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Journal articles on the topic 'Transformation Equations To Two Fermat's Equations'

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Published: 7 June 2025

Last updated: 16 July 2025

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1

P.N, Seetharaman. "An Alternative Elementary Proof for Fermat's Last Theorem." International Journal of Basic Sciences and Applied Computing (IJBSAC) 11, no. 8 (2025): 11–16. https://doi.org/10.35940/ijbsac.H0534.11080425.

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<strong>Abstract:</strong> Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connect these two equations by using transformation

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2

P., N. Seetharaman. "A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation." Indian Journal of Advanced Mathematics (IJAM) 4, no. 2 (2024): 19–24. https://doi.org/10.54105/ijam.A1182.04021024.

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<strong>Abstract:</strong> Fermat’s Last Theorem states that there exists no three positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Fermat and Euler had already proved the theorem for the exponents n = 4 and n = 3 in the equations x 4 + y 4 = z 4 and x 3 + y 3 = z 3 respectively. Hence taking into account of the same, it is enough to prove the theorem for the exponent n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + s p = t p where p is any prime >3 a

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3

P. N Seetharaman. "A Proof for Fermat's Last Theorem using an Auxiliary Fermat's Equation." Indian Journal of Advanced Mathematics 4, no. 2 (2024): 19–24. https://doi.org/10.54105/ijam.a1182.04021024.

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Fermat’s Last Theorem states that there exists no three positive integers x, y and z satisfying the equation xn + yn = zn, where n is any integer > 2. Fermat and Euler had already proved the theorem for the exponents n = 4 and n = 3 in the equations x4 + y4 = z4 and x3 + y3 = z3 respectively. Hence taking into account of the same, it is enough to prove the theorem for the exponent n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation rp + sp = tp where p is any prime >3 and prove the theorem by the method of contradi

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4

P., N. Seetharaman. "In Search of an Elementary Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics (IJAM) 4, no. 1 (2024): 35–39. https://doi.org/10.54105/ijam.A1190.04010424.

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<strong>Abstract:</strong> Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connects these two equations by using the transform

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5

P.N Seetharaman. "An Alternative Elementary Proof for Fermat's Last Theorem." International Journal of Basic Sciences and Applied Computing 11, no. 8 (2025): 11–16. https://doi.org/10.35940/ijbsac.h0534.11080425.

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Abstract:

Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connect these two equations by using transformation equations. On solving the transf

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6

P.N. Seetharaman. "In Search of an Elementary Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics 4, no. 1 (2024): 35–39. https://doi.org/10.54105/ijam.a1190.04010424.

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Abstract:

Fermat’s Last Theorem states that the equation x n + y n = z n has no solution for x, y and z as positive integers, where n is any positive integer > 2. Taking the proofs of Fermat and Euler for the exponents n = 4 and n = 3, it would suffice to prove the theorem for the exponent n = p, where p is any prime > 3. We hypothesize that r, s and t are positive integers satisfying the equation r p + s p = t p and establish a contradiction in this proof. We include another Auxiliary equation x 3 + y 3 = z 3 and connects these two equations by using the transformation equations. On solving the t

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7

P.N, Seetharaman. "An Elementary Proof for Fermat's Last Theorem using Three Distinct Odd Primes F, E and R." Indian Journal of Advanced Mathematics (IJAM) 5, no. 1 (2025): 22–26. https://doi.org/10.54105/ijam.A1191.05010425.

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<strong>Abstract:</strong> In number theory, Fermat’s Last Theorem states that no three positive integers a, b and c satisfy the equation a n + b n = c n where n is any integer > 2. Fermat and Euler had already proved that there are no integral solutions to the equations x 3 + y3 = z3 and x4 + y4 = z4 . Hence it would suffice to prove the theorem for the index n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + sp = tp where p is any prime >3 and prove the theorem using the method of contradiction. We h

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8

P.N Seetharaman. "An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation." Indian Journal of Advanced Mathematics 5, no. 1 (2025): 27–31. https://doi.org/10.54105/ijam.a1192.05010425.

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Fermat’s Last Theorem states that there are no positive integers x, y and z satisfying the equation x n + y n = z n, where n is any integer > 2. Around 1637 Fermat proved that there are non-zero solutions to the above equation with n = 4. In the 18th century Euler treated the case n = 3, thereby reducing the proof for the case of a prime exponent ≥ 5 in this proof we hypothesize that r, s and t are positive integers satisfying the equation rp + sp = tp , where p is any prime >3 and establish a contradiction. We use an Auxiliary equation x 3 + y3 = z3 and create transformation equations.

