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In this study based on the accelerated over relaxation (AOR) method we make an iterative scheme for solving generalized Lyapunov matrix equation ... over complex or real matrices. Then we analyze the convergence of the new iterative method in detail. There have been discussions for the calculation of optimal parameters. Finally a numerical example is given to demonstrate the capability of the new method. [ABSTRACT FROM AUTHOR]

Abstract In the current investigation, an error estimate has been proposed to solve the two-dimensional weakly singular integro-partial differential equation with space and time fractional derivatives based on the finite element/finite difference scheme. The time and space derivatives are based on the Riemann–Liouville and Riesz fractional derivatives, respectively. At first, the temporal variable has been discretized by a second-order difference scheme and then the space variable has been approximated by the finite element method (FEM). The analytical study shows that the presented scheme is unconditionally stable and convergent. Finally, some examples have been introduced to confirm the theoretical results and efficiency of the proposed technique. [ABSTRACT FROM AUTHOR]

This paper presents 2 new classes of the Bessel functions on a compact domain [0, T] as generalized-tempered Bessel functions of the first- and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders. [ABSTRACT FROM AUTHOR]

In the current manuscript, we consider a fractional partial integro-differential equation that is called fractional evolution equation. The fractional evolution equation is based on the Riemann–Liouville fractional integral. The presented numerical algorithm is based on the following procedures: at first a difference scheme has been used to discrete the temporal direction and secondly the spectral element method is applied to discrete the spatial direction and finally these procedures are combined to obtain a full-discrete scheme. For the constructed numerical technique, we prove the unconditional stability and also obtain an error bound. We use the energy method to analysis the full-discrete scheme. We employ some test problems to show the high accuracy of the proposed technique. Also, we compare the obtained numerical results using the present method with the existing methods in the literature. [ABSTRACT FROM AUTHOR]

This paper gives an error analysis for the method of lines (MOL) using generalized interpolating moving least squares (GIMLS) approximation. In this study, error bound for the time-dependent linear and nonlinear second-order differential equations in d -dimension will be obtained, when the GIMLS method is used for approximating the spatial variables. Also, the well-known Courant–Friedrichs–Lewy (CFL) condition will be derived in both cases (linear and nonlinear equations). Finally, numerical examples will be reported to confirm the ability of the proposed technique. [ABSTRACT FROM AUTHOR]

In this article, we discuss the application of two important numerical methods, Ritz-Galerkin and Method of Fundamental Solutions (MFS), for solving some inverse problems, arising in the context of two-dimensional elliptic equations. The main incentive for studying the considered problems is their wide applications in engineering fields. In the previous literature, the use of these methods, particularly MFS for right hand side reconstruction has been limited, partly due to stability concerns. We demonstrate that these diculties may be surmounted if the aforementioned methods are combined with techniques such as dual reciprocity method(DRM). Moreover, we incorporate some iterative regularization techniques. This fact is especially veried by taking into account the noisy data with boundary conditions. In addition, parts of this article are dedicated to the problem of boundary data approximation and the issue of numerical stability, ending with a general discussion on the advantages and disadvantages of various methods. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1995-2026, 2015 [ABSTRACT FROM AUTHOR]

Three inverse problems of reconstructing the time-dependent, spacewise- dependent and both initial condition and spacewise-dependent heat source in the one-dimensional heat equation are considered. These problems are reformulated by eliminating the unknown functions using some special assumptions concerning the points in space or time as additional measurements. Then direct techniques are proposed to solve the non-classical boundary value problems. For obtaining the robust and stable approximations, Bernstein multi-scaling and B-spline basis functions in the context of the Ritz–Galerkin method are utilized to immediate passage from differential equations to algebraic equations and afterwards, a Newton-type method is used to produce the admissible solution. The numerical convergence and stability are discussed in the test examples to show that the presented schemes provide accurate and acceptable approximations. [ABSTRACT FROM AUTHOR]

In this paper we first introduce fractional orthogonal Jacobi functions then we obtain a new fractional derivative operational matrix for these orthogonal functions. It is based on the relationship between the coefficients of the fractional Taylor series and fractional Jacobi function expansions. We also apply this new operational matrix to the collocation method for solving general multi-order fractional differential equations (FDEs) and nonlinear fractional integro–differential equations (FIDEs). We also present several test problems. The numerical results show that our new scheme is very effective and convenient for solving FDEs and FIDEs. [ABSTRACT FROM PUBLISHER]

