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Differential Transformation Method and Variation Iteration Method for Cauchy Reaction-diffusion Problems Differential Transformation Method and Variation Iteration Method for Cauchy Reaction-diffusion Problems en en In this chapter, we will compare the differential transform method (DTM) and variational iteration method (VIM) for solving the one-dimensional, time dependent reaction-diffusion equations. Different cases of the equation are discussed and analytical solution in series form can be derived. The results obtained by the proposed method (DTM) are compared with the results obtained by (VIM). Some examples are presented to show the ability of the methods for such problems. 61 75 Mohamed I. A. Othman A. M. S. Mahdy Differential transformation Variation iteration method Cauchy reaction- diffusion problems Taylor’s series expansion. Article.1.pdf  M. A. Abdou, A. A. Soliman, Variational iteration method for solving Burger's and coupled Burger's equations, J. Comput. Appl. Math., 181 (2005), 245-251 ## E. M. Abulwafa, M. A. Abdou, A. A. Mahmoud, The solution of nonlinear coagulation problem with mass loss, Chaos Solitons Fractals, 29 (2006), 313-330 ## A. Arikoglu, I. Ozkol, Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput., 168 (2005), 1145-1158 ## A. Abbasov, A. R. Bahadir, The investigation of the transient regimes in the nonlinear systems by the generalized classical method, Mathematical Problems in Engineering, 5 (2005), 503-519 ## F. Ayaz, Solutions of the systems of differential equations by differential transform method, Appl. Math. Comput., 147 (2004), 547-567 ## F. Ayaz, Applications of differential transform method to differential-algebraic equations, Appl. Math. Comput., 152 (2004), 649-657 ## I. H. Abdel-Halim Hassan, On solving some eigenvalue problems by using differential transformation, Appl. Math. Comput., 127 (2002), 1-22 ## I. H. Abdel-Halim Hassan, Different applications for the differential transformation in the differential equations, Appl. Math. Comput., 129 (2002), 183-201 ## I. H. Abdel-Halim Hassan, Differential transformation technique for solving higher-order initial value problems, Appl. Math. Comput., 154 (2004), 299-311 ## I. H. Abdel-Halim Hassan, Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos Solitons Fractals, 36 (2008), 53-65 ## I. H. Abdel-Halim Hassan, Application to differential transformation method for solving systems of differential equations, Appl. Math. Model., 32 (2008), 2552-2559 ## I. H. Abdel-Halim Hassan, Vedat Suat Ertrk, Applying differential transformation method to the one-dimensional planar bratu problem, Int. J. Contemp. Math. Sci., 2 (2007), 1493-1504 ## I. H. Abdel-Halim Hassan, M. I. A Othman, A. M. S Mahdy, Variational iteration method for solving twelve order boundary value problems, Int. J. Math. Anal., (), - ## J. Biazar, H. Ghazvini , He’s variational iteration method for solving linear and non-linear systems of ordinary differential equations, Appl. Math. Comput., 191 (2007), 287-297 ## N. Bildik, A. Konuralp , The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 65-70 ## C. L. Chen, S. H. Lin, C. K. Chen, Application of Taylor transformation to nonlinear predictive control problem, Appl. Mathe. Model., 20 (1996), 699-710 ## C. K. Chen, S. H. Ho, Application of differential transformation to eigenvalue problems, Appl. Math. Comput., 79 (1996), 173-188 ## C. K. Chen, S. H. Ho, Solving partial differential equations by two dimensional differential transform method, Appl. Math. Comput., 106 (1999), 171-179 ## C. L. Chen, Y. C. Liu, Solution of two point boundary value problems using the differential transformation method, J. Optim. Theory Appl., 99 (1998), 23-35 ## P. A. Ebadian, A method for the numerical solution of the integro-differential equations, Appl. Math. Comput., 188 (2007), 657-668 ## A. Golbabai, M. Javidi, A variational iteration method for solving parabolic partial differential equations, Comput. Math. Appl., 