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9

P.N, Seetharaman. "An Elementary Proof for Fermat's Last Theorem using a Transformation Equation to Fermat's Equation." Indian Journal of Advanced Mathematics (IJAM) 5, no. 1 (2025): 27–31. https://doi.org/10.54105/ijam.A1192.05010425.

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<strong>Abstract: </strong>Fermat’s Last Theorem states that there are no positive integers x, y and z satisfying the equation x n + y n = z n , where n is any integer > 2. Around 1637 Fermat proved that there are non-zero solutions to the above equation with n = 4. In the 18th century Euler treated the case n = 3, thereby reducing the proof for the case of a prime exponent ≥ 5 in this proof we hypothesize that r, s and t are positive integers satisfying the equation rp + sp = tp , where p is any prime >3 and establish a contradiction. We use an Auxiliary equation x 3 + y3 = z3

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10

P. N Seetharaman. "A Comprehensible Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics 4, no. 1 (2024): 29–34. https://doi.org/10.54105/ijam.a1181.04010424.

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Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation r p + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation x 3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero integer

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11

P., N. Seetharaman. "A Comprehensible Proof for Fermat's Last Theorem." Indian Journal of Advanced Mathematics (IJAM) 4, no. 1 (2024): 29–34. https://doi.org/10.54105/ijam.A1181.04010424.

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Abstract:

<strong>Abstract:</strong> Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation r p + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation x 3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zer

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12

P.N Seetharaman. "An Elementary Proof for Fermat's Last Theorem using Three Distinct Odd Primes F, E and R." Indian Journal of Advanced Mathematics 5, no. 1 (2025): 22–26. https://doi.org/10.54105/ijam.a1191.05010425.

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In number theory, Fermat’s Last Theorem states that no three positive integers a, b and c satisfy the equation a n + b n = c n where n is any integer > 2. Fermat and Euler had already proved that there are no integral solutions to the equations x 3 + y3 = z3 and x4 + y4 = z4 . Hence it would suffice to prove the theorem for the index n = p, where p is any prime > 3. In this proof, we have hypothesized that r, s and t are positive integers in the equation r p + sp = tp where p is any prime >3 and prove the theorem using the method of contradiction. We have used an Auxiliary equations x

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13

Панчук, К., K. Panchuk, Е. Любчинов, and E. Lyubchinov. "Cyclographic Interpretation and Computer Solution of One System of Algebraic Equations." Geometry & Graphics 7, no. 3 (2019): 3–14. http://dx.doi.org/10.12737/article_5dce5e528e4301.77886978.

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The subject of this study is an algebraic equation of one form and a system of such equations. The peculiarity of the subject of research is that both the equation and the system of equations admit a cyclographic interpretation in the operational Euclidean space, the dimension of which is one more than the dimension of the subspace of geometric images described by the original equations or system of equations. The examples illustrate the advantages of cyclographic interpretation as the basis of the proposed solutions, namely: it allows you to get analytical, i.e. exact solutions of the complet

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14

Seetharaman, Palamadai N. "Direct Proof for Fermat's Last Theorem using Ramanujan-Nagell Equation." European Journal of Mathematics and Statistics 3, no. 6 (2022): 1–7. http://dx.doi.org/10.24018/ejmath.2022.3.6.176.

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Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation: An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3 [1]. We hypothesize that all r, s and t are non-zero integers in the equation: rp + sp = tp and establish contradiction. Just for supporting the proof in the above equation, we have another equation: x3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3 a non-zero in

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15

Fine, Benjamin. "Cyclotomic equations and square properties in rings." International Journal of Mathematics and Mathematical Sciences 9, no. 1 (1986): 89–95. http://dx.doi.org/10.1155/s016117128600011x.

