space is a normed vector space. We especially show that any motion planning algorithm in a compact sub-configuration space can be approximated by some piecewise affine ones.

Cite this paper:

Derfoufi Y., Mamouni M.I., Motion planning algorithms, topological properties and affine approximation. Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 3, No. 1, Jan-Jun, pp.7-12, 2015

**Matrix completion via a low rank factorization model and an Augmented Lagrangean Succesive Overrelaxation Algorithm**

H. Lara, H. Oviedo, J. Yuan

The matrix completion problem (MC) has been approximated by using the nuclear norm relaxation. Some algorithms based on this strategy require the computationally expensive singular value decomposition (SVD) at each iteration. One way to avoid SVD calculations is to use alternating methods, which pursue the completion through matrix factorization with a low rank condition. In this work an augmented Lagrangean-type alternating algorithm is proposed. The new algorithm uses duality information to define the iterations, in contrast to the solely primal LMaFit algorithm, which employs a Successive Over Relaxation scheme. The convergence result is studied. Some numerical experiments are given to compare numerical performance of both proposals.

Cite this paper:

Lara H., Oviedo H., Yuan J., Matrix completion via a low rank factorization model and an Augmented Lagrangean Succesive Overrelaxation Algorithm, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 2, No. 2, Jul-Dec, pp.21-46, 2014

M. Murugan

Let G(V,E) be a simple, finite, connected, undirected graph. Distance two labeling or L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of non-negative integers such that |f(x)-f(y)| ≥ 2 if x and y are adjacent and |f(x)-f(y)| ≥ 1 if x and y are at distance 2, for all x and y in V(G). The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling f with max{f(v):v in V(G)}=k. In this paper, we construct L(2,1)-labeling of subdivisions of cycle dominated graphs like subdivided Double Fans, subdivided nC

_{α}with a common vertex and subdivided Books B_{n}and hence we find the λ-number of these graphs.
Cite this paper:

Murugan M., L(2,1)-Labeling for Subdivisions of cycle dominated graphs, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 2, No. 2, Jul-Dec, pp.7-19, 2014

W. La Cruz

This paper presents a residual approach of the square root balanced truncation algorithm for model order reduction of continuous, linear and time-invariante compartmental systems. Specifically, the new approach uses a residual method to approximate the controllability and observability gramians, whose resolution is an essential step of the square root balanced truncation algorithm, that requires a great computational cost. Numerical experiences are included to highlight the efficacy of the proposed approach.

Cite this paper:

La Cruz W., A residual approach for Balanced Truncation Model Reduction of compartmental systems, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 2, No. 1, Jan-Jun, pp.7-23, 2014

M. Remili, L.D. Oudjedi

By constructing a Lyapunov functional, we obtain some sufficient conditions which guarantee the stability and boundedness of solutions for some nonlinear differential equations of third order with delay. Our results improve and extend some well known results in the literature and one example is given for illustration of the subject.

Cite this paper:

Remili M., Oudjedi L.D., Uniform stability and boundedness of a kind of third order delay differential equations, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 2, No. 1, Jan-Jun, pp.25-35, 2014

**Oceanic influence on extreme rainfall trends in the north central coast of Venezuela: present and future climate assessments**

L. Guenni, C. Nobre, J. Marengo, G. Huerta, B. Sansó

Extreme events are an important part of climate variability and their intensity and persistence are often modulated by large scale climatic patterns which might act as forcing drivers affecting their probability of occurrence. When the North Tropical Atlantic (NTA) and the Equatorial Pacific (Niño 3 region) sea surface temperature (SST) anomalies are of opposite signs and the first one is positive while the second one is negative, the rainfall response is stronger in the northern coast of Venezuela as well as in the Pacific coast of Central America during the Nov-Feb period. The difference between these two SST anomaly time series (NTA-Niño3) is used in this analysis and it is called the Atlantic-Pacific Index or API. By fitting a dynamic generalized extreme value (GEV) model to station based daily rainfall at different locations and to the Xie and Arkin dataset for the Vargas state, we found the API index to be an adequate index to explain the probabilistic nature of rainfall extremes in the northern Venezuelan coast for the months Nov-Feb. Dependence between the Atlantic-Pacific index and the probabilistic behavior of extreme rainfall was also explored for simulations from two global coupled General Circulation Models for the 20th century climate (20C3M experiment) and the 21st century climate (SRES A2 experiment): the Echam5 model and the HadCM3 model. A significant dependence of extreme rainfall on the Atlantic-Pacific index is well described by the GEV dynamic model for the Echam5 20C3M experiment model outputs. When looking at future climates under the SRES A2 experiment, the dependence of extreme rainfall from the API index is still significant for the middle part of the 21st century (2046-2064), while this dependence fades off for the latest part of the century (2081-2099).

Cite this paper:

Guenni L., Nobre C., Marengo J., Huerta G., Sansó B., Oceanic influence on extreme rainfall trends in the north central coast of Venezuela: Present and future climate assessments, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 2, Jul-Dec, pp.7-45, 2013

M. Andrade, R. Escalante, R. Espitia

In this paper we extend the application of the alternating projection algorithm to solve the problem of finding a point in the intersection of n sets (n ≥ 2), which are not all of them convex sets. Here we term such method as alternating generalized projection (AGP) method. In particular, we are interested in addressing the problem of avoiding the so-called trap points, which may prevent an algorithm to obtain a feasible solution in two or more sets not all convex. Some strategies that allow us to reach the feasible solution are established and conjectured. Finally, we present simple numerical results that illustrate the efficiency of the iterative methods considered.

