Functional separation of variables in nonlinear PDEs: General approach, new solutions of diffusion-type equations

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A number of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.


A brief overview of modifications of the method of functional separation of variables
The methods of generalized and functional separation of variables (and their various modifications) are among the most effective methods for constructing exact solutions to various classes of nonlinear equations of mathematical physics and mechanics (including partial differential equations of fairly general forms that involve arbitrary functions). In , many exact solutions to equations of heat and mass transfer theory, wave theory, hydrodynamics, gas dynamics, nonlinear optics, and mathematical biology were obtained using these methods.
To be specific, we will consider nonlinear PDEs of mathematical physics with two independent variables F (x, t, u x , u t , u xx , u xt , u tt , . . . where u = u(x, t) is the unknown function.
The methods of generalized and functional separation of variables are based on setting a priori a structural form of u that depends on several free functions (the specific form of these functions is determined subsequently by analyzing the arising functional differential equations).
Exact solutions in the form of the sum or product of two functions that depend on different arguments, u = ξ(x) + η(t) or u = ξ(x)η(t), are called ordinary separable solutions. Examples of nonlinear PDEs with such solutions can be found in [10,18,23].
Often (in a narrow sense) the term 'solution with functional separation of variables' (or 'functional separable solution') is used for exact solutions of the form (e.g., see [1-4, 7, 8, 18, 22, 23]) where the functions ϕ(z), ξ(x), and η(t) are determined in a subsequent analysis. Sometimes the external function ϕ(z) is specified from a priori considerations, while the internal functions ξ(x) and η(t) are to be found [19,23].
The studies [33][34][35] described a new direct method for constructing exact solutions with functional separation of variables. It is based on an implicit integral representation of solutions in the form ζ(u) du = ξ 1 (x)η(t) + ξ 2 (x), where the functions ζ(u), ξ 1 (x), ξ 2 (x), and η(t) are determined by the splitting method in the subsequent analysis. This method allowed to find more than 40 exact solutions of nonlinear reaction-diffusion equations and wave type equations with variable coefficients involving one or more arbitrary functions. In [36], it was shown that some of the solutions given in [34,35] cannot be obtained using the nonclassical method of symmetry reductions [38][39][40][41][42][43][44][45] (see also [18,23]) based on the use of the invariant surface condition (a first-order differential constraint equivalent to the relation (4)).
Note that constructing solutions in implicit form with the integral term (4) often allows us to reduce the order of the resulting functional differential equations [33,34].
In the general case, the term 'functional separable solution' with regard to nonlinear PDEs (1) will be used for exact solutions that can be represented as where the desired functions ϕ(z) and Q(x, t) are described respectively by overdetermined systems of ODEs and PDEs. In the simplest cases, each of these functions can be described by a single equation. Representation (5) was used in [30][31][32] to construct exact solutions with functional separation of variables to some classes of nonlinear reaction-diffusion, convective-diffusion, and wave type equations. It is necessary to distinguish between direct and indirect functional separation of variables based on one of the representations of solutions (2), (3), (4), or (5).
At the first stage of direct functional separation of variables, the representation of solution is substituted into the original PDE, after which the resulting equation is analyzed (e.g., see [23,30,31,[33][34][35]). At the first stage of indirect functional separation of variables, the representation of solution is replaced by one or more equivalent differential constraints, and then the overdetermined system of PDEs obtained in this way is analyzed for compatibility (e.g., see [9,20,22,36]).
To construct exact solutions of nonlinear partial differential equations, this paper proposes to use a direct method based on a special transformation with an integral term as well as the generalized splitting principle. This approach is technically simpler and more convenient than finding a solution in the form (5); it generalizes the dependence (4) and allows one to find various solutions in a uniform manner without specifying their structure a priori.