54 (2007), 987-992 ## J. H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg., 167 (1998), 69-73 ## J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 167 (1998), 57-68 ## J. H. He, Variational theory for linear magneto-electro-elasticity, Int. J. Nonlinear Sci. Numer. Simul., 2 (2001), 309-316 ## J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114 (2000), 115-123 ## J. H. He, A new approach to nonlinear partial differential equations, Comm. Nonlinear Sci. Numer. Simul., 2 (1997), 203-205 ## J. H. He, A variational iteration approach to nonlinear problems and its applications, Mech. Appl., 20 (1998), 30-31 ## J. H. He, A generalized variational principle in micromorphic thermoelasticity, Mech. Res. Comm., 32 (2005), 93-98 ## J. H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B, 20 (2006), 1141-1199 ## J. H. He, X. H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos Solitons Fractals, 29 (2006), 108-113 ## J. H. He, Variational iteration method. Some recent results and new interpretations, J. Comput. Appl. Math., 207 (2007), 3-17 ## J. H. He, Variational iteration methoda kind of nonlinear analytical technique: Some examples, Int. J. Nonlinear Mech., 34 (1999), 699-708 ## M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, Variational Methods in the Mechanics of Solids, 1978 (1978), 156-162 ## M. J. Jang, C. L. Chen, Y. C. Liy, On solving the initial value problems using the differential transformation method, Appl. Math. Comput., 115 (2000), 145-160 ## M. J. Jang, C. L. Chen, Y. C. Liy, Two-dimensional differential transform for partial differential equations, Appl. Math. Comput., 121 (2001), 261-270 ## M. J. Jang, C. L. Chen, Analysis of the response of a strongly non-linear damped system using a differential transformation technique, Appl. Math. Comput., 88 (1997), 137-151 ## M. Javidi, A. Golbabai, Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos Solitons Fractals, Chaos Solitons Fractals, 36 (2008), 309-313 ## D. Lesnic, The decomposition method Cauchy reaction-diffusion problems, Appl. Math. Lett., 20 (2007), 412-418 ## H. Liu, Y. Z. Song, Differential transform method applied to high index differential algebraic equations, Appl. Math. Comput., 184 (2007), 748-753 ## S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A, 355 (2006), 271-279 ## S. Momani, S. Abusaad, Application of He's variational-iteration method to Helmholtz equation, Chaos Solitons Fractals, 27 (2006), 1119-1123 ## V. Marinca, An approximate solution for one-dimensional weakly nonlinear oscillations, Int. J. Nonlinear Sci. Numer. Simul., 3 (2002), 107-120 ## Z. M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 27-36 ## A. A. Soliman, Numerical simulation of the generalized regularized long wave equation by He's variational iteration method, Math. Comput. Simulation, 70 (2005), 119-124 ## A. M. Wazwaz, A comparison between the variational iteration method and Adomian decomposition method, J. Comput. Appl. Math., 207 (2007), 129-136 ## A. M. Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Appl., 54 (2007), 895-902 ## J. K. Zhou, Differential transformation and its applications for electrical circuits, Huarjung University Press, China (1986)
Basic Unary Transformations and Functions Operating in Fuzzy Plane Basic Unary Transformations and Functions Operating in Fuzzy Plane en en In this paper first a series of basic transformation such integral, Rising and Falling has been defined. then the integrals have been proved. So falling and rising planes have been studied and a theorem about it has been proved. At the end, operations fuzzy time planes is shown and related proposition to it is proved. 