Cite this paper:

Andrade M., Escalante R., Espitia R., Some convergence strategies for the Alternating Generalized Projection Method, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 2, Jul-Dec, pp.47-77, 2013

**Numerical solution for a family of delay functional differential equations using step by step Tau approximations**

R. Escalante

We use the segmented formulation of the Tau method to approximate the solutions of a family of linear and nonlinear neutral delay differential equations

a_{1}(t)y'(t) = y(t)(a_{2}(t) y(t-t) + a_{3}(t) y'(t-t) + a_{4}(t)) + a_{5}(t) y(t-t) + a_{6}(t)y'(t-t) + a_{7}(t), t ≤ 0

y(t) = Y(t), t < 0

which represents, for particular values of a

_{i}(t), i=1,...,7, and t, functional differential equations that arise in a natural way in different areas of applied mathematics. This paper means to highlight the fact that the step by step Tau method is a natural and promising strategy in the numerical solution of functional differential equations.
Cite this paper:

Escalante R., Numerical solution for a family of delay functional differential equations using step by step Tau approximations, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 2, Jul-Dec, pp.81-91, 2013

J. L. Palacios

We show that for any natural number n, an exponential distribution can be written as the sum of n discontinuous variables and another exponential distribution, all of them independent.

Cite this paper:

Palacios J.L., The exponential distribution as the sum of discontinuous distributions, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 1, Jan-Jun, pp.7-10, 2013

**Global improvements of a protein alignment algorithm and comparison with a global optimization solver**

P. Gouveia, R. Andreani, A. Friedlander, J.M. Martínez, L. Martínez

The LovoAlign method for Protein Alignment, based on the Low-Order Value Optimization theory, is recalled. The method is modified in order to improve global convergence properties and compared against other global minimization procedures.

Cite this paper:

Gouveia P., Andreani R., Friedlander A., Martínez J.M., Martínez L., Global improvements of a protein alignment algorithm and comparison with a global optimization solver, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 1, Jan-Jun, pp.11-24, 2013

O. Cahueñas, L.M. Hernández-Ramos, M. Raydan

We address the issue of approximating the pseudoinverse of the coefficient matrix for dynamically building preconditioning strategies for the numerical solution of large dense linear least-squares problems. The new preconditioning strategies are embedded into simple and well-known iterative schemes that avoid the use of the, usually ill-conditioned, normal equations. We analyze a scheme to approximate the pseudoinverse, based on Schulz iterative method, and also different iterative schemes, based on extensions of Richardson's method, and the conjugate gradient method, that are suitable for preconditioning strategies. We present preliminary numerical results to illustrate the advantages of the proposed schemes.

Cite this paper:

Cahueñas O., Hernández-Ramos L.M., Raydan M., Pseudoinverse preconditioners and iterative methods for large dense linear least-squares problems, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 1, Jan-Jun, pp.25-47, 2013

**(Free) Software for general partial differential equation problems in non-rectangular 2D and 3D regions**

G. Sewell

PDE2D is a general-purpose partial differential equation solver which solves very general systems of nonlinear, steady-state, time-dependent and eigenvalue PDEs in 1D intervals, general 2D regions (see Figure 1), and a wide range of simple 3D regions (see Figure 2), with general boundary conditions. It uses a collocation finite element method [2] for 3D problems, and either a collocation or Galerkin finite element method can be used for 1D and 2D problems. It has been sold commercially for 30 years, but recently a version has been made available, which can be downloaded at no cost from www.pde2d.com.

Cite this paper:

Sewell G., (Free) Software for general partial differential equation problems in non-rectangular 2D and 3D regions, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 1, Jan-Jun, pp.51-54, 2013

G. Sewell

The Kadomtsev-Petviashvili I (KPI) equation is the difficult nonlinear wave equation U

_{xt}+ 6U_{x}^{2}+ 6U_{xx}+ U_{xxxx}= 3U_{yy}. We solve this equation using PDE2D (www.pde2d.com) with initial conditions consisting of two lump solitons, which collide and reseparate. Since the solution has steep, moving, peaks, an adaptive finite element grid is used with a grading which moves with the peaks.
Cite this paper:

Sewell G., Solving the KPI wave equation with a moving adaptive FEM grid, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 1, Jan-Jun, pp.55-71, 2013

M. Sajo-Castelli, B. Feijoo

We present an open-source software package written for GNU Octave. The software is an implementation of the Simplex algorithm for the minimal cost network flow problem oriented towards the academic environment. The implementation supports the use of Big-M and Phase I/Phase II methods and it can also start from a given feasible solution. Flexibility of the package's output configuration provides many attractive possibilities. The outputs are plain editable \LaTeX\ files that can be modified and orchestrated to fit most academic needs. It can be used in examination materials, homework assignments or even form part of a project. The format used to describe the network is the DIMACS min file format to which a simple extension was added in order to support the description of feasible trees in the file.

Cite this paper:

Sajo-Castelli M., Feijoo B., GNU Oflox: An Academic Software for the Minimal Cost Network Flow Problem, Bull. Comput. Appl. Math. (Bull CompAMa), Vol. 1, No. 1, Jan-Jun, pp.73-77, 2013