In this paper we present two pseudospectral methods based on Fourier series and rational Chebyshev functions for solving the Nagumo equation. In each of the two presented methods the problem is reduced to a system of ordinary differential equations that is solved by the leapfrog difference scheme and the fourth-order Runge-Kutta method, respectively. We compare the numerical solutions with the exact solution to validate the numerical methods. Numerically comparing of the two methods also will be considered. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, the variational iteration method is implemented to give approximate solution of the Euler-Lagrange, Euler-Poisson and Euler-Ostrogradsky equations as ordinary (or partial) differential equations which arise from the variational problems. In this method, general Lagrange multipliers are introduced to construct correction functional for the variational problems. The multipliers in the functionals can be identified optimally via the variational theory. The initial approximations can be freely chosen with possible unknown constant, which can be determined by imposing the boundary or initial conditions. Illustrative examples are included to demonstrate the validity and applicability of the variational iteration method. [ABSTRACT FROM AUTHOR]

A numerical technique is presented for the solution of fourth-order integro-differential equations. This method uses the Chebyshev cardinal functions. The method consists of expanding the required approximate solution as the elements of Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. [ABSTRACT FROM AUTHOR]

Third-order boundary value problems (BVPs) were discussed in many papers in recent years. The existence of solutions to these problems is shown under certain conditions. Finding the exact or approximate solution of these problems is not easy especially when the boundary conditions are non-linear. In this paper, we present a method for solving a class of third-order BVPs with non-linear boundary conditions. The decomposition procedure of Adomian is used for solving these problems and the advantages of this method over the other existing methods are shown. Numerical results are presented to show the efficiency of the developed method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

Abstract: Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. This paper deals with the numerical solution of a class of fractional differential equations. The fractional derivatives are described in the Caputo sense. Our main aim is to generalize the Legendre operational matrix to the fractional calculus. In this approach, a truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem. The method is applied to solve two types of fractional differential equations, linear and nonlinear. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. [Copyright &y& Elsevier]

Abstract: The variational iteration method is applied to solve the generalized pantograph equation. This technique provides a sequence of functions which converges to the exact solution of the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. Employing this technique, it is possible to find the exact solution or an approximate solution of the problem. Some examples are given to demonstrate the validity and applicability of the method and a comparison is made with existing results. [Copyright &y& Elsevier]

Abstract: A homotopy perturbation method (HPM) is proposed to solve non-linear systems of second-order boundary value problems. HPM yields solutions in convergent series forms with easily computable terms, and in some cases, yields exact solutions in one iteration. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The results reveal that the method is very effective, straightforward and simple. [Copyright &y& Elsevier]

Abstract: In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation. [Copyright &y& Elsevier]

Abstract: In this paper, by using the extended Sturm–Liouville theorem for symmetric functions, we introduce the differential equation as a generalization of the differential equation of trigonometric sequences and for and obtain its explicit solution in a simple trigonometric form. We then prove that the obtained sequence of solutions is orthogonal with respect to the constant weight function on and compute its norm square value explicitly. One of the important advantages of this generalization is to find some new infinite series. A practical example is given in this sense. [Copyright &y& Elsevier]

Abstract: Hyperbolic partial differential equations with an integral condition serve as models in many branches of physics and technology. Recently, much attention has been expended in studying these equations and there has been a considerable mathematical interest in them. In this work, the solution of the one-dimensional nonlocal hyperbolic equation is presented by the method of lines. The method of lines (MOL) is a general way of viewing a partial differential equation as a system of ordinary differential equations. The partial derivatives with respect to the space variables are discretized to obtain a system of ODEs in the time variable and then a proper initial value software can be used to solve this ODE system. We propose two forms of MOL for solving the described problem. Several numerical examples and also some comparisons with finite difference methods will be investigated to confirm the efficiency of this procedure. [Copyright &y& Elsevier]

Abstract: In this work, a system of two nonlinear integro-differential equations which arises in biology is considered and the well-known variational iteration method is implemented for finding the solution of this system. This method is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. This technique reduces the volume of calculations by not requiring discretization of the variables, linearization or small perturbations and constructs a sequence which converges to the exact solution rapidly. Also a numerical technique based on the pseudospectral Legendre method is developed to solve the model. Several test problems are given and the results are compared with Adomian decomposition method and the variational iteration technique. [Copyright &y& Elsevier]