76 79 A. Taleshian S. Rezvani fuzzy plane Y-function operations fuzzy time planes Extend Shift Exp Integrate. Article.2.pdf  H. J. Ohlbach, Modelling periodic temporal notions by labelled partitionings of the real numbers, University of Munich, 2004 (2004), 1-42 ## H. J. Ohlbach, Calendrical calculations with time partitionings and fuzzy time intervals, in: Principles and Practice of Semantic Web Reasoning, 2004 (2004), 118-133 ## H. J. Ohlbach, Fuzzy time intervals and relations–the FuTIRe library, Institute for Computer Science, Munich (2004) ## H. J. Ohlbach, Relations between fuzzy time intervals, 11th International Symposium on Temporal Representation and Reasoning, 2004 (2004), 44-51 ## H. J. Ohlbach, The role of labelled partitionings for modeling periodic temporal notions, 11th International Symposium on Temporal Representation and Reasoning, 2004 (2004), 60-63 ## F. Bry, B. Lorenz, H. J. Ohlbach, S. Spranger, On Reasoning on Time and Location on the Web, in: Principles and Practice of Semantic Web Reasoning, 2003 (2003), 69-83 ## J. F. Allen, Maintaining knowledge about temporal intervals, Communications of the ACM, 26 (1983), 832-843 ## F. Baader, D. Calvanese, D. M. Guinness, D. Nardi, P. P. Schneider, The description logic handbook: Theory, implementation and applications, Cambridge university press, Cambridge (2003) ## B. T. Lee, M. Fischetti, Weaving the Web: the original design and ultimate destiny of the World Wide Web by its inventor, Harper, San Francisco (1999) ## D. R. Cukierman, A Formalization of structured temporal objects and Repetition, Simon Fraser University (PhD thesis), Vancouver (2003) ## D. Dubois, H. Prade, Fundamentals of fuzzy sets, Kluwer Academic Publishers, Boston (2000) ## J. O'Rourke, Computational geometry in C, Cambridge University Press, Cambridge (1998) ## G. Nagypal, B. Motik, A fuzzy model for representing uncertain, subjective and vague temporal knowledge in ontologies, Proceedings of the International Conference on Ontologies, Databases and Applications of Semantics (ODBASE), 2003 (2003), 906-923 ## K. U. Schulz, F. Weigel, Systematic and architecture for a resource representing knowledge about named entities, in: Principles and Practice of Semantic Web Reasoning, 2003 (2003), 189-207 ## , The ACM Computing Classification System, , (2001), - ## N. Dershowitz, E. M. Reingold, Calendrical Calculations, Cambridge University Press, Cambridge (1997) ## H. J. Ohlbach, About real time, calendar systems and temporal notions, in: Advances in Temporal Logic, 2000 (2000), 319-338 ## H. J. Ohlbach, Calendar logic, in: Temporal Logic: Mathematical Foundations and Computational Aspects, 2000 (2000), 489-586 ## H. J. Ohlbach, D. Gabbay, Calendar logic, Journal of Applied Non-Classical Logics, 8 (1998), 291-323 ## L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353 ## J. E. Goodman, J. O’Rourke, Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton (1997)
Gram-schmidt Approach for Linear System of Equations with Fuzzy Parameters Gram-schmidt Approach for Linear System of Equations with Fuzzy Parameters en en In this paper, we focus on solving linear system of equations with fuzzy parameters. We employ Dubois and Prades approximate arithmetic operators on LR fuzzy numbers to find a positive fuzzy vector $$\tilde{x}$$ which satisfies $$\tilde{A}\otimes \tilde{x}=\tilde{b}$$, where $$\tilde{A}$$ and $$\tilde{b}$$ are the fuzzy matrix and vector, respectively. We shall illustrate our method by solving some numerical examples. 80 89 S. H. Nasseri M. Sohrabi Fully fuzzy linear system Fuzzy number QR-decomposition Gram-Schmidt method. Article.3.pdf  H. J. Ohlbach, Modelling periodic temporal notions by labelled partitionings of the real numbers, University of Munich, 2004 (2004), 1-42 ## H. J. Ohlbach, Calendrical calculations with time partitionings and fuzzy time intervals, in: Principles and Practice of Semantic Web Reasoning, 2004 (2004), 118-133 ## H. J. Ohlbach, Fuzzy time intervals and relations–the FuTIRe library, Institute for Computer Science, Munich (2004) ## H. J. Ohlbach, Relations between fuzzy time intervals, 11th International Symposium on Temporal Representation and Reasoning, 2004 (2004), 44-51 ## H. J. Ohlbach, The role of labelled partitionings for modeling periodic temporal notions, 11th International Symposium on Temporal