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In recent years the use of invariant imbedding in the solution of a variety of problems has been increasing. In this paper, application of this method to problems in heat conduction and radiation is demonstrated. No previous knowledge of invariant imbedding is assumed. The method is applied to several relatively simple problems. The initial value problem obtained by the method is numerically stable. Sample calculations are presented which demonstrate the accuracy of the algorithm.

In this paper, it is shown that the method introduced recently by Kalaba and Scott for solving nonlinear characteristic value problems for integral operators is equivalent to a nonlinear characteristic value problem for a matrix. This is demonstrated by finding the explicit solution to the set of differential equations used by them. The results generalize those proved recently by the author for the linear problem.

A novel general direct iteration algorithm for the evaluation of branching points of nonlinear boundary-value problems is suggested. The method proposed takes advantage of the GPM concept. The analysis presented in the paper encompasses also the determination of branching points in nonlinear algebraic equations. The method can be applied to problems arising in a number of physical and mathematical applications. The technique is tailored to problems of diffusion and heat conduction accompanied by a chemical reaction.

In this paper we describe the solution of a rather intractable differential eigenvalue problem arising from a stability problem in fluid dynamics. Techniques used for solving the problem include finite differences, a variety of shooting methods and Riccati transformations (invariant imbedding). A description is given of the difficulties encountered. The invariant imbedding technique was found to be the most efficient method for obtaining the solution accurately. Though our results are for one difficult problem, we anticipate that our conclusions will be widely applicable to problems of similar type.

This paper deals with the following problems: (1) the development of a general theory which incorporates several methods as special cases; (2) the applicability of moment methods to a class of nonlinear problems, (3) the specification of the class of admissible weighting functions; (4) the estimation of the number of parameters; (5) the elimination of the cutoff error in the analysis of fluorescence decay data.

A code has been developed which will automatically locate and analyze points of Hopf bifurcation in autonomous ordinary differential systems. The code first locates critical value(s) v"c of a user-specified parameter v (the bifurcation parameter) such that a stationary (equilibrium) solution x"*(v) loses linear stability by virtue of a complex conjugate pair of eigenvalues. The code computes x"*(v) during the location of v"c. Then the code computes the various coefficients in a local approximation to the family of periodic solutions which arise, a process which involves computation of second and third partial derivatives by numerical differencing of the user-supplied Jacobian matrix. The current version of the code, called BIFOR2, is fully described in Hassard, Kazarinoff, and Wan, Theory and Applications of Hopf Bifurcation, Cambridge U.P., 1981. In this paper we demonstrate the code in applications to two systems drawn from chemical reactor theory. The first application is to a 4th order ordinary differential system modeling a coupled tank reactor. The second application is to a partial differential system modeling a catalyst particle system. These represent the first applications of the code to chemically reacting systems other than the Brusselator. The second application demonstrates how collocation methods may be used in conjunction with BIFOR2 to perform Hopf bifurcation analysis of partial differential systems.

The steady-state equations of inviscid fluid flow, the Euler equations, are a nonlinear nonelliptic system of equations admitting solutions with dis continuities (for example, shocks). The efficient numerical solution of these equations poses a strenuous challenge to multigrid methods. A multigrid code has been developed for the numerical solution of the Euler equations. In this paper some of the factors that had to be taken into account in the design and development of the code are reviewed. These factors include the importance of choosing an appropriate difference scheme, the usefulness of local mode analysis as a design tool, and the crucial question of how to treat the nonlinearity. Sample calculations of transonic flow about airfoils will be presented. No claim is made that the particular algorithm presented is optimal.

In the case of degenerate covariance an initial-value solution of the optimal estimate of a signal in the presence of white Gaussian noise can be evaluated with the aid of a Cauchy system for a matrix Riccati differential equation, whose nonlinear form is convenient for computer-aided determination. In the present paper, with the aid of invariant imbedding, we show how to get an initial-value solution ofthe impulse response function in the case of degenerate covariance. Furthermore, it is also shown that a real-time solution of the linear least-squares filtering problem is given by the Cauchy system for the newly introduced function, which is a time integral of the stochastic process weighted by the auxiliary function.

In this paper we study the periodic boundary value problem for first order differential equations by combining techniques of the theory of differential inequalities, namely the method of upper and lower solutions, and the alternative method for nonlinear problems at resonance. The results obtained are in terms of the behavior of the nonlinear part at infinity.