Abstract: Delay differential equations (denoted as DDE) have a wide range of application in science and engineering. They arise when the rate of change of a time-dependent process in its mathematical modeling is not only determined by its present state but also by a certain past state. Recent studies in such diverse fields as biology, economy, control and electrodynamics have shown that DDEs play an important role in explaining many different phenomena. In particular they turn out to be fundamental when ODE-based models fail. In this research, the solution of a delay differential equation is presented by means of a homotopy perturbation method and then some numerical illustrations are given. These results reveal that the proposed method is very effective and simple to perform. [Copyright &y& Elsevier]

Abstract: The Chebyshev finite difference method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and yields very accurate results. [Copyright &y& Elsevier]

Abstract: In this paper, the solution of Cauchy reaction–diffusion problem is presented by means of variational iteration method. Reaction–diffusion equations have special importance in engineering and sciences and constitute a good model for many systems in various fields. Application of variational iteration technique to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computations. [Copyright &y& Elsevier]

A Chebyshev finite difference method has been proposed in order to solve linear and nonlinear second-order Fredholm integro-differential equations. The approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a nonuniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

Abstract: The sinc-collocation method is presented for solving a nonlinear system of second-order boundary value problems. Some properties of the sinc-collocation method required for our subsequent development are given and are utilized to reduce the computation of solution of the system of second-order boundary value problems to some algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results. The method is easy to implement and yields very accurate results. [Copyright &y& Elsevier]

Abstract: Determination of an unknown time-dependent function in parabolic partial differential equations, plays a very important role in many branches of science and engineering. In the current investigation, the Adomian decomposition method is used for finding a control parameter in the quasilinear parabolic equation , in with known initial and boundary conditions and subject to an additional condition in the form of which is called the boundary integral overspecification. The main approach is to change this inverse problem to a direct problem and then solve the resulting equation using the well known Adomian decomposition method. The decomposition procedure of Adomian provides the solution in a rapidly convergent series where the series may lead to the solution in a closed form. Furthermore due to the rapid convergence of Adomian’s method, a truncation of the series solution with sufficiently large number of implemented components can be considered as an accurate approximation of the exact solution. This method provides a reliable algorithm that requires less work if compared with the traditional techniques. Some illustrative examples are presented to show the efficiency of the presented method. [Copyright &y& Elsevier]

Abstract: In the modeling of a large class of problems in science and engineering, the minimization of a functional is appeared. Finding the solution of these problems needs to solve the corresponding ordinary differential equations which are generally nonlinear. In recent years He''s variational iteration method has been attracted a lot of attention of the researchers for solving nonlinear problems. This method finds the solution of the problem without any discretization of the equation. Since this method gives a closed form solution of the problem and avoids the round off errors, it can be considered as an efficient method for solving various kinds of problems. In this research He''s variational iteration method will be employed for solving some problems in calculus of variations. Some examples are presented to show the efficiency of the proposed technique. [Copyright &y& Elsevier]

Abstract: In this article, we introduce a suitable preconditioner for the class of fourth order approximations employed for solving the elliptic partial differential equations Au xx + Bu yy + Cu zz = f(x, y, z, u, u x , u y , u z ), where A, B and C are constants. Numerical results are given to show the efficiency of the presented approach. [Copyright &y& Elsevier]

Abstract: In this work, the method of radial basis functions is used for finding the solution of an inverse problem with source control parameter. Because a much wider range of physical phenomena are modelled by nonclassical parabolic initial-boundary value problems, theoretical behavior and numerical approximation of these problems have been active areas of research. The radial basis functions (RBF) method is an efficient mesh free technique for the numerical solution of partial differential equations. The main advantage of numerical methods which use radial basis functions over traditional techniques is the meshless property of these methods. In a meshless method, a set of scattered nodes are used instead of meshing the domain of the problem. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes. [Copyright &y& Elsevier]

Abstract: In this paper, we study the qualitative behavior of solutions of the class of delay difference equationwhere the initial conditions x −2k+1,…, x −1, x 0 are positive, k ∈{1,2,…}, and the parameters β, γ, A, B are positive. Our concentration is on invariant intervals and the global stability of the above mentioned equation. We obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients. Furthermore, the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed. It is also illustrated that for some special case of parameters, the solution will be either a period-2 solution, or will converge to the equilibrium point, or will have unbounded solutions. [Copyright &y& Elsevier]