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IfRis a ring, the structure of the projective special linear groupPSL2(R)is used to investigate the existence of sum of square properties holding inR. Rings which satisfy Fermat's two-square theorem are called sum of squares rings and have been studied previously. The present study considers a related property called square property one. It is shown that this holds in an infinite class of rings which includes the integers, polynomial rings over many fields andZpnwherePis a prime such that−3is not a squaremodp. Finally, it is shown that the class of sum of squares rings and the class satisfying

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16

P. N Seetharaman. "An Elementary Proof for Fermat's Last Theorem using Ramanujan-Nagell Equation." Indian Journal of Advanced Mathematics 4, no. 2 (2024): 10–15. http://dx.doi.org/10.54105/ijam.b1180.04021024.

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Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation rp + sp = tp and establish a contradiction in this proof. Just for supporting the proof in the above equation, we have used another equation x3 + y3 = z3 Without loss of generality, we assert that both x and y as non-zero integers; z3

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17

P., N. Seetharaman. "An Elementary Proof for Fermat's Last Theorem using Ramanujan-Nagell Equation." Indian Journal of Advanced Mathematics (IJAM) 4, no. 2 (2024): 10–15. https://doi.org/10.54105/ijam.B1180.04021024.

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<strong>Abstract: </strong>Fermat’s Last Theorem states that it is impossible to find positive integers A, B and C satisfying the equation An + Bn = Cn where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3. We hypothesize that all r, s and t are non-zero integers in the equation rp + sp = tp and establish a contradiction in this proof. Just for supporting the proof in the above equation, we have used another equation x3 + y3 = z3 Without loss of generality, we assert that both

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18

Dolan, Stan. "Fermat and the difference of two squares." Mathematical Gazette 96, no. 537 (2012): 480–91. http://dx.doi.org/10.1017/s0025557200005118.

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In a previous note [1], Fermat's method of descente infinie was used to prove that the equations.have no positive integer solutions. The geometrically based proof of [1] masked the underlying use of the difference of two squares. In the proofs of this article we shall make its use explicit, just as Fermat did [2, pp. 293-294].We shall use the elementary idea of the difference of two squares to develop a powerful technique for solving equations of the form ax4 + bx2y2 + cy4 = z2. This will then be applied to three problems of historical interest.

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19

Khadka, Chandra Bahadur. "Geometrical Interpretation of Space Contraction in Two-dimensional Lorentz Transformation." BIBECHANA 21, no. 2 (2024): 103–12. http://dx.doi.org/10.3126/bibechana.v21i2.62271.

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This paper points out that the transformation equations for the spatial and temporal coordinates between two frames of reference in the existing generally accepted version of the Lorentz transformation are deficient, since transformation equations are based on one dimensional motion between inertial frames. Therefore, all possible space-time coordinate transformation equations between moving and stationary frames by prolonging Lorentz transformation in a two-dimensional plane are thoroughly proposed in this article. If denotes the relative velocity between stationary frame (x, y) and moving fr

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20

Miyazaki, Takafumi, and István Pink. "Number of solutions to a special type of unit equations in two unknowns." American Journal of Mathematics 146, no. 2 (2024): 295–369. http://dx.doi.org/10.1353/ajm.2024.a923236.

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abstract: For any fixed relatively prime positive integers $a$, $b$ and $c$ with $\min\{a,b,c\}>1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x$, $y$ and $z$, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.~A. Bennett

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21

Mansour, Eman A., and Emad A. Kuffi. "SEE Transform in Difference Equations and Differential-Difference Equations Compared With Neutrosophic Difference Equations." International Journal of Neutrosophic Science 22, no. 4 (2023): 36–43. http://dx.doi.org/10.54216/ijns.220403.

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The Sadiq-Emad-Emann (SEE) transform, also known as operational calculus, has gained significant importance as a fundamental component of the mathematical knowledge necessary for physicists, engineers, mathematicians, and other scientific professionals. This is because the SEE transform offers accessible and efficient resources for resolving several applications and challenges encountered in diverse engineering and science domains. This study aims to introduce the fundamental principles of SEE transformation and establish the validity of two statements and associated attributes. The objective

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22

Orverem, Joel Mvendaga. "A Linear Approximation of the Non-linear Modified Langumir and Van der Pol Differential Equations by the Application of the Generalized Sundman Transformation." UMYU Scientifica 3, no. 4 (2024): 441–47. https://doi.org/10.56919/usci.2434.038.