In this paper, we proposed a new numerical scheme to solve the time fractional nonlinear KleinGordon equation. The fractional derivative is described in the Caputo sense. The method consists of expanding the required approximate solution as the elements of Sinc functions along the space direction and shifted Chebyshev polynomials of the second kind for the time variable. The proposed scheme reduces the solution of the main problem to the solution of a system of nonlinear algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces accurate results.

In this paper, two methods based on the symmetric marching technique (SMT) are presented for the solution of elliptic equations over nonrectangular regions. Method I illustrates the direct adaptation of the SMT to irregular geometries. In method II, an efficient implementation of the capacitance matrix method has been considered using SMT. The favorable characteristics of the SMT for solving the Poisson equation with several right hand side functions and different boundary conditions without extra computational effort have been exploited for the last generation of the capacitance matrix. The successful application of the SMT combined with quasilinearization to solve mildly nonlinear elliptic equations is also described. Several test examples have been solved to demonstrate the efficiency of the proposed methods.

This paper is concerned with the existence and uniqueness of solutions of reaction-diffusion equations arising in many applications and often coupled. Our approach to the problem is based on converting the coupled equations into a system of Volterra-Fredholm type integral equations by means of the Green's function representation and using the Banach fixed point theorem. An application to nonlinear second order evolution equations is also indicated.

Various properties of nonlinear difference equations are analysed within the framework of a model of endogenous preference change. Theorems by Sarkovskii and Li and Yorke show that simple first order nonlinear difference equations exhibit a wide range of dynamical behavior: Stable points, a hierarchy of stable cycles, unstable cycles, and an uncountable set of aperiodic points. The economic analysis in this paper focuses in particular on the sequence of stable cycles and questions about rationality and consistent behavior over time within a structure of utility maximization.

Equidistribution of a weight function over a mesh is the main concept employed in the recent development of adaptive grid schemes. This idea is reviewed in this paper, and several examples of grids produced using equidistribution are presented. The shortcomings of this approach are identified, and its usefulness is evaluated. Next, the traditional Poisson grid generators are written as nonlinear equidistribution schemes. In this form, the relationship between the numerical solution of a physical problem, the source terms in the grid generation equations, and the weight functions in equidistribution schemes are identified. Examples are given illustrating the case of providing a useful adaptive grid with this approach.

Unlike uniform stability, the linearized system inherits the property of uniform Lipschitz stability from its original nonlinear system. In this paper we investigate sufficient conditions for uniform Lipschitz stability of nonlinear integro-differential systems through their associated variational systems.

This paper gives a parallel computing scheme for minimizing a twice continuously differentiable function with the form @?f(x) = @?i = [email protected]?"i(x"i) + @?i = [email protected]?j = 1(j > i)m @?"i"j(x"i, x"j),where x = (x^T"1,...,x^T"m)^T and x"i @? R^n^"^i, @?^m"i" "=" "1n"i = n, and n a very big number. It is proved that we may use m parallel processors and an iterative procedure to find a minimizer of @?(x). The convergence and convergence rate are given under some conditions. The conditions for finding a global minimizer of @?(x by using this scheme are given, too. A similar scheme can also be used parallelly to solve a large scale system of nonlinear equations in the similar way. A more general case is also investigated.

Let f: X->X be an order convex operator, where X is partially ordered Banach space. The purpose of this paper is to describe a method which, under frequently satisfied hypotheses, permits two-sided bounds on zero of f to be obtained using any member of a wide class of procedures for determining one-sided bounds. The method is illustrated for systems of nonlinear algebraic equations which arise from the discretization of nonlinear two-point boundary-value problems. Numerical results for two such systems are given.

The paper generalizes the Eckalbar inventory model by introducing a nonlinear adjustment function of production. We especially concern ourselves with the existence of limit cycles in the system. It is shown that economic fluctuations can be created endogeneously, rather than by switching the control equation of the economic system. The Hopf bifurcation theorem is applied to establish the existence of limit cycles.

In this paper we extend Levin and Sidi's nonlinear D- and d-transformations for accelerating convergence of infinite integrals and series. The simple extension is made by viewing all these transformations as generalized jackknifes. The new transformations appear to be more accurate in general and are as easily implemented as D- and d-transformations. The new transformations have also some computational advantages. As a result more accurate approximations can be obtained in many cases, as is shown in the examples which are included.