The concept of 'exact solution' for nonlinear PDEs
In what follows, the term 'exact solution' with regard to nonlinear partial differential equations is used in the following cases: (iii) the solution is expressible in terms of solutions to ordinary differential equations or systems of such equations.
Combinations of cases (i) and (iii) as well as (ii) and (iii) are also allowed. To construct exact solutions of equation (1), we first introduce a new dependent variable ϑ using the nonlinear transformation Both functions ϑ = ϑ(x, t) and ζ = ζ(u) will be found simultaneously in the subsequent analysis. Once these functions are determined, the integral relation (6) will specify an exact solution of the equation in question in implicit form (in some cases, the solution may be represented explicitly).
Differentiating (6) with respect to the independent variables, we find the partial derivatives We assume that after substituting expressions (7) into (1), the resulting equation can be converted to the following form: where To construct exact solutions of equation (8)-(9), we use the splitting principle described below.
The generalized splitting principle. We consider linear combinations of two sets of elements {Φ j } and {Ψ j } included in (8), which are connected by relations N n=1 α ni Φ n = 0, i = 1, . . . , l; N n=1 β nj Ψ n = 0, j = 1, . . . , m, where 1 ≤ l ≤ N − 1 and 1 ≤ m ≤ N − 1. The constants α ni and β nj in (10) are chosen so that the bilinear equality (8) is satisfied identically (this can always be done as shown below). Importantly, relations (10) are purely algebraic in nature and are independent of any particular expressions of the differential forms (9).
Once relations (10) are obtained, we substitute the differential forms (9) into them to arrive at systems of differential equations (often overdetermined) for the unknown functions ϑ = ϑ(x, t) and ζ = ζ(u) that appear in (6).
Remark 1. Degenerate cases where one or more of the differential forms Φ n and/or Ψ n vanish in addition to the linear relations (10) must be treated separately.
Remark 2. The main ideas of the direct method of functional separation of variables based on transformation (6) were expressed in the brief note [46], where four exact solutions of a generalized porous medium equation with a nonlinear source were obtained. The present paper demonstrates the effectiveness of this method by constructing a large number of solutions (more than 30 solutions have been obtained in total) to a nonlinear diffusion-type equation involving several arbitrary functions. In addition, it will be shown that the direct method is more efficient than indirect methods. Remark 3. Bilinear functional-differential equations that are similar in appearance to (8)-(9) arise when one searches for exact solutions to nonlinear equations of mathematical physics using the methods of generalized and functional separation of variables with a priori given solution structure. However, there is a fundamental difference in this case: the differential forms Φ n and Ψ n in (9) depend, in view of transformation (6), on the same independent variables x and t, whereas when the methods of generalized and functional separation of variables [18,23,[34][35][36] (see also [47]) are used, the differential forms depend on different independent variables. This circumstance significantly expands the possibilities of constructing exact solutions by switching to equivalent equations (see Section 2.3 for details).
Remark 4. Instead of transformation (6), we can use the transformation ϑ = Z(u), which leads to slightly more complex equations. The method for constructing functional separable solutions described above is more convenient and is based on a substantial generalization of traveling wave type solutions of various classes of nonlinear PDEs. To illustrate this, consider the nonlinear heat equation For arbitrary f (u), equation (11) admits the traveling wave solution where κ and λ are arbitrary constants. Substituting (12) in (11) yields the ODE z , the integration of which gives its solution in implicit form where C 1 and C 2 are arbitrary constants. On the left-hand side of (13), z has been replaced with the original variables using (12). The representation of the solution in the form (6) is an essential generalization of the traveling wave solution (13), which is carried out as follows:

Some formulas allowing the satisfaction of relation (8) identically
1. For any N , equality (8) can be satisfied if all Φ i but one are put proportional to a selected element Φ j (j = i). As a result, we get . . , j − 1, j + 1, . . . , N ; where A i are arbitrary constants. In formulas (14), the symbols can be swapped, In this case, we have the relations where A ij are arbitrary constants and the indices i and j together take all values from 1 to N .
3. For N ≥ 3, equality (8) is also satisfied identically if we choose the linear relations where A i and B i are arbitrary constants. In formulas (15), the symbols can be swapped, Φ n ⇄ Ψ n , or simultaneous pairwise substitutions Φ i ⇄ Φ j and Ψ i ⇄ Ψ j can be made.
To construct more complex linear combinations of the form (10) that would identically satisfy the bilinear relation (8) for any N , one can use the formulas for the coefficients α ni and β nj given in the books [18,23] (in sections devoted to generalized separation of variables).

Possible generalizations based on the use of equivalent equations
Other exact solutions of equation (1) can be obtained if, instead of (8)-(9), we consider equivalent differential equations that reduce to (8)-(9) on the set of functions satisfying relation (6). Indicated below are two classes of such equations, which will be used later in Section 3.3.

One can use equations of the form
which for any functions G(x, t, u, ϑ) are equivalent to equation (8)- (9). Further, in Section 3.3, specific examples of using equations of the form (17) for G(x, t, u, ϑ) = λ(x, t, u)ϑ will be given. The functions G and λ can explicitly depend on ϑ and ζ (and their derivatives) and the functional coefficients of the original PDE (which suggests implicit dependence on the original variables x, t, and u).
In the generic case, applying the splitting principle to equations (16) and (17) will lead to other exact solutions of the original equation (1) than applying this principle to equation (8).
Using the method described in Section 2, we further obtain a number of new exact solutions to equations of the form (18), in which at least two functional coefficients a(x) and f (u) are given arbitrarily (and the others are expressed through them). Below, for brevity, the arguments of the functions included in transformation (6) and equation (18) will often be omitted.
Having made the transformation (6), we substitute the derivatives (7) in (18). After simple rearrangements we get For ζ = 1, equation (19) coincides with the original equation (18), where u = ϑ. Therefore, at this stage, no solutions are lost. We introduce the following notation: As a result, equation (19) can be represented in the bilinear form (8) with N = 5: We now turn to the construction of exact solutions of nonlinear equations of the form (18) based on relations (20) and (21) using the approach described in Section 2.1.