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Study’s Excerpt: The modified Langumir and Van der Pol nonlinear ordinary differential equations are solved by the GST. These equations are very significant and useful in many areas of human life. The GST method linearizes the equations into forms that can be solved, resulting in 3u''+4u'+2=0 and u''+u'+2=0, respectively. Here, the GST technique yielded innovative and workable analytical solutions. Full Abstract: The non-linear ordinary differential equations of Langumir and Van der Pol are challenging to solve analytically. Thus, this work aims to convert these non-linear equations into linea

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23

Qin, Huizeng, and Bin Zheng. "Oscillation of a Class of Fractional Differential Equations with Damping Term." Scientific World Journal 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/685621.

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We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

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24

Neuman, František. "Transformation and canonical forms of functional-differential equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 115, no. 3-4 (1990): 349–57. http://dx.doi.org/10.1017/s0308210500020692.

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SynopsisFunctional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.

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25

Fan, En Gui, and Man Wai Yuen. "Similarity reductions and exact solutions for two-dimensional Euler–Boussinesq equations." Modern Physics Letters B 33, no. 27 (2019): 1950328. http://dx.doi.org/10.1142/s0217984919503287.

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In this paper, by introducing a stream function and new coordinates, we transform classical Euler–Boussinesq equations into a vorticity form. We further construct traveling wave solutions and similarity reduction for the vorticity form of Euler–Boussinesq equations. In fact, our similarity reduction provides a kind of linearization transformation of Euler–Boussinesq equations.

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26

Ma, Wen-Xiu, Yuan Zhou, and Rachael Dougherty. "Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations." International Journal of Modern Physics B 30, no. 28n29 (2016): 1640018. http://dx.doi.org/10.1142/s021797921640018x.

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Lump-type solutions, rationally localized in many directions in the space, are analyzed for nonlinear differential equations derived from generalized bilinear differential equations. By symbolic computations with Maple, positive quadratic and quartic polynomial solutions to two classes of generalized bilinear differential equations on [Formula: see text] are computed, and thus, lump-type solutions are presented to the corresponding nonlinear differential equations on [Formula: see text], generated from taking a transformation of dependent variables [Formula: see text].

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27

Gong, Shengbo, Yan Guo, and Ya-Guang Wang. "Boundary layer problems for the two-dimensional compressible Navier–Stokes equations." Analysis and Applications 14, no. 01 (2016): 1–37. http://dx.doi.org/10.1142/s0219530515400011.

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We study the well-posedness of the boundary layer problems for compressible Navier–Stokes equations. Under the non-negative assumption on the laminar flow, we investigate the local spatial existence of solution for the steady equations. Meanwhile, we also obtain the solution for the unsteady case with monotonic laminar flow, which exists for either long time small space interval or short time large space interval. Moreover, the limit of these solutions with vanishing Mach number is considered. Our proof is based on the comparison theory for the degenerate parabolic equations obtained by the Cr

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28

Teia, Luis. "Fermat's Theorem -- a Geometrical View." Journal of Mathematics Research 9, no. 1 (2017): 136. http://dx.doi.org/10.5539/jmr.v9n1p136.

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Fermat's Last Theorem questions not only what is a triple, but more importantly, what is an integer in the context of equations of the type $x^n+y^n=z^n$. This paper explores these questions in one, two and three dimensions. It was found that two conditions are required for an integer element to exist in the context of the Pythagoras' theorem in 1D, 2D and 3D. An integer must satisfy the Pythagoras' theorem of the respective dimension -- condition 1. And, it must be completely successfully split into multiple unit scalars -- condition 2. In 1D, the fundamental unit scalar is the line length 1.

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29

Zhaqilao and Zhijun Qiao. "Darboux transformation and explicit solutions for two integrable equations." Journal of Mathematical Analysis and Applications 380, no. 2 (2011): 794–806. http://dx.doi.org/10.1016/j.jmaa.2011.01.078.

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RAMOS, JAIRZINHO, and ROBERT GILMORE. "DERIVATION OF THE SOURCE-FREE MAXWELL AND GRAVITATIONAL RADIATION EQUATIONS BY GROUP THEORETICAL METHODS." International Journal of Modern Physics D 15, no. 04 (2006): 505–19. http://dx.doi.org/10.1142/s021827180600822x.