This paper is concerned with nonlinear integer programming problems- unconstrained and constrained as well as mixed. The problems are transformed into nonlinear global optimization problems and then solved by the filled function transformation method. This approach should be efficient, since it has been shown that the filled function transformation method is efficient in solving global optimization problems with large numbers (up to 30^2^5 in the present examples) of local minimizers. However, the sense of ''efficient'' as ''having polynomial complexity in the worst or average case'' is not suitable for nonlinear integer programming problems, since the complexity of a nonlinear programming algorithm is in general nonpolynomial.

Requirements for the evaluation of high-order partial derivatives arise in many applications, including optimal control, system identification, and the numerical solution of nonlinear algebraic equations, two-point boundary-value problems, and integral equations. These evaluations have long been shunned by numerical analysts because of the difficulty in forming symbolic expressions for the derivatives. This paper describes in detail an implementation in ADA of a set of simple procedures for evaluating derivatives exactly, without the requirement to form their symbolic expressions and without resorting to finite-difference methods. An expression whose derivatives are desired is written in the usual way, using standard mathematical operators and intrinsic functions, and values of all desired derivatives are returned automatically. Two examples of parameter identification are shown, one from economics and one from guidance and control. A complete listing of all ADA code is included.

E.R. Hansen and R.I. Greenberg have presented an interval Newton method to solve a system of n nonlinear equations in n real variables. Following R.E. Moore, they have linearized the system using a mean value expansion, and used preconditioning technique due to Hansen and R.R. Smith to modify the system. The modified system is subjected to a Hansen-Sengupta step to obtain an updated interval containing the solution. Hansen and Greenberg have in fact used the subalgorithms of preconditioning and Hansen-Sengupta step witha real (local, noninterval) iteration and an elimination procedure to provide an algorithm of greater efficiency. In this paper, we indicate procedures which will further improve the efficiency of the Hansen-Greenberg algorithm. Firstly, a better approximating matrix for the identity is obtained. Secondly, a successive overrelaxation (SOR) technique is introduced to replace the Hansen-Sengupta step if necessary. Finally, an interval iteration is suggested to provide an alternative to the real (noninterval) inner iteration introduced by Hansen and Greenberg. Examples are included to show improvement in results and/or the forms the new procedures would take. the additional procedures provide more efficient results besides improving the techniques.

This paper is addressed to an old but difficult topic, viz. how to find more and more solutions of a system of nonlinear equations. We find a transformation by using so-called locally affective functions to transform the original system to a new system, which has not the already found solutions of the original system, but only the unfound solutions, and so we can find a new solution of the original system through finding a solution of the transformed system. We also discuss some likely uses of locally affective functions in the solution of other nonlinear problems.

The paper presents a new algorithm to determine Volterra kernels of discrete nonlinear systems. The algorithm is error-free, i.e., it does not introduce calculation errors into the solution. Its derivation is based on a polynomial algebra concept, which permits one to minimize the number of necessary calculation steps.

Standard homotopy continuation methods for solving systems of nonlinear equations require the continuation parameter to move from 0 to 1 along the real line. Difficulties can occur, however, if a point of singularity is encountered during the course of the integration. To ameliorate these difficulties, this paper proposes extending the continuation parameter to complex values and adaptively computing a continuation path in the complex plane that avoids points giving rise to singularities. Specifically, it is proposed that the continuation parameter move from 0 + 0i to 1 + 0i along a spider-web grid centered at 1 + 0i in the complex plane. The actual path through the grid is determined step by step in accordance with two objectives: short path length, and avoidance of singular points. A two-phase homotopy continuation is used to study the implementation of this idea. Numerical examples are presented which indicate the effectiveness of the approach.

Several sophisticated and efficient algorithms are now available in the literature for solving nonlinear function inequalities in a finite number of iterations. This paper addresses an alternative approach to this class of problems. The approach is relatively straightforward and easy to use. Essentially, the nonlinear inequality constrained problem is reformulated as a standard unconstrained optimization problem via a differentiable transcription. Thus, any existing efficient unconstrained optimization software packages can be used to solve the corresponding unconstrained optimization problem. Due to the special structure of the differentiable transcription, a solution of the nonlinear inequality constraints can be obtained after a finite number of iterations in the process of solving the unconstrained optimization problem. The main aim of this paper is, however, to extend the approach to solving nonlinear functional, rather than function, inequalities in a finite number of iterations. For illustration, several examples are computed to demonstrate the feasibility and versatility of the approach.