Representation and Reasoning, 2004 (2004), 60-63 ## F. Bry, B. Lorenz, H. J. Ohlbach, S. Spranger, On Reasoning on Time and Location on the Web, in: Principles and Practice of Semantic Web Reasoning, 2003 (2003), 69-83 ## J. F. Allen, Maintaining knowledge about temporal intervals, Communications of the ACM, 26 (1983), 832-843 ## F. Baader, D. Calvanese, D. M. Guinness, D. Nardi, P. P. Schneider, The description logic handbook: Theory, implementation and applications, Cambridge university press, Cambridge (2003) ## B. T. Lee, M. Fischetti, Weaving the Web: the original design and ultimate destiny of the World Wide Web by its inventor, Harper, San Francisco (1999) ## D. R. Cukierman, A Formalization of structured temporal objects and Repetition, Simon Fraser University (PhD thesis), Vancouver (2003) ## D. Dubois, H. Prade, Fundamentals of fuzzy sets, Kluwer Academic Publishers, Boston (2000) ## J. O'Rourke, Computational geometry in C, Cambridge University Press, Cambridge (1998) ## G. Nagypal, B. Motik, A fuzzy model for representing uncertain, subjective and vague temporal knowledge in ontologies, Proceedings of the International Conference on Ontologies, Databases and Applications of Semantics (ODBASE), 2003 (2003), 906-923 ## K. U. Schulz, F. Weigel, Systematic and architecture for a resource representing knowledge about named entities, in: Principles and Practice of Semantic Web Reasoning, 2003 (2003), 189-207 ## , The ACM Computing Classification System, , (2001), - ## N. Dershowitz, E. M. Reingold, Calendrical Calculations, Cambridge University Press, Cambridge (1997) ## H. J. Ohlbach, About real time, calendar systems and temporal notions, in: Advances in Temporal Logic, 2000 (2000), 319-338 ## H. J. Ohlbach, Calendar logic, in: Temporal Logic: Mathematical Foundations and Computational Aspects, 2000 (2000), 489-586 ## H. J. Ohlbach, D. Gabbay, Calendar logic, Journal of Applied Non-Classical Logics, 8 (1998), 291-323 ## L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353 ## J. E. Goodman, J. O’Rourke, Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton (1997)
Differential Transformation Method and Variation Iteration Method for Cauchy Reaction-diffusion Problems Differential Transformation Method and Variation Iteration Method for Cauchy Reaction-diffusion Problems en en The paper reads the flow of an incompressible, unidirectional, steady third grade non-Newtonian fluid between two infinite planes. The flow is symmetric with respect to $$x$$− axis with constant pressure gradient. The governing equations for the flow are second order nonlinear differential equations. Homotopy Analysis Method is applied to obtain the solution. 90 101 M. R. Mohyuddin M. A. Sadiq A. M. Siddiqui Two layer flow non-Newtonian flow HAM solution Article.4.pdf  C. L. M. H. Navier, Me´moire sur les lois du mouvement des fluides, C. R. Acad. Sci., 6 (1827), 389-440 ## G. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Philos. Soc., 9 (1851), 8-106 ## H. Schlichting, Boundary layer theory, McGraw-Hill, New York (1968) ## M. E. Erdogan, A note on an unsteady flow of a viscous fluid due to an oscillating plane wall , Int. J. Non-Linear Mech., 35 (2000), 1-6 ## M. R. Mohyuddin, Stokes' problem for an oscillating plate in a porous medium with Hall effects, Journal of Porous Media, 9 (2006), 195-205 ## M. R. Mohyuddin, Few Exact Solution of the Stokes' Problem with Slip at the Wall in Case of Suction/blowing, J. Mech. Sci. Technol., 21 (2007), 829-836 ## M. R. Mohyuddin, Unsteady MHD flow due to eccentric rotating disks for suction and blowing, Turk. J. Phys., 31 (2007), 123-135 ## R. B. Bird, R. C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids: Vol. 1, Wiley, New York (1977) ## R. Rivlin, J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. Anal., 4 (1955), 323-425 ## C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics, Springer, Berlin (1965) ## S. Asghar, M. R. Mohyuddin, T. Hayat, Effects of Hall current and heart transfer on flow due to a pull of