Exact solutions obtained by analyzing equation (19)
Solution 1. Equation (21) can be satisfied identically if we use the linear relations where k is an arbitrary constant. Substituting (20) into (22), we arrive at the The solution of the overdetermined system consisting of the first three equations (23) has the form where a(x) is an arbitrary function and b 0 , c 0 , and C 1 are arbitrary constants. The solution of the system consisting of the last two equations in (23) can be written as follows: where f (u) and g(u) are arbitrary functions. From formulas (24) and (25) for b 0 = c 0 = 1 we obtain the equation which admits a generalized traveling wave solution in the implicit form Note that equation (26) contains three arbitrary functions a(x), f (u), and g(u) and two arbitrary constants C 2 and k. (21) can be satisfied if we take

Solution 2. Equation
where k is an arbitrary constant. Substituting (20) into (28), we arrive at the equations The solutions of the first three equations (29) are where a(x) is an arbitrary function and C 1 , C 2 , and λ are arbitrary constants.
The last two equations (29) give two functions where f = f (u) and h = h(u) are arbitrary functions and C 3 is an arbitrary constant.
Setting C 1 = 1 in (30) and (31), we obtain the equation where a(x), f (u), and h(u) are arbitrary functions, while k and λ are arbitrary constants. This equation admits two exact solutions where C 2 and C 3 are arbitrary constants.
Solution 3. Equation (21) can be satisfied by setting where k 1 , k 2 , and k 3 are arbitrary constants. Substituting (20) in (34), we get The solution of the overdetermined system consisting of the first three equations (35) can be represented as where a(x) is an arbitrary function, while c 0 , C 1 , and C 2 are arbitrary constants. From the last two equations (35) we obtain where f = f (u) and g = g(u) are arbitrary functions.
For c 0 = k 3 = 1, formulas (36) and (37) lead to the equation which has the generalized traveling wave solution

Solution 4. Equation (21) holds if we set
where k is an arbitrary constant. Substituting (20) into (38) yields The general solution of the overdetermined system consisting of the first two equations (39) has the form where a = a(x) and b = b(x) are arbitrary functions, while C 1 , C 2 , C 3 , and C 4 are arbitrary constants. The solution of the system consisting of the last three equations (39) is given by Given relations (40) and (41), we obtain the equation which admits an exact solution in the implicit form Here a(x), b(x), and f (u) are arbitrary functions, and the functions c(x) and s(x) are defined in (40). In particular, for C 2 = λ, C 1 = 0, and k = 1, we get the which has the solution Solution 5. Equation (21) can be satisfied by setting where k is an arbitrary constant. Substituting (20) in (46), we get The first two equations (47) admit the solution where a = a(x) and r = r(x) are arbitrary functions, while λ and C 1 are arbitrary constants. From the last equation (47) we get kζ −3 ζ ′ u = −h, which gives two solutions where h = h(u) is an arbitrary function and C 2 is an arbitrary constant.