This paper presents iterative methods for nonlinear differential systems with finite delay. The basic idea of the iterative method is the monotone approach, which involves the notion of upper and lower solutions and the construction of monotone sequences. This approach provides constructive proof of existence as well as numerical procedures for computation of solutions.

This paper revives a familiar idea for use in solving optimization problems, particularly unstructured nonlinear programs with many variables. The idea involves partitioning variables into groups and successively performing a sectional search with respect to one group at a time. The paper addresses various difficulties normally associated with this type of procedure, and proposes ways to solve them. These include testing optimality of the point at termination, and developing ways to restart the process upon finding that the terminating point is not optimal. The paper also briefly discusses some useful implications of the results on how to group variables in unstructured nonlinear programs so as to improve convergence.

Dayawansa and Martin have shown in a previous paper that several classes of nonobservable linear systems can be continuously observed by nonlinear observers. In this paper we show that a modification of their polynomial output function may be used so as to be able to observe continuously certain linear systems. Then we show that the corresponding problem with discrete sampling of the output may be solved by solving for the zeros of a certain functions, F:R^n->R^n, constructed from the sampling data, by use of continuation methods.

The areas of modern control, estimation, and systems identification theory often result in the need for solution of nonlinear two-point boundary-value problems. These problems are generally difficult to handle both analytically and computationally. Due to the split boundary conditions, a simple integration is not possible, and an iterative technique is time-consuming and often complicated to solve. High-speed computational approaches will often enable solutions if the algorithm to be processed is efficient and effective with regard to computer solution time. The invariant-imbedding procedure is an approach whereby the missing initial (or terminal) conditions are obtained in a direct manner. The original problem is thereby effectively reduced to an initial-value problem. Comprehensive discussions of the principle of invariant imbedding in the context of control-systems engineering first appeared in various works by Bob Kalaba, Richard Bellman, and others. In this paper, we provide an overview of these contributions, together with some extensions in the area of systems control, estimation, and identification.

An account is given of how a relationship was found between some equations of Chandrasekhar for solving the Wiener-Hopf integral equations and the nonlinear Riccati differential equations encountered in linear least-squares estimation problems for systems described in state-space form. A paper by Casti, Kalaba, and Murthy (1972) provided the inspiration for seeking such a relationship and then extending it and applying it in many other fields, using the concept of displacement structure.

In the present paper it is shown that, assuming the stochastic process of particle diffusion in plane Couette flow and furthermore making use of the modified Chapman-Kolmogorov equation, the source function is readily evaluated by solving the Cauchy system of a nonlinear integrodifferential equation, whose form is similar to the Kolmogorov-Feller equation. The initial value is expressed in terms of the scattering function, whose numerical value can be computed by solving a Riccati-type nonlinear integrodifferential equation. Once the source function has been determined, we can compute successively the one-dimensional perturbation velocity distribution function, mass density, flow velocity, temperature, and other quantities. It is thus shown how such significant physical quantities in the linearized plane Couette flow can be readily evaluated by computing the lower-order moments of the distribution functions.

We recently formulated and solved a successful model of the renal concentrating mechanism, a problem of long-standing interest in mathematical biology [33, 34]. The model is a 39th-order nonlinear two-point boundary-value problem with separated linear boundary conditions. Although quasilinearization and superposition with Gram-Schmidt orthonormalization successfully solved the equations, these methods will probably not be sufficient to solve more detailed models of the urine concentrating mechanism. In this paper we review the current literature on solving two-point boundary-value problems and find a number of promising alternatives and improvements to our current methods. In addition we derive a new numerical method for calculating the parameter sensitivity of the solution to a nonlinear two-point boundary-value problem coupled to a system of algebraic relations. This new method is faster and more accurate than the method previously employed.