eccentric rotating disks, Int. J. Heat Mass Transfer, 48 (2005), 599-607 ## S. Asghar, M. R. Mohyuddin, P. D. Ariel, T. Hayat, On Stokes' problem for the flow of a third-grade fluid induced bya variable shear stress, Can. J. Phys., 84 (2006), 945-958 ## K. R. Rajagopal, F. Mollica, Secondary flows due to axial shearing of a third-grade fluid between two eccentrically placed cylinders, Int. J. Engng. Sci., 37 (1999), 411-429 ## K. R. Rajagopal, P. N. Kaloni, Some Remarks on Boundary Conditions for Fluids of the Differential Type, In: Continuum Mechanics and Its Applications, 1989 (1989), 935-942 ## M. R. Mohyuddin, A. A. Mirza, A. M. Siddiqui, Hydromagnetic unsteady Rivlin-Ericksen third grade flow for Stokes second problem, Chem. Eng. Commun., 194 (2007), 1201-1214 ## T. Hayat, M. R. Mohyuddin, S. Asghar, A. M. Siddiqui, The flow of a viscoelastic fluid on an oscillating plate, ZAMM Z. Angew. Math. Mech., 84 (2004), 65-70 ## M. R. Mohyuddin, Resonance and Viscoelastic Poiseuille Flow in a Porous Medium, Journal of Porous Media, 9 (2006), 799-811 ## K. R. Rajagopal, A note on unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Non-Linear Mech., 17 (1982), 369-373 ## M. E. Erdogan, Plane surface suddenly set into motion in a non-Newtonian fluid, Acta Mechanica, 108 (1995), 179-187 ## S. Asghar, M. R. Mohyuddin, T. Hayat, A. M. Siddiqui, The flow of a non-Newtonian fluid induced due to the oscillations of a porous plate, Mathematical Problems in Engineering, 2004 (2004), 133-143 ## S. Asghar, M. R. Mohyuddin, T. Hayat, Unsteady flow of a third-grade fluid in the case of suction, Math. Comput. Modelling, 38 (2003), 201-208 ## S. J. Liao, Homotopy analysis method: a new analytic method for nonlinear problems, Appl. Math. Mech., 19 (1998), 885-890 ## S. J. Liao, An approximate solution technique not depending on small parameters: a special example, Int. J. Non-Linear Mech., 30 (1995), 371-380 ## S. J. Liao, A kind of approximate solution technique which does not depend upon small parameters-II: an application in fluid mechanics, Int. J. Non-Linear Mech., 32 (1997), 815-822 ## S. J. Liao, An explicit, totally analytical solution of laminar viscous flow over a semi-infinite flat plate, Comm. Nonl. Sci. Num. Sim., 3 (1998), 53-57 ## S. J. Liao, A. T. Chwang, Application of homotopy analysis method in nonlinear oscillations, ASME J. Appl. Mech., 65 (1998), 914-922 ## S. J. Liao, A uniformly valid analytic solution of two dimensional viscous flow over a semi-infinite plat plate, J. Fluid Mech., 385 (1999), 101-128 ## S. J. Liao, Beyond perturbation: Introduction to the homotopy analysis method, CRC press, Boca Raton (2004)
Some Remarks on Convexity of Čebyšev Sets Some Remarks on Convexity of Čebyšev Sets en en In this paper, we study a part of approximation theory that presents the conditions under which a Čebyšev set in a Banach space is convex. To do so, we use Gateaux differentiability of the distance function. 102 106 Hossein Asnaashari Distance function nearest point Cebyšev set strictly convex space smooth space Gateaux differentiability. Article.5.pdf  V. S. Balaganski, L. P. Vlasov, The problem of the convexity of Čebyšev sets, Russian Math. Surveys, 51 (1996), 1127-1190 ## J. M. Borwein, Proximality and Čebyšev sets, Optim. Lett., 1 (2007), 21-32 ## J. M. Borwein, S. P. Fitzpatrick, J. R. Giles, , J. Math. Anal. Appl., 128 (1987), 512-534 ## J. R. Giles, Convex analysis with applications in differentiation of convex functions, Pitman, London (1982) ## G. G. Johnson, A nonconvex set which has the unique nearest point property, J. Approx. Theory, 51 (1987), 289-332