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We derive source-free Maxwell-like equations in flat space–time for any helicity j by comparing the transformation properties of the 2(2j+1) states that carry the manifestly covariant representations of the inhomogeneous Lorentz group with the transformation properties of the two helicity j states that carry the irreducible representations of this group. The set of constraints so derived involves a pair of curl equations and a pair of divergence equations. These reduce to the free-field Maxwell equations for j = 1 and the analogous equations coupling the gravito-electric and the gravito-magnet

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31

Zhao, Dan, and Zhaqilao. "Darboux transformation approach for two new coupled nonlinear evolution equations." Modern Physics Letters B 34, no. 01 (2019): 2050004. http://dx.doi.org/10.1142/s0217984920500049.

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A new coupled Burgers equation and a new coupled KdV equation which are associated with [Formula: see text] matrix spectial problem are investigated for complete integrability and covariant property. For integrability, Lax pair and conservation laws of the two new coupled equations with four potentials are established. For covariant property, Darboux transformation (DT) is used to construct explicit solutions of the two new coupled equations.

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32

Zhang, Yi. "Theory of Generalized Canonical Transformations for Birkhoff Systems." Advances in Mathematical Physics 2020 (May 26, 2020): 1–10. http://dx.doi.org/10.1155/2020/9482356.

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Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and

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33

Bekova, G. T., and A. A. Zhadyranova. "MULTI-LINE SOLITON SOLUTIONS FOR THE TWO-DIMENSIONAL NONLINEAR HIROTA EQUATION." PHYSICO-MATHEMATICAL SERIES 2, no. 336 (2021): 172–78. http://dx.doi.org/10.32014/2021.2518-1726.38.

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At present, the question of studying multidimensional nonlinear integrable equations in the framework of the theory of solitons is very interesting to foreign and Kazakh scientists. Many physical phenomena that occur in nature can be described by nonlinearly integrated equations. Finding specific solutions to such equations plays an important role in studying the dynamics of phenomena occurring in various scientific and engineering fields, such as solid state physics, fluid mechanics, plasma physics and nonlinear optics. There are several methods for obtaining real and soliton, soliton-like so

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34

Zhang, Ronghang, Yue Lan, and Zhenyu Xu. "Equivalent transformation of differential and integral equations in hyperbolic plane." Highlights in Science, Engineering and Technology 120 (December 25, 2024): 783–89. https://doi.org/10.54097/28hwdh93.

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Hyperbolic numbers, serving as an extension of real numbers, exhibit an algebraic structure akin to complex numbers, constituting commutative rings with zero divisors that stem from two real numbers. This paper delves into the exploration of the equivalence relation between the solutions of differential equations and integral equations within the realm of the hyperbolic plane. The pivotal finding of this study reveals that the solutions of hyperbolic differential equations and integral equations are inherently equivalent. Such a discovery not only solidifies the groundwork for delving into hyp

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35

Zhang, Li-Hua, and Xi-Qiang Liu. "A Direct Transformation Method and its Application to Variable Coefficient Nonlinear Equations of Schrödinger Type." Zeitschrift für Naturforschung A 64, no. 11 (2009): 697–708. http://dx.doi.org/10.1515/zna-2009-1105.

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In this paper, the generalized variable coefficient nonlinear Schrödinger (NLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients are directly reduced to simple and solvable ordinary differential equations by means of a direct transformation method. Taking advantage of the known solutions of the obtained ordinary differential equations, families of exact nontravelling wave solutions for the two equations have been constructed. The characteristic feature of the direct transformation method is, that without much extra effort, we circumvent the integ

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Zheng, Pengshe, Jing Luo, Shunchu Li, and Xiaoxu Dong. "Elastic transformation method for solving ordinary differential equations with variable coefficients." AIMS Mathematics 7, no. 1 (2021): 1307–20. http://dx.doi.org/10.3934/math.2022077.

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<abstract><p>Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be o

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Li, Biao, Yong Chen, and Hongqing Zhang. "Auto-B¨Acklund Transformations And Exact Solutions For The Generalized Two-Dimensional Korteweg-De Vries-Burgers-Type Equations And Burgers-Type Equations." Zeitschrift für Naturforschung A 58, no. 7-8 (2003): 464–72. http://dx.doi.org/10.1515/zna-2003-7-813.

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In this paper, based on the idea of the homogeneous balance method and with the help of Mathematica, we obtain a new auto-Bäcklund transformation for the generalized two-dimensional Kortewegde Vries-Burgers-type equation and a new auto-Bäcklund transformation for the generalized twodimensional Burgers-type equation by introducing two appropriate transformations. Then, based on these two auto-Bäcklund transformation, some exact solutions for these equations are derived. Some figures are given to show the properties of the solutions.