Abstract: The aim of this work is to generalize techniques of difference equations in order to apply them to a wider class. We start by establishing the boundedness, the oscillatory character, and the global stability of the equationwith positive parameters, k ∈{1,2,…}, and nonnegative initial conditions. [Copyright &y& Elsevier]

Partial integro-differential equations occur in many fields of science and engineering. This study presents various numerical schemes for solving a partial integro-differential equation with a weakly singular kernel. These schemes are presented for determining the solution of a partial integro-differential equation subject to an initial condition and given boundary conditions. We construct several computational techniques based on finite-difference schemes and the product trapezoidal numerical integration rule. This problem can be found in the modelling of physical phenomena involving viscoelasticity forces. The finite-difference procedures developed are based on the forward Euler explicit scheme, the backward Euler implicit technique, the Crank–Nicolson implicit formula and Crandall’s implicit method. Three of the methods have second-order accuracy with respect to the space variable. The order of accuracy of the Crandall’s scheme is higher than that of the others. The numerical results of a test problem are given to verify the stability and accuracy of the procedures discussed. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the qualitative behavior of solutions to the difference equationwhere the initial conditions x −k ,…, x −1, x 0 are non-negative, k ∈{1,2,3,…}, and the parameters p, q are non-negative. We start by establishing the periodicity, the character of semicycles, the global stability, and the boundedness of the above mentioned equation. We also present solutions that have unbounded behavior. It is worth to mention that this difference equation is a special case of an open problem was introduced by M.R.S. Kulenovic and G. Ladas [Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, 2002]. Several computational examples are given to support our theoretical discussions. The presented numerical tests represent different types of qualitative behavior of solutions to our nonlinear difference equation. [Copyright &y& Elsevier]

Many physical phenomena are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary conditions have received much attention in the last twenty years. Most of the papers were directed to the second-order parabolic equation, particularly to the heat conduction equation. One could generically classify these problems into two types; boundary value problems with nonlocal initial conditions, and boundary value problems with nonlocal boundary conditions. We will deal here with the second type of nonlocal boundary value problems that is the solution of nonlocal boundary value problems with standard initial condition. The main difficulty in the implicit treatment of the nonlocal boundary value problems is the nonclassical form of the resulting matrix of the system of linear algebraic equations. In this paper, various approaches for the numerical solution of the one-dimensional heat equation subject to the specification of mass which have been considered in the literature, are reported. Several methods have been proposed for the numerical solution of this boundary value problem. Some remarks comparing our work with earlier work will be given throughout the paper. Numerical examples are given at the end of this paper to compare the efficiency of the new techniques. Some specific applications in engineering models are introduced. [Copyright &y& Elsevier]

In this work we are concerned with the solution of a nonlocal boundary value problem. An approach is presented for solving the two-dimensional parabolic partial differential equation subject to integral boundary specifications. The main objective is to propose an alternative method of solution, one not based on finite difference methods or finite element schemes or spectral techniques. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the two-dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions. The Adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques. The introduction of this idea as will be discussed, not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume. The solutions will be handle more easily, quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent power series with easily computed components. The Adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization. To give a clear overview of the methodology, we have selected illustrative example. [Copyright &y& Elsevier]

The study of advection–diffusion equation continues to be an active field of research. The subject has important applications to fluid dynamics as well as many other branches of science and engineering. In this paper several different numerical techniques will be developed and compared for solving the three-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level fully explicit and fully implicit finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. Another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties. The new second-order accurate methods are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared. [Copyright &y& Elsevier]

In this paper special attention is devoted to one particular problem, the diffusion equation subject to the specification of mass in a portion of the domain, which has been studied quite extensively, both analytically and numerically, in recent years. Parabolic partial differential equations with a non-local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena. In this paper the application of the method of lines (MOL) to such problems is considered. The MOL semi-discretization approach will be used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs). The MOL is a method of solving PDEs by discretizing the equation with respect to all but one variable (usually time). The spatial partial derivative is approximated by a finite-difference method. The solution of the resulting system of first-order ODEs satisfies a recurrence relation which involves a matrix exponential function. Numerical techniques are developed by approximating the exponential matrix function in this recurrence relation. The matrix exponential function is approximated by a rational approximation consisting of four parameters. New parallel algorithms are developed using the resulting approximation. Numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms. [Copyright &y& Elsevier]