Solution 6. Equation (21) holds if we set
where k 1 , k 2 , and λ are arbitrary constants. Substituting (20) into (50), we obtain The solution of the first three equations (51) is expressed as where a(x) is an arbitrary function, while C 1 and C 2 are arbitrary constants. The solution of the last two equations (51) is given by where f = f (u) and g = g(u) are arbitrary functions, while C 3 is an arbitrary constant.
Setting k 1 = k and k 2 = 1 in (52) and (53), we arrive at the equation where a(x), f (u), and g(u) are arbitrary functions, while C 1 , C 3 , k, and λ are arbitrary constants. This equation admits the exact solution in implicit form Solution 7. Equation (21) can be satisfied by setting where k 1 , k 2 , and k 3 are arbitrary constants. Substituting (20) in (54) yields The solutions of the first three equations (55) can be represented as where a = a(x) is an arbitrary constant, while C 1 and λ are arbitrary constants. The solutions of the last two equations (55) are given by where f = f (u) and h = h(u) are arbitrary functions. (56) and (57), we arrive at the equation which contains three arbitrary functions a(x), f (u), and h(u) and has the exact solution Solution 8. Equation (21) can be satisfied if we take where k 1 and k 2 are arbitrary constants. Substituting (20) into (60), we arrive at the equations The solutions of the first three equations (61) are where a(x) is an arbitrary function, while C 1 , C 2 , and λ are arbitrary constants. The last two equations (61) give two solutions where f = f (u) and h = h(u) are arbitrary functions and C 3 is an arbitrary constant.
Setting C 1 = s, k 1 = −1, k 2 = k, and λ = k in (62) and (63), we obtain the equation where a(x), f (u), and h(u) are arbitrary functions, while k and s are arbitrary constants. This equation admits the exact solutions where C 2 and C 3 are arbitrary constants.
In the special case k = −1, f (u) = 1, and s = 0, equation (64) is reduced to a simpler equation, which was considered in [34]. Setting h(u) = 0, C 3 = 0, and s = 0 in (64), and renaming a(x) to xa(x), we obtain the equation whose solutions are Solution 9. Equation (21) holds if we set where k 1 and k 2 are arbitrary constants. Substituting (20) into (66), we obtain the equations In the special case k 1 = k 2 = 0, the solution of system (67) leads to the equation where a(x) and f (u) are arbitrary functions, while β and λ are arbitrary constants. This equation admits two exact solutions Solution 10. Equation (21) can be satisfied if we take where k 1 , k 2 , and k 3 are arbitrary constants. Substituting (20) into (70), we arrive at the equations The solutions of the first three equations (71) are where a(x) is an arbitrary function, while C 1 and C 2 are arbitrary constants. The solutions of the last two equations (71) are given by where f (u) and g(u) are arbitrary functions and C 3 is an arbitrary constant.
In particular, setting a(x) = x n , C 1 = C 2 = 0, C 3 = m, k 1 = k, k 2 = 1 − n, and k 3 = 1 in (72)-(73), we obtain the equation Solution 11. Equation (21) can be satisfied if we use the relations where k 1 and k 2 are arbitrary constants. Substituting (20) in (74) yields The first three equations (75) admit two solutions, which are given by where a = a(x) is an arbitrary function and C 1 is an arbitrary constant (in both formulas, the upper or lower signs are taken simultaneously). From the last equation (75) we get g = −f /k 2 ; then the penultimate equation, which serves to determine the function ζ, is converted to the Abel equation of the second kind Setting k 1 = ±k and k 2 = 1 in (76) and (77), we obtain the equation which has two exact solutions that can be represented in implicit form where the function ξ = ξ(u) is described by the Abel equation Exact solutions of the Abel equations for various functions f (u) and h(u) can be found in [62].
Solution 12. We set a = b = c = 1 in (19) and then make the substitution where α and β are free parameters, to obtain Below we give three solutions of equation (80), which lead to different solutions of the original PDE (18).

A particular solution to equation (80) is sought in the form
where C 1 and C 2 are arbitrary constants. We get which leads to the defining system of equations By virtue of the second equality (81), the solutions of these equations are Thus, we arrive at the equation which depends on an arbitrary function f = f (u) and admits the exact solution in implicit form Setting λ/γ = σ, β − (αλ/γ) = µ, and αγ = ε in (84) and (85), we obtain the more compact equation which has the exact solution 2. For g ≡ 0, equation (80) has the steady-state particular solution where f = f (u) is an arbitrary function, while C and k are arbitrary constants. This leads to the PDE [23] u This equation admits a solution in the implicit form f (u) du = −kx 2 + αx + βt + C.
Solution 13. In (19) we set ζ = f and then make the transformation where β and k are free parameters, to obtain We are looking for a steady-state solutionθ =θ(x) of equation (92). After the splitting procedure, we get the equations where µ, λ, γ, and σ are arbitrary constants. These equations admit the solution where C 1 , C 2 , and µ are arbitrary constants. Taking into account relation (91), we obtain the equation which admits the exact solution

Solution 14.
We seek a particular solution to equation (92) as the product of functions with different arguments ϑ = e λt ξ(x). (94) As a result, we arrive at the equations For g = const, we obtain f = const and h = const, which corresponds to a linear equation. Therefore, we further assume that g = const.
The first equation (95) is satisfied if we put where A and B are arbitrary constants (A = 0). The first two equations (96) involve three functions a = a(x), b = b(x), and ξ = ξ(x), one of which can be considered arbitrary.
Assuming that the function ξ = ξ(x) in (96) is given, we find that If we assume that the function b = b(x) is given, then the solutions of the first two equations (96) can be written as where C 1 and C 2 are arbitrary constants (C 2 = 0). In particular, for B = 1 and b(x) = 1, from (98) we find that and for B = 1 and b(x) = x we get The last equation (95) can be satisfied in two cases, which are considered below. 1. For β = 0, the solution of the last equation (95) is given by in the derivation of which the last relation in (96) was taken into account. Thus, the equation where b(x) and f (u) are arbitrary functions, and a = a(x) is expressed via b = b(x) by (98), admits the solution 2. For k = 0, the solution of the last equation (95) is As a result, we obtain the equation where b(x) and f (u) are arbitrary functions, and a = a(x) is expressed via b = b(x) by (98), which has the solution Solution 15. Equation (21) can be satisfied if we take Φ i (i = 1, 2, 3, 4) proportional to Φ 5 . As a result, we get Substituting (20) in (101) yields Consider two cases.