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Aerts, Diederik, and Marek Czachor. "Two-State Dynamics for Replicating Two-Strand Systems." Open Systems & Information Dynamics 14, no. 04 (2007): 397–410. http://dx.doi.org/10.1007/s11080-007-9064-0.

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We propose a formalism for describing two-strand systems of a DNA type by means of soliton von Neumann equations, and illustrate how it works on a simple example exactly solvably by a Darboux transformation. The main idea behind the construction is the link between solutions of von Neumann equations and entangled states of systems consisting of two subsystems evolving in time in opposite directions. Such a time evolution has analogies in realistic DNA where the polymerazes move on leading and lagging strands in opposite directions.

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Elías-Zúñiga, Alex, Daniel Olvera Trejo, Inés Ferrer Real, and Oscar Martínez-Romero. "A Transformation Method for Solving Conservative Nonlinear Two-Degree-of-Freedom Systems." Mathematical Problems in Engineering 2014 (2014): 1–14. http://dx.doi.org/10.1155/2014/237234.

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A nonlinear transformation approach based on a cubication method is developed to obtain the equivalent representation form of conservative two-degree-of-freedom nonlinear oscillators. It is shown that this procedure leads to equivalent nonlinear equations that describe well the numerical integration solutions of the original equations of motion.

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Huang, Sui Liang. "Two-dimensional numerical modeling of chemical transport–transformation in fluvial rivers: formulation of equations and physical interpretation." Journal of Hydroinformatics 11, no. 2 (2009): 106–18. http://dx.doi.org/10.2166/hydro.2009.025.

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Based on previous work on the transport–transformation model of heavy metal pollutants in fluvial rivers, this paper presents the formulation of a two-dimensional model to describe chemical transport–transformation in fluvial rivers by considering basic principles of environmental chemistry, hydraulics and mechanics of sediment transport and recent developments along with three very simplified test cases. The model consists of water flow governing equations, sediment transport governing equations, transport–transformation equation of chemicals and convection–diffusion equations of sorption–des

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Vitanov, Nikolay K., and Zlatinka I. Dimitrova. "Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity." Entropy 23, no. 12 (2021): 1624. http://dx.doi.org/10.3390/e23121624.

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We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-li

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Eliazar, Iddo. "Selfsimilar stochastic differential equations." Europhysics Letters 136, no. 4 (2021): 40002. http://dx.doi.org/10.1209/0295-5075/ac4dd4.

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Abstract Diffusion in a logarithmic potential (DLP) attracted significant interest in physics recently. The dynamics of DLP are governed by a Langevin stochastic differential equation (SDE) whose underpinning potential is logarithmic, and that is driven by Brownian motion. The SDE that governs DLP is a particular case of a selfsimilar SDE: one that is driven by a selfsimilar motion, and that produces a selfsimilar motion. This paper establishes the pivotal role of selfsimilar SDEs via two novel universality results. I) Selfsimilar SDEs emerge universally, on the macro level, when applying scal

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Ghaffar, M., K. Ali, A. Yasmin, and M. Ashraf. "Unsteady Flow between Two Orthogonally Moving Porous Disks." Journal of Mechanics 31, no. 2 (2015): 147–51. http://dx.doi.org/10.1017/jmech.2014.90.

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ABSTRACTThe unsteady laminar incompressible flow of a fluid between two orthogonally moving porous coaxial disks is considered numerically. A transformation is used to reduce the governing partial differential equations (PDEs) to a set of nonlinear coupled ordinary differential equations. The effects of physical parameters of interest such as the wall expansion ratio and the permeability Reynolds number on the velocity are discussed in detail.

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Abed, Alaa Mohsin, Hossein Jafari, and Mohammed Sahib Mechee. "Some numerical methods for solving fractional partial difference equations in the discrete domain." Journal of Interdisciplinary Mathematics 27, no. 4 (2024): 737–43. http://dx.doi.org/10.47974/jim-1763.

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This paper proposes a method for solving fractional partial difference equations using discrete Sumudu transformation. Some properties of discrete Sumudu transformation were used. The applications of the test examples for the initial value problems of fractional partial difference equations were tested using two methods: the successive approximation method. The second, homotopy perturbation, was proven efficient with the proposed method.