Differential equations of the form y^(^n^)=f (t, y, y',..., y^(^n^), where f is not necessarily linear in its arguments, represent certain physical phenomena (e.g., free oscillations of positively damped systems, nonlinear systems subject to harmonic excitations, motion of a particle on a rotating parabola) and have been known for quite some time. Earlier we established the existence of a unique solution of the nonstandard first order initial value problem y' =f (t, y, y'), y(t"0) = y"0 under certain natural hypotheses on f and developed some linear and quadratic convergent numerical schemes for the construction of approximate solution of the above problem. In this paper we establish existence results and develop some linearly convergent numerical schemes for the construction of solution of nonstandard nth order initial value problems, and we solve two physical examples.

Most activities, or events, in discrete event simulation models are probabilistic in nature and describable by density functions. Consequently, persons developing simulation models are frequently required to select and fit density functions to sample data and, correspondingly, to determine the process, or random-event, generator. This paper seeks to establish that the graduation process involved with simulation modeling may be expedited: generalized Weibull functions numerically fitted by nonlinear least-squares or maximum-likelihood procedures are comparable in performance to the fits of more traditional functions. The use of a single function simplifies the ambiguous process of function selection, while offering, in addition, the important advantages of a closed-form inverse process generator.

In this paper, the successive overrelaxation iteration and the variation of parameters formula are used to discuss the stability of linear and nonlinear Volterra integrodifferential equations. Some sufficient conditions are obtained, such that the trivial solution of the Volterra integrodifferential equations is asymptotically stable; these stability conditions are given directly from the coefficients of the equations.

Previously we have shown that for a particular choice of origin point data, the nonintegrable Aesthetic Field Equations collapse into a simpler set of equations, called the A, B, J, L equations. These equations lead to sinusoidal variation along any path segment for all the field quantities. In this paper we find, for a different choice of origin point data, that the Aesthetic Field Equations simply into a nonlinear system that describes a sine curve within a sine curve along any path segment. These results are obtained from visual inspection of the computer plots as well as numerical fitting to the data. We examine two dimensional maps, first when we specify an integration path, and then when we make use of a superposition principle at each point. The superposition principle arises as a consequence of the theory of nonintegrable systems, which we have developed on an earlier occasion. We find two dimensional maps in both instances that do not appear regular in the small. However, in both cases, we were able to observe large scale regularities. In our studies, we make use of computer techniques that focus on the large wavelength oscillations (big picture) as well as techniques that focus on the small wavelength oscillations (small picture). When the superposition principle is used, the small as well as big oscillations no longer have a sinusoidal appearance for y equals constant lines. This contrasts with the case when we specify a path where we see a sine curve within a sine curve along these y equals constant lines. In addition, we obtain a set of linear equations that describes a sine curve within a sine curve that can be considered in its own right. In this case, in addition to numerical integrations, we show analytically that a sine curve within a sine curve is a solution to the equations.

The paper first points out the connection between the problem of finding a set of Pareto-optimal paths in the presence of multiple criteria and the problem of finding the optimal path in the presence of a single multiattribute criterion. A survey of literature indicates that the former problem has received much attention and several exact (but nonpolynomial) and approximate algorithms are available for finding all and nearly all Pareto-optimal paths, respectively. The latter problem of finding the optimal path that minimizes a nonlinear cost function of multiple attributes has received less attention. The paper examines the properties of the optimal path when the cost function is monotonic and concave in the attributes, especially how it relates to the set of “efficient” paths within the nondominated set. When the cost function in convex in two attributes, a line-search algorithm is developed that finds a good, if not optimal, path without using any assumptions or information on the derivatives of the cost function.

In this paper, a computation method for the nonlinear problem with a small perturbation parameter via domain decomposition is presented. We apply a combination of the two approaches to solving the problem: iterative algorithms for domain decomposition and a grid refinement technique on subdomains. Two iterative algorithms are examined: the first one is the Schwarz alternating procedure and the second algorithm is highly suitable for parallel computing. Numerical examples are provided.

Several new and significant observations are presented pertaining to the classical problem of single-population growth with hereditary influences. In its conventional form, the resulting equation with heredity is mathematically represented by a nonlinear Volterra integro-differential equation. In this paper, we propose a new differential formulation where the dependent variable is now defined in terms of the integral of the unknown population. This formulation allows us to develop novel analyses leading to enlightening results. Some particular findings include: the development and analysis of an integrated phase-plane; the elucidation of the exact value for the extremum of the population and several other important functional relations at that corresponding time; the development of two analytic expressions for determining the time at which the population peaks; the determination of the upper asymptote for the cumulative population; and the development of an accurate early-time solution as obtained from a Riccati equation. Additionally, we illustrate that an analytical solution, based on Taylor Series expansions, can be developed with the aid of Mathematica TM . A pure numerical solution is offered for comparison with the analytic solution.