The Flows of a Third Grade Fluid Through Infinite Planes The Flows of a Third Grade Fluid Through Infinite Planes en en The paper reads the flow of an incompressible, unidirectional, steady third grade non-Newtonian fluid between two infinite planes. The flow is symmetric with respect to $$x$$− axis with constant pressure gradient. The governing equations for the flow are second order nonlinear differential equations. Homotopy Analysis Method is applied to obtain the solution. 107 122 M. R. Mohyuddin M. A. Sadiq A. M. Siddiqui Two layer flow non-Newtonian flow HAM solution Article.6.pdf  C. L. M. H. Navier, Me´moire sur les lois du mouvement des fluides, C. R. Acad. Sci., 6 (1827), 389-440 ## G. G. Stokes, On the effect of the internal friction of fluids on the motion of pendulums, Trans. Cambridge Philos. Soc., 9 (1851), 8-106 ## H. Schlichting, Boundary layer theory, McGraw-Hill, New York (1968) ## M. E. Erdogan, A note on an unsteady flow of a viscous fluid due to an oscillating plane wall , Int. J. Non-Linear Mech., 35 (2000), 1-6 ## M. R. Mohyuddin, Stokes' problem for an oscillating plate in a porous medium with Hall effects, Journal of Porous Media, 9 (2006), 195-205 ## M. R. Mohyuddin, Few Exact Solution of the Stokes' Problem with Slip at the Wall in Case of Suction/blowing, J. Mech. Sci. Technol., 21 (2007), 829-836 ## M. R. Mohyuddin, Unsteady MHD flow due to eccentric rotating disks for suction and blowing, Turk. J. Phys., 31 (2007), 123-135 ## R. B. Bird, R. C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids: Vol. 1, Wiley, New York (1977) ## R. Rivlin, J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. Anal., 4 (1955), 323-425 ## C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics, Springer, Berlin (1965) ## S. Asghar, M. R. Mohyuddin, T. Hayat, Effects of Hall current and heart transfer on flow due to a pull of eccentric rotating disks, Int. J. Heat Mass Transfer, 48 (2005), 599-607 ## S. Asghar, M. R. Mohyuddin, P. D. Ariel, T. Hayat, On Stokes' problem for the flow of a third-grade fluid induced bya variable shear stress, Can. J. Phys., 84 (2006), 945-958 ## K. R. Rajagopal, F. Mollica, Secondary flows due to axial shearing of a third-grade fluid between two eccentrically placed cylinders, Int. J. Engng. Sci., 37 (1999), 411-429 ## K. R. Rajagopal, P. N. Kaloni, Some Remarks on Boundary Conditions for Fluids of the Differential Type, In: Continuum Mechanics and Its Applications, 1989 (1989), 935-942 ## M. R. Mohyuddin, A. A. Mirza, A. M. Siddiqui, Hydromagnetic unsteady Rivlin-Ericksen third grade flow for Stokes second problem, Chem. Eng. Commun., 194 (2007), 1201-1214 ## T. Hayat, M. R. Mohyuddin, S. Asghar, A. M. Siddiqui, The flow of a viscoelastic fluid on an oscillating plate, ZAMM Z. Angew. Math. Mech., 84 (2004), 65-70 ## M. R. Mohyuddin, Resonance and Viscoelastic Poiseuille Flow in a Porous Medium, Journal of Porous Media, 9 (2006), 799-811 ## K. R. Rajagopal, A note on unsteady unidirectional flows of a non-Newtonian fluid, Int. J. Non-Linear Mech., 17 (1982), 369-373 ## M. E. Erdogan, Plane surface suddenly set into motion in a non-Newtonian fluid, Acta Mechanica, 108 (1995), 179-187 ## S. Asghar, M. R. Mohyuddin, T. Hayat, A. M. Siddiqui, The flow of a non-Newtonian fluid induced due to the oscillations of a porous plate, Mathematical Problems in Engineering, 2004 (2004), 133-143 ## S. Asghar, M. R. Mohyuddin, T. Hayat, Unsteady flow of a third-grade fluid in the case of suction, Math. Comput. Modelling, 38 (2003), 201-208 ## S. J. Liao, Homotopy analysis method: a new analytic method for nonlinear problems, Appl. Math. Mech., 19 (1998), 885-890 ## S. J. Liao, An approximate solution technique not depending on small parameters: a special example, Int. J. Non-Linear Mech., 30 (1995), 371-380 ## S. J. Liao, A kind of approximate solution technique which does not depend upon small parameters-II: an application in fluid mechanics, Int. J. Non-Linear Mech., 32 (1997), 815-822 ## S. J. Liao, An explicit, totally analytical solution of laminar viscous flow over a semi-infinite flat plate, Comm. Nonl. Sci. Num. Sim., 3 (1998), 53-57 ## S. J. Liao, A. T. Chwang, Application of homotopy analysis method in nonlinear oscillations, ASME J. Appl. Mech., 65 (1998), 914-922 ## S. J. Liao, A uniformly valid analytic solution of two dimensional viscous flow over a semi-infinite plat plate, J. Fluid Mech., 385 (1999), 101-128 ## S. J. Liao, Beyond perturbation: Introduction to the homotopy analysis method, CRC press, Boca Raton (2004)