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HALEEM, EZATULLAH, AHAD KHAN PYAWARAI, MOHAMMAD DAUD AHMADZAI, and IRSHAD SALARZAI. "Convergence Analysis For Numerical Solution of Integral Equations Using Galarkin Method with two Orthogonal Polynomials." Cognizance Journal of Multidisciplinary Studies 5, no. 4 (2025): 1015–30. https://doi.org/10.47760/cognizance.2025.v05i04.032.

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This research explores the role of orthogonal, Chebyshev, and Hermite polynomials in the numerical solution of integral equations, focusing on convergence analysis. The study employs the Galerkin method to solve Volterra integral equations, providing a numerical approach that ensures accuracy and efficiency. The research emphasizes the importance of integral equations in various scientific and engineering applications, highlighting their transformation from differential equations in physics, biology, and chemistry. MATLAB was used to implement the proposed numerical methods, verifying converge

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BRACKEN, PAUL. "ON TWO-DIMENSIONAL MANIFOLDS WITH CONSTANT GAUSSIAN CURVATURE AND THEIR ASSOCIATED EQUATIONS." International Journal of Geometric Methods in Modern Physics 09, no. 03 (2012): 1250018. http://dx.doi.org/10.1142/s0219887812500181.

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The components for the frame field of a two-dimensional manifold with constant Gaussian curvature are determined for arbitrary nonzero curvature. The components of the frame fields are found from the structure equations and lead to specific nonlinear equations which pertain to surfaces with specific values of the Gaussian curvature. For negative curvature, the equation is of sine-Gordon type, and for positive curvature it is of sinh-Gordon type. The integrability and Bäcklund properties of these equations are then investigated by studying a differential ideal of two-forms which leads to the eq

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Orlov, Aleksandr I. "ALGORITHM OF ELECTRIC CIRCUIT STATE MATRIX TRANSFORMATION USING THE MODIFIED NODAL ANALYSIS." Vestnik Chuvashskogo universiteta, no. 3 (September 29, 2022): 73–80. http://dx.doi.org/10.47026/1810-1909-2022-3-73-80.

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The article proposes the algorithm of electric circuit state matrix transformation obtained by the modified nodal analysis. The algorithm is aimed at reducing the system of electrical circuit differential equations to Cauchy problem form for the purpose of subsequent solution. Application of the modified nodal analysis to an electric circuit with energy inertial elements leads to a system of ordinary differential equations, which contains a singular matrix before the derivatives vector in matrix form if there are ungrounded capacitors in a circuit. For this reason, direct reduction of this equ

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Chen, Jing, Ling Liu, and Li Liu. "Separation Transformation and a Class of Exact Solutions to the Higher-Dimensional Klein-Gordon-Zakharov Equation." Advances in Mathematical Physics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/974050.

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The separation transformation method is extended to then+1-dimensional Klein-Gordon-Zakharov equation describing the interaction of the Langmuir wave and the ion acoustic wave in plasma. We first reduce then+1-dimensional Klein-Gordon-Zakharov equation to a set of partial differential equations and two nonlinear ordinary differential equations of the separation variables. Then the general solutions of the set of partial differential equations are given and the two nonlinear ordinary differential equations are solved by extendedF-expansion method. Finally, some new exact solutions of then+1-dim

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M. Turq, Saed, and Emad A. Kuffi. "On the Double of the Emad - Falih Transformation and Its Properties with Applications." Ibn AL-Haitham Journal For Pure and Applied Sciences 35, no. 4 (2022): 220–34. http://dx.doi.org/10.30526/35.4.2938.

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In this paper, we have generalized the concept of one dimensional Emad - Falih integral transform into two dimensional, namely, a double Emad - Falih integral transform. Further, some main properties and theorems related to the double Emad - Falih transform are established. To show the proposed transform's efficiency, high accuracy, and applicability, we have implemented the new integral transform for solving partial differential equations. Many researchers have used double integral transformations in solving partial differential equations and their applications. One of the most important uses

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Li, Xinyue, Zhixin Zhang, Qiulan Zhao, and Chuanzhong Li. "Darboux transformation of two novel two-component generalized complex short pulse equations." Reports on Mathematical Physics 90, no. 2 (2022): 157–84. http://dx.doi.org/10.1016/s0034-4877(22)00063-5.

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