In this paper, we consider a class of nonlinear regulator optimal control problems with an infinite planning horizon. By assuming a specific feedback structure for the controller, the original problem is reduced to a constrained optimal parameter selection problem. Because the reduced problem is on an infinite planning horizon, we construct a sequence of finite time optimal parameter selection problems in which each is readily solvable using existing numerical software. The question concerning the asymptotical stability of the dynamical system under the obtained optimal feedback controller is then investigated. For illustration, two numerical examples are considered.

In this paper, we present a nonlinear trapezoidal formula for the solution of the Goursat problem. The new scheme implements the harmonic mean (HM) averaging of the functional values rather than the arithmetic mean (AM) or the geometric mean (GM) averaging. A comparison is made with the existing techniques, and the results obtained show better approximations related to the accuracy level in favor of the HM strategy.

This paper is devoted to the study of structure variable homotopy methods for solving nonlinear systems. A general structure variable homotopy algorithm is described in Section 2 and its relation with Newton's method is indicated. It is proved that a modified structure variable homotopy algorithm, called the descent structure variable homotopy, or DSVH, algorithm, converges globally and quadratically in the neighborhood of the solution under the hypothesis of the nonsingularity of the nonlinear systems. In Section 4, another structure variable homotopy algorithm is developed and the global convergence is also given. Finally, three numerical examples are given to demonstrate the effectiveness of our algorithms.

In this paper, we study the oscillatory behavior of solutions of the forced second-order nonlinear ordinary functional differential equation x + φ(t)g(x(t)) = p(t) , ·= d dt Sufficient conditions are established for all solutions of the equation to be oscillatory as well as bounded.

In this paper, we shall present two new nonlinear variation of parameters formulae for the summary difference equations in several independent variables. A corrected variation of parameters formula for the corresponding integrodifferential equation is also provided. Some applications that dwell upon the importance of these new results are also included.

This paper deals with the Cauchy problem for nonlinear impulsive partial differential equations of first order. Theorems on impulsive differential inequalities are obtained. Comparison results implying uniqueness criteria are proved.

When a real homotopy is used for solving a system of nonlinear equations of complex variables, especially polynomial systems, bifurcation at singular points is inevitable. In this paper, the notion of a k th order turning point, a singular point with k bifurcation branches, is introduced. It is shown that the tangent lines of those k solution paths at a k th order turning point lie in the same one-dimensional complex plane and equally divide this plane. This result generalizes the phenomena taking place at a quadratic turning point where tangeant lines of the solution paths, only two of them, are perpendicular to each other. Moreover, our result provides a condition to verify the true multiplicity of a solution when several homotopy paths converge to it.

We consider a (0, 2) lacunary interpolation problem, with prescribed nonlinear endpoint conditions, solving it in the class of quartic splines of deficiency 2. Under suitable assumptions, we show existence and uniqueness of the solution. We provide a convergence analysis, showing that the method is of order four. These results are then applied to a two-point boundary value problem. If the latter is solved with sufficiently high accuracy, we show that the smoothing method based on the first part of the paper is fourth-order accurate.

This paper considers the design problem of adaptive filters based on the state-space models for linear discrete-time stationary stochastic signal processes. The adaptive state estimator consists of both the predictor and the sequential prediction error estimator. The discrete Chandrasekhar filter developed by author is employed as the predictor and the nonlinear least-squares estimator is used as the sequential prediction error estimator. Two models are presented for calculating the parameter sensitivity functions in the adaptive filter. One is the exact model called the linear innovations model and the other is the simplified model obtained by neglecting the sensitivities of the Chandrasekhar X and Y functions with respect to the unknown parameters in the exact model.

In this paper, the Adomian's decomposition method is effectively implemented to establish an analytic solution and a reliable numerical approximation to the Goursat partial differential equation. Linear and nonlinear Goursat examples are examined, and the method supplied quantitatively reliable results for these types of problems. The accuracy level of the results obtained indicates the superiority of the decomposition method over existing numerical methods that were applied to the Goursat problem.