The Commuting Graphs on Groups D2n and Qn The Commuting Graphs on Groups D2n and Qn en en Given group $$G$$, the commuting graph of $$G$$, is defined as the graph with vertex set $$G-Z(G)$$, and two distinct vertices $$x$$ and $$y$$ are connected by an edge, whenever they commute, that is $$xy=yx$$. In this paper we get some parameters of graph theory, as independent number and clique number for groups $$D_{2n},Q_n$$. 123 127 J. Vahidi A. Asghar Talebi independent number clique number generalized quaternion group Article.7.pdf  A. Abdollahi, S. Akbary, H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006), 468-492 ## J. A. Bondy, U. S. R. Murty, Graph theory with applications, American Elsevier Publishing Co., New York (1976) ## J. R. Moghadamfar, W. J. Shi, W. Zhou, A. R. Zokayi, On the non-commuting graph associated with a finite group, Sib. Math. J., 46 (2005), 325-332 ## Y. Segev, On finite homomorphic image of the multiplicative group of a division algebra, Ann. of Math., 149 (1999), 219-251 ## Y. Segev, G. M. Seitz, Anisotropic groups of type $$A_n$$ and the commuting graph of finite simple groups, Pacific J. Math., 202 (2002), 125-225 ## A. Asghar Talebi, On the Non-commuting graphs of group $$D_{2n}$$, International Journal of Algebra, International Journal of Algebra, 2 (2008), 957-961
On the Solving Nonlinear Approximate Long Wave Equations On the Solving Nonlinear Approximate Long Wave Equations en en In this letter, He's Variational Iteration Method (VIM) is implemented for solving the nonlinear Whitham-Broer-Kaup partial differential equations in the special case is named approximate long wave equations (ALW). This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which those obtained by the Adomian decomposition method (ADM). The work confirms that the VIM method is superior and very faster to the ADM . 128 135 M. Matinfar A. Fereidoon A. Aliasghartoyeh Whitham-Broer-Kaup equations Approximate Long Wave equations Variational Iteration Method Adomian Decomposition Method. Article.8.pdf  M. A. Abdou, A. A. Soliman, Variational iteration method for solving Burger's and coupled Burger's equation, J. Comput. Appl. Math., 18 (2005), 245-251 ## N. Bildik, A. Konuralp, The use of variational interation method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, Int. J. Nonlinear Sei. Numer. Simul. , 7 (2006), 65-70 ## S. M. El-Sayed, D. Kaya, Exact and numerical traveling wave solutions of Whitham-Broer-Kaup equations, Appl. Math. Comput., 167 (2005), 1339-1349 ## J.-H. He, X.-H. Wu, Construction of solitary solution and compacton-like solution by variational iteration method, Chaos Solitons Fractals, 29 (2006), 108-113 ## J.-H. He, Variational iteration method –a kind of nonlinear analytical technique : some examples, Int. J. Non-Linear Mech., 34 (1999), 699-708 ## J.-H. He, Variational iteration method for autonomous ordinary differential system, Appl. Math. Comput., 114 (2000), 115-123 ## J.-H. He, Variational theory for linear magneto--electro--elasticity, Int. J. Nonlinear Sci. Numerical, Simul., 2 (2001), 309-316 ## M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: Variational method in the mechanics of solids , 1978 (1978), 156-162 ## S. Momani, S. Abuasad, Application of He's variational iteration method to Helmoltz equation, Chaos Solitons Fractals, 27 (2006), 1119-1123 ## Z. M. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 27-34 ## M. Rafei, H. Daniali, Application of the variational iteraton method to the whitham-Broerkaup equation, Comput. Math. Appl., 54 (2007), 1079-1085 ## A. A. Soliman, A numerical simulation and explicit solutions of Kdv-Burgers and Lax's sventh-order KdV equation, Chaos Solitions Fractals, 29 (2006), 294-302 ## F. Xie, Z. Yan, H. Zhang, Explicit and exact traveling wave solutions of Whitham–Broer–Kaup shallow water equations, Phys. Lett. A, 285 (2001), 76-80