Abstract. In this work, arecently developed semi-analytic technique, so called the residualpower series method, is generalized to process higher-dimensional linear andnonlinear partial differential equations. The obtained solution takes a form ofan infinite power series which can, in turn, be expressed in a closed exactform. The results reveal that the proposed generalization is very effective,convenient and simple.
This isachieved by handling the -dimensional Burgers equation.Key words. -dimensional nonlinear partial differential equations;Generalized residual power series method; Convergence analysis; Exact solution;Burgers equation.1.
Introduction Over the last four years, arecent developed technique, namely the residual power series method (RPSM), forsolving linear and nonlinear ordinary differential equations of integer andfractional orders have been proposed 1-4. In the current work, we improvethis method to process linear and nonlinear partial differential equations(NPDEs) of higher dimensional and orders.Due to their broad varietyof relevance, comparative studies, using the Adomian decomposition method,homotopy perturbation method, variational iteration method, differentialtransform method and its reduction, were discussed deeply to tackle higherdimensional Burgers type equations 5-7. 2. The GPSMConsider the -dimensional,th – order NPDE in general form, , . 1)Where is assumed to besufficiently smooth on the indicated domain contains the starting values. represents the th derivative of the analytic function with respect toindependent variable , and in the same way for other independent variables.The generalized RPSM(shortly GRPSM) assumes the solution , of Eq.
(1), in a form of power series. 2)In more compact form, , 3)where and .Subject to the initialconditions, , … , , 4)the initial approximation of will be. 5)The th-order approximate solution is defined bythe truncated series, . 6)subject to . 7)Where is the well-defined th- residual function defined by, 8)which represents the basicidea of the GRPSM.
The exact analytic solution of the initial-value problem,Eq.(1) and Eq.(4), is given by. 9)Provided that the series hasexact closed form.3. Convergence Analysis In this part, convergenceanalysis and error estimating for using GRPSM are studied. The presentedscheme, under considerations mentioned in the previous section, approaches theexact analytic solution as more and more terms found.Theorem 1.
If is an analyticoperator on an open interval containing , then the residual function vanishes as approaches theinfinity.Proof: Since isassumed to be analytic as well as for . It is obvious by the definition of residual functionEq.(7).
Lemma 1. Suppose that , then, . 10)Proof: The th-derivative of with respect to is continuous at . Therefore ,.Theorem 2. The approximate truncated series solution defined in Eq.
(6) andobtained by applying the GRPSM for solving the Eqs.(1) and (4) is the th Taylor polynomial of about . In general, as , the series solution in Eq.(9) concise the Taylor seriesexpansion of centered at .Proof: For , it is clear from the initial approximation of in Eq.
(5). For , it suffices to prove that . Applying Eq.
(7) to the th-order approximate solution given in Eq.(6) ,and using the result of Theorem 1, we getAs a result of Lemma 1, .Which completes the proof.Corollary 1. Suppose that the truncated series defined in Eq.
(6) is used as anapproximation to the solution of problem Eqs.(1)- (4) on ,then numbers , satisfies , and exist with.Proof: Theorem 1 implies that.
Followingthe proof of Taylor’s Theorem 8, a number exists with.Sincethe st-derivative of the analytic function with respect to is bounded on , a number also exists with for all . Hence the result.
Corollary 2. The GRPSM results the exact analytic solution if it is a polynomial of .4. Numerical Illustration To illustratethe technique discussed in Sections 2, we consider the ()-dimensional nonlinear Burgers’ equation 9-10 , , 11)subject to the initial condition. 12)Eq. (11) is alsoknown as Richard’s equation, which is used in the study of cellular automata,and interacting particle systems. It describes the flow pattern of the particlein a lattice fluid past an impenetrable obstacle; it can be also used as amodel to describe the water flow in soils.Applying thegeneralized residual power series mechanism to suggested problem, the initialapproximation is , and the th-order approximate solution has the form, , 13)whichsatisfies.
14)For , we have and.Hence we get and therefore . Repeating this procedure for , we obtain that .As , the solution takes the form.The seriessolution leads to the exact solution obtained by Taylor’s expansion.5.
ConclusionsIn this work, we have improved an analyticsolution procedure, called the generalized residual power series method, forsolving higher dimensional partial differential equations. The resultsvalidate the efficiency and reliability of the aforesaid technique thatare achieved by handling the (m+1)-dimensional Burgers equation. The method is a powerful mathematical tool forsolving a wide range of problems arising in engineering and sciences.References1. O. Abu Arqub, Z.
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