This study presents several finite difference schemes for solving the three-dimensional parabolic equation with a control function. These techniques are developed for indentifying u=u(x,y,z,t) and p=p(t), the temperature parameter which satisfy ∂u/∂t=(∂2u/∂x2)+(∂2u/∂y2)+(∂2u/∂z2)+p(t)u+φ(x,y,z,t), in R×(0,T], u(x,y,z,0)=f(x,y,z), (x,y,z)∈R=[0,1]3, u is known on the boundary of R and also u(x0,y0,z0,t)=E(t), 0⩽t⩽T, where E(t) is known and (x0,y0,z0) is a given point of R. These schemes are considered for determining the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the forward time centred space fully explicit finite difference technique and the backward time centred space (BTCS) fully implicit formula, and the (7,7) Crank–Nicolson implicit finite difference scheme. These methods have second-order accuracy with respect to the space variable. The BTCS finite difference technique and the (7,7) formula are unconditionally stable. The results of a numerical experiment are also presented. [Copyright &y& Elsevier]

Parabolic partial differential equations with nonstandard initial condition, feature in the mathematical modelling of many phenomena. While a significant body of knowledge about the theory and numerical methods for parabolic partial differential equations with classical initial condition has been accumulated, not much has been extended to parabolic partial differential equations with nonstandard initial condition. In this paper finite difference approximations to the solution of one-dimensional parabolic equation with nonstandard initial condition are studied. Several finite difference schemes are presented for solving a parabolic partial differential equation with nonlocal time weighting initial condition. These schemes are based on the forward time centred space explicit formula, the backward time centred space implicit technique, the Crank–Nicolson implicit formula, the Crandall’s implicit scheme and the Saulyev’s (2,2) explicit methods. The results of a numerical experiment are presented. [Copyright &y& Elsevier]

Numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial-boundary value problem for the two-dimensional parabolic partial differential equation (PDE) with a non-linear boundary condition. This class of parabolic PDEs plays a very important role in many branches of science and engineering. The non-linear condition is in the form of a double integral giving the specification of mass in the solution domain. Not only the problem has both Neumann and Dirichlet boundary conditions but the Dirichlet boundary condition is in a non-standard form. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Due to the structure of the boundary conditions a reduced one-dimensional parabolic equation for the new unknown v(y,t)=∫10u(x,y,t) dx can be formulated. The resulting problem has a non-local boundary condition. The new two-dimensional parabolic PDE with Neumann’s boundary conditions will be solved numerically by using the method of lines semi-discretization approach. The space derivatives in the PDE are approximated by finite difference replacements. The solution of the resulting system of first-order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time is controlled by choosing several subdiagonal Pade approximants to replace this matrix exponential term. Numerical techniques are developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem from the literature. The article concludes with the results of some numerical experiments. [Copyright &y& Elsevier]

Numerical solution of the parabolic partial differential equations with an unknown parameter play a very important role in engineering applications. This study presents numerical schemes for solving two two-dimensional parabolic inverse problems, i.e. for the determination of some unknown functions in the parabolic partial differential equations. These schemes are developed for determining the parameter p(t) which satisfy ut=uxx+uyy+p(t)u+φ, in R×(0,T], u(x,y,0)=f(x,y),(x,y)∈R=[0,1]2. It is assumed that u is known on the boundary of R and subject to the integral overspecification over a portion of the spatial domain ∫01∫01u(x,y,t) dx dy=E(t), 0⩽t⩽T, or to the overspecification at a point in the spatial domain u(x0,y0,t)=E(t), 0⩽t⩽T, where E(t) is known and (x0,y0) is a given point of R. These schemes are considered for identifying the control parameter which produces, at any given time, a desired energy distribution in the spacial domain, or a desired temperature distribution at a given point in the spacial domain. The Saulyev’s finite difference techniques are used to compute the solution. Although the resulting difference equations do not appear explicit, a suitable use of the equations make them explicit. These explicit schemes are unconditionally stable. Numerical experiments are presented to demonstrate the accuracy of the finite difference schemes and the efficiency of the proposed computational procedures. [Copyright &y& Elsevier]