The simplest solution of the first four equations (102),
leads to a traveling wave solution of the original reaction-diffusion equation (18) (this solution will not be discussed here).
2. The first four equations (102) also admit a different solution Setting k = k 1 and k 2 = 1 in (103) and using the last equation in (102), we arrive at the reaction-diffusion type equation where and f = f (u), g = g(u), and ξ = ξ(u) are arbitrary functions. This equation admits the exact invariant solution f (u) ξ(u) du = −kt + ln x + C 1 .
Note that the invariant solution (106) of equation (104) can be obtained in the standard way in the form u = U (z) with z = −kt + ln x (in this case, relation (105) between f , g, h, and ξ is not used). The function U (z) is described by the ordinary differential equation

Solution 16. Equation (21) holds if we set
where k 1 , k 2 , and k 3 are arbitrary constants. Thus, we obtain the equations The solutions of the first two equations (108) are given by where c(x) and c(x) are arbitrary functions, while C 1 and C 2 are arbitrary constants. From the last three equations (108) we get Substituting C 1 = C 3 = 0 and k 3 = 1 in (109) and (110), we obtain the equation which has the solution where C 4 is an arbitrary constant. When deriving formula (112), the equality

Exact solutions obtained by analyzing equivalent equations
Now, using the considerations outlined in Section 2.3, we will obtain some other exact solutions to equation (1). To this end, instead of (8)-(9), we consider equivalent differential equations that reduce to (8)-(9) on the set of functions satisfying relation (6).
where Z = ζ du and λ is an arbitrary constant. Equations (19) and (113) (21) where As previously, equation (21) can be satisfied using relations (22). Substituting (114) into (22), we arrive at the equations which for λ = 0 coincide with (23). The solution of the overdetermined system consisting of the first three equations (115) has the form where a(x) is an arbitrary function and b 0 , C 1 , C 2 , k, and λ are arbitrary constants.
The solution of the system consisting of the last two equations (115) is written as where f (u) and g(u) are arbitrary functions. (21) can also be satisfied using relations (34). Substituting (114) into (34) yields

Solution 18. Equation
The solution of the overdetermined system consisting of the first three equations (118) is where c(x) is an arbitrary function (other than a constant), while C 1 , C 2 , and λ are arbitrary constants. The solutions of the last two equations (118) are expressed as where f = f (u) and g = g(u) are arbitrary functions.
Solution 19. As before, equation (21) can be satisfied using relations (38). Substituting (114) into (38), we get the equations The solution of the overdetermined system consisting of the first two equations (121) can be written as where a = a(x) and b = b(x) are arbitrary functions, while C 1 , C 2 , C 3 , k, and λ are arbitrary constants. The solution to the system consisting of the last three equations (121) is given by where m = 0 is an arbitrary constant.
Solution 20. Substituting (114) into (101), we arrive at the equations The first four equations of system (124) admit a solution for the functional coefficients in exponential form: Using the last equation in (124), we obtain the reaction-diffusion type equation where and f = f (u), g = g(u), and ζ = ζ(u) are arbitrary functions. Equation (126) admits the exact invariant solution Note that the invariant solution (128) of equation (126) can be represented in the standard form u = U (z) with z = 1 λ ln t + x (in this case, relation (127) linking f , g, h and ζ is not used). The function U (z) is described by the ordinary differential equation Solution 21. The first four equations of system (124) also admit a solution for power-law functional coefficients: Using the last equation in (124), we arrive at the reaction-diffusion type equation where and f = f (u), g = g(u), and ζ = ζ(u) are arbitrary functions. Equation (130) admits the exact invariant solution The self-similar solution (132) of equation (130) can be sought in the standard form u = U (z) with z = xt 1/(n−2) (in this case, relation (131) linking f , g, h, and ζ is not used). The function U (z) is described by the ODE (21) can be satisfied if we take Ψ i (i = 1, 3, 4, 5) proportional to Ψ 2 . As a result, we get

Solution 22. Equation
Substituting (114) into (133), we obtain the equations The first four equations of system (134) admit a solution for exponential functional coefficients: In this case, we obtain the reaction-diffusion type equation which has the exact solution with additive separation of variables with the function η = η(x) described by the ordinary differential equation Equations (136) and (138) contain three arbitrary functions a(x), b(x), and c(x).
Note that equation (138) reduces with the substitution ξ = e βη to the linear second-order ODE Solution 23. The first four equations of system (134) also admit a solution for the power-law functional coefficients f (u) = u n , g(u) = u n , h(u) = u n+1 , ζ(u) = 1/u, Z = ln u, λ = −n, k 1 = k 3 = k 4 = 1, k 2 = n + 1. (139) In this case, the solution of the last equation in (134) is determined by the formula ϑ = −(1/n) ln t + η(x), with the function η = η(x) satisfying the ODE As a result, we get the reaction-diffusion type equation the exact solution of which can be represented as the product of functions with different arguments u = t −1/n ξ(x), with the function ξ(x) = e η described by ODE Solution 24. Let us return to the class of reaction-diffusion equations of the form (18). Having made the substitution (6), instead of equation (19), we consider the more complex equation where Z = ζ du. Equations (19) and (142) are equivalent, because, by virtue of transformation (6), the relation ϑ = Z holds. (21), where

Equation (142) can be represented in bilinear form
Equation (21) can be satisfied by using the relations (34). Substituting (143) in (34), we get ϑ t = k 1 cϑ, (aϑ x ) x = −k 2 cϑ, bϑ x = −k 3 cϑ; where k 1 , k 2 , and k 3 are arbitrary constants. Let a = a(x), f = f (u), and g = g(u) be arbitrary functions. Then the solutions of equations (144) are given by where λ is an arbitrary constant, and the function ω = ω(x) solves the linear In the special case a(x) = const and k 3 = 0, formulas (145) lead to the nonlinear reaction-diffusion equation and its solution, which were considered in [23].

Solution 25. Consider the special case
We look for a solution of equation (142) under conditions (146) in the form Substituting (147) into (142) and taking into account (146), we obtain where F = f du. Equating the expressions in square brackets in (189) with zero, we arrive at the equations Solving these equations for g and h, we get As a result, we obtain the equation which has the exact solution where γ and δ are arbitrary constants.
Omitting the intermediate calculations, we arrive at the equation which has the solution where A and B are arbitrary constants.
Example 1. In the special case γ = −α and δ = β, equation (151) simplifies and takes the form and its solution is written as
Solution 28. Instead of equation (142), we can look at the more complex equation where Z = ζ du and n is an arbitrary constant. Equations (19) and (155) are equivalent, since, by virtue of transformation (6), the relation ϑ = Z holds. Equation (155) can be represented in the bilinear form (21) where Equation (21) can be satisfied by using the relations (34). Substituting (156) in (34), we get ϑ n ϑ t = k 1 cϑ, (aϑ x ) x = −k 2 cϑ, bϑ x = −k 3 cϑ; where k 1 , k 2 , and k 3 are arbitrary constants. Let a = a(x), f = f (u), and g = g(u) be arbitrary functions. Then the solutions of equations (157) are expressed as where the function ω = ω(x) is a solution of a second-order nonlinear ODE of the Emden-Fowler type: We set k 3 = k 1 n and k = k 2 /(k 1 n). From relations (158) it follows that the nonlinear reaction-diffusion type equation where f (u), g(u), and a(x) are arbitrary functions, k and n are arbitrary constants, and F (u) = f (u) du, admits the functional separable solution in implicit form The function ω = ω(x) in (160) and (161) is described by the nonlinear ordinary differential equation Note that for n = −1, the general solution of equation (162) is where C 1 and C 2 are arbitrary constants.
Example 2. Substituting a(x) = 1 and k = 0 into (160)-(162), we get the equation which admits the exact solution in implicit form (161). This solution is noninvariant and it is of a self-similar type; when substituted into equation (163), it causes the term [f (u)u x ] x to vanish.

Equation (164) is invariant under the transformation
where C 1 is an arbitrary constant.
It is easy to verify that for constant a, b, and c, which without loss of generality can be set equal to 1, equation (164) has the particular solution where f = f (u) is an arbitrary function, while C 2 , β, and µ are arbitrary constants.
Given (165), we obtain the equation which has the exact solution in the implicit form Setting β = λ − σµ, equation (167) can be rewritten in the more compact form In this case, its solution is f (u) du = C 1 e λt + C 2 e (λ−σµ)t−µx .
Solution 30. We look for a steady-state particular solution ϑ = ϑ(x) of equation (164). In this case, we have where k 1 , k 2 , and k 3 are arbitrary constants. The solution of the first three equations (169) with k 1 k 2 = 0 can be represented as where C 2 and C 3 are arbitrary constants. The solution to the system consisting of the last two equations (169) is written as follows: where f = f (u) and g = g(u) are arbitrary functions, C 4 is an arbitrary constant.
Substituting C 2 = 1/k, C 3 = C 4 = 0, k 1 = k, k 2 = k 3 = 1, and λ = kσ in (170) and (171), we arrive at the equation For k = 0, this equation admits the exact solution in the construction of which the invariance of equation (164) with respect to transformation (165) was taken into account.
Solution 31. In equation (164), we set ζ = f and λ = p(x)f (u) (recall that λ can be any function dependent on x, t, and u; see Item 2 in Section 2.3).
On dividing by f , we get where F = f (u) du.
Assuming the function f to be given arbitrarily, we look for the functions g and h in the form where k i and m i are some constants (i = 1, 2, 3). Substituting (175) into (174), we arrive at the equations Equations (176) admit the following exact solution where the three functions a = a(x), b = b(x), and c = c(x) are connected by one equation and the function η are described by the linear ODE Note that for given functions a and c, equation (178) is algebraic with respect to b, for given b and c it is a first-order linear ODE with respect to a (which is readily integrated), and for given a and b it is a second-order ODE with a quadratic nonlinearity with respect to c.
To sum up, we have obtained the nonlinear reaction-diffusion type equation where f (u) is an arbitrary function, and any two of the three functions a = a(x), b = b(x), and c = c(x) can be given arbitrarily, while the remaining function satisfies equation (178) with k 1 = 1. Equation (180) has the exact solution in implicit form where the function η(x) is determined by ODE (179) with k 1 = 1.
Remark 7. The more general equation where f = f (u) and m = m(x) are arbitrary functions, and the four functions a = a(x), b = b(x), c = c(x), and n = n(x) are connected by one equation (algebraic in b and n, and differential in a and c) admits the exact solution with the function η(x) determined by the ODE Solution 32. Solutions of equation (174) can be sought in the form where k n are some constants; the last relation (182) is used to determine the function f . By setting k 1 = 0, k 2 = 1, k 5 = 2, and k 6 = 0 in (182), we obtain f = g = u −1/2 , h = k 3 u 1/2 + k 4 , and F = 2u 1/2 . The corresponding nonlinear reaction-diffusion type equation where a(x), b(x), and c(x) are arbitrary functions, while and k 3 and k 4 are arbitrary constants, has an exact solution in implicit form F = ξ(x)t + η(x), which can be expressed in explicit form as The functions ξ = ξ(x) and η = η(x) are determined by solving the ordinary differential equations For c(x) = 1, the first equation (185) can be satisfied if we take ξ(x) = k 3 .

Remark 8. The equation
which is more general than (183), has an exact solution of the form (184). In the case d(x)/c(x) = const, equation (186) belongs to the class of equations (18) in question.
Remark 9. The nonlinear delay PDE where τ is the delay time and a 1 (x), a 2 (x), b 1 (x), b 2 (x), c 1 (x), c 2 (x), and d(x) are arbitrary functions, also admits an exact solution of the form (184).
Solution 33. Now we consider the equation which, by virtue of (6), is equivalent to equation (19) for a = c = 1 and b = 0. An exact solution of equation (188) is sought in the form where A, B, C, and λ are constants to be found. Omitting the intermediate calculations, we ultimately arrive at the equation which has two exact solutions where β, γ, and λ are arbitrary constants.
Remark 10. The described approach also makes it possible to obtain other exact solutions of equation (18), which are not considered here (recall that in this article we only look at nonlinear equations of a fairly general form that depend on arbitrary functions).
Remark 11. This approach can also be used to construct exact solutions of nonlinear ordinary differential equations with variable coefficients.

Using transformation (6) to simplify equations
Transformation (6) can also be used to simplify nonlinear PDEs. To illustrate this, we consider the equation where a is a constant.
Transformation (6) reduces equation (190) to the form In (191), we set As a result, we obtain the nonlinear equation where b(x), c(x), and f (u) are arbitrary functions, which can be reduced with the to the linear equation Some exact solutions of this equation can be found in [63].
Remark 12. Note that in equations (192) and (194), the functional coefficients a(x) and b(x) can be replaced with a(x, t) and b(x, t).
Example 3. In the special case a = 1 and f (u) = 1, equation (192) becomes and transformation (193) can be written in explicit form as u = ln ϑ. As a result, we obtain equation (194) with a = 1.

Functional separation of variables based on the nonclassical method of symmetry reductions
The method of functional separation of variables based on transformation (6) is closely related to the nonclassical method of symmetry reductions which is based on an invariant surface condition [38]. To show this, we differentiate formula (6) with respect to t to obtain where Q(x, t) = ϑ t and φ(u) = 1/ζ(u). Relation (195) can be treated as a first-order differential constraint or an invariant surface condition of a special form (in general, an invariant surface condition is a quasilinear first-order PDE of general form), which can be used to find exact solutions of equation (18) through a compatibility analysis of the overdetermined pair of differential equations (18) and (195) with the single unknown u. The invariant surface condition (195) is equivalent to relation (6); at the initial stage, both functions Q(x, t) and φ(u) included on the right-hand side of (195) are considered arbitrary, and the specific form of these functions is determined in the subsequent analysis. A description of the nonclassical method of symmetry reductions and examples of its application to construct exact solutions of nonlinear PDEs can be found, for example, in [9,18,23,[38][39][40][41][42][43][44]. Although the invariant surface condition (195) is equivalent to the functional relation (6), the subsequent procedure for finding exact solutions by the nonclassical method of symmetry reductions (or by the method of differential constraints) and that by the direct method for seeking functional separable solutions differ significantly. Let us compare the effectiveness of these methods by the example of the reaction-diffusion type equation (18) (since its functional separable solutions have already been obtained in Sections 3.2 and 3.3). To construct exact solutions by the nonclassical method of symmetry reductions, we will use relation (195) as an invariant surface condition.
Remark 13. The nonclassical method of symmetry reductions, based on the invariant surface condition (195), and the method of differential constraints [64], based on the single differential constraint (195), end up in the same result. A description of the method of differential constraints and examples of its application to construct exact solutions of nonlinear PDEs can be found, for example, in [7,18,23,45,50,[65][66][67][68][69].
Taking into account the last remark, below we use the method of differential constraints [23]. We solve equation (18) for the highest derivative u xx and eliminate (199) is differentiated with respect to t, after which the derivatives u t and u xt are eliminated from the resulting expression using relation (195) and the first formula of (197). As a result, we obtain P u 2 x + Qu x + R = 0, where P = K t + U K u + 2U u K, For brevity, short notations are used here: Equation (207) is a very complex nonlinear PDE, which includes third-order derivatives Q xxt and φ ′′′ uuu (recall that Q and φ are both unknown functions), whose length in expanded form (taking into account the relations (199) and (206)) occupies almost an entire page. In addition, equation (207), which includes one or more arbitrary functions f (u), g(u), etc., must be solved together with the equations (195) and (196) (or the original equation). As a result, instead of one equation (18) (or equation (19) together with (6)), it is necessary in this case to deal with a much more complex system of coupled nonlinear PDEs. In this case, one has to substitute into equation (207) the following functions: K = U uu , M = 2U xu , N = U xx − U t , U = Qφ; P = U tuu + U U uuu + 2U u U uu , Q = 2(U xtu + U U xuu + U u U xu + U x U uu , One can see that the nonlinear third-order equation (207)-(208) becomes isolated (can be solved independently of the original equation); it is far more complicated than the linear heat equation under consideration.
The degenerate case of MP − KQ ≡ 0 can be treated likewise.
It is apparent from the above examples that the method in question, based on the analysis of three PDEs (195), (196), and (199), is extremely difficult for practical use.
The second algorithm. In this case, we differentiate formula (6) with respect to t and x. As a result, we obtain two relations which can be interpreted as two compatible differential constraints, where the functions ϑ = ϑ(x, t) and φ(u) = 1/ζ(u) are to be determined. Differentiating the second relation (209) with respect to x, we find the second derivative Next, we substitute the derivatives (209) and (210) in (18). As a result, we arrive at an equation that is equivalent to equation (19). Using further the generalized splitting principle described in Section 2.1, one can find the exact solutions obtained in Section 3.2. However, it will not be possible to find the solutions obtained in Section 3.3 in this way. In order to find these solutions, one must first integrate the differential relations (209) and return to the original relation (6), and then consider the equivalent equations described in Section 2.3.
Thus, it seems that the use of transformation (6) is more effective for constructing exact solutions than the use of one or two equivalent differential constraints.

Brief conclusions
A general method for constructing exact solutions of nonlinear PDEs has been described, which is based on nonlinear transformations with an integral term in combination with the generalized splitting principle. The high productivity of the method has been illustrated by nonlinear equations of the reaction-diffusion type with variable coefficients that depend on one, two or three arbitrary functions. Many new exact functional separable solutions and generalized traveling wave solutions have been obtained. The effectiveness of various methods for constructing exact solutions of nonlinear differential equations has been compared.
The direct method of functional separation of variables based on transformation (6), in addition to diffusion-type equations, is also applicable to other classes of PDEs. In particular, these include nonlinear wave equations, nonlinear Klein-Gordon type equations, nonlinear telegraph-type equations, and others; these also include some third-and higher-order PDEs. This method is easy to generalize to equations with